Kazuhiro Ishige

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On the Existence of Solutions of the Cauchy Problem for a Doubly Nonlinear Parabolic Equation

1996-09-01

Kazuhiro Ishige

For the existence of weak solutions of \[ \frac{\partial } {{\partial t}}\left( {|u|^{\beta - 1} u} \right) = {\operatorname{div}}\left( {|\nabla u|^{p - 2} \nabla u} \right)\quad {\text{with}}\,|u|^{\beta - 1} u( … For the existence of weak solutions of \[ \frac{\partial } {{\partial t}}\left( {|u|^{\beta - 1} u} \right) = {\operatorname{div}}\left( {|\nabla u|^{p - 2} \nabla u} \right)\quad {\text{with}}\,|u|^{\beta - 1} u( \cdot ,0) = \mu ( \cdot ), \] we give a sufficient condition for the growth order of the initial data $\mu (x)$ as $|x| \to \infty $.

Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains

2001-01-01

Kazuhiro Ishige , Minoru Murata

We study uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on non-compact Riemannian manifolds or domains in We introduce two notions: (1) the parabolic Harnack principle with … We study uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on non-compact Riemannian manifolds or domains in We introduce two notions: (1) the parabolic Harnack principle with scale function p concerning inhomogeneity at infinity of manifolds and the second order terms of equations; and (2) the relative boundedness with scale function p concerning growth order at infinity of the lower terms of equations. In terms of this scale function, we give a general and sharp sufficient condition for the uniqueness of nonnegative solutions to hold. We also give a Tacklind type uniqueness theorem for solutions with growth conditions, which plays a crucial role in establishing our Widder type uniqueness theorem for nonnegative solutions. Our Tacklind type uniqueness theorem is of independent interest. It is new even for parabolic equations on in regard to growth rates at infinity of their lower order terms.

Existence of solutions for a fractional semilinear parabolic equation with singular initial data

2018-06-14

Kotaro Hisa , Kazuhiro Ishige

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On the behavior of the solutions of degenerate parabolic equations

1999-01-01

Kazuhiro Ishige

Abstract In this paper we consider degenerate parabolic equations, and obtain an interior and a boundary Harnack inequalities for nonnegative solutions to the degenerate parabolic equations. Furthermore we obtain boundedness … Abstract In this paper we consider degenerate parabolic equations, and obtain an interior and a boundary Harnack inequalities for nonnegative solutions to the degenerate parabolic equations. Furthermore we obtain boundedness and continuity of the solutions.

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Supersolutions for a class of nonlinear parabolic systems

2015-12-30

Kazuhiro Ishige , Tatsuki Kawakami , Mikołaj Sierżęga

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The decay of the solutions for the heat equation with a potential

2009-01-01

Kazuhiro Ishige , Michinori Ishiwata , Tatsuki Kawakami

We study the large time behavior of the solutions for the Cauchy problem, ∂ t u = Δu + a(x, t)u in ℝ N × (0, ∞), u (x, 0) … We study the large time behavior of the solutions for the Cauchy problem, ∂ t u = Δu + a(x, t)u in ℝ N × (0, ∞), u (x, 0) = Q(x) in ℝ N , where Q ∈ L 1 (ℝ N , (1+|x| K ) dx) with K ≥ 0 and ∥a(t)∥ L ∞ (ℝ N ) = O(t -A ) as t → ∞ for some A > 1. In this paper we classify the decay rate of the solutions and give the precise estimates on the difference between the solutions and their asymptotic profiles. Furthermore, as an application, we discuss the large time behavior of the global solutions for the semilinear heat equation, ∂ t u = Δu + λ|u| p-1 u, where λ ∈ ℝ and p > 1.

Is quasi-concavity preserved by heat flow?

2008-04-16

Kazuhiro Ishige , Paolo Salani

Parabolic quasi‐concavity for solutions to parabolic problems in convex rings

2010-09-22

Kazuhiro Ishige , Paolo Salani

Abstract We investigate some geometric properties of level sets of the solutions of parabolic problems in convex rings. We introduce the notion of parabolic quasi‐concavity , which involves time and … Abstract We investigate some geometric properties of level sets of the solutions of parabolic problems in convex rings. We introduce the notion of parabolic quasi‐concavity , which involves time and space jointly and is a stronger property than the spatial quasi‐concavity, and study the convexity of superlevel sets of the solutions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Global solutions for a nonlinear integral equation with a generalized heat kernel

2014-01-01

Kazuhiro Ishige , Tatsuki Kawakami , Kanako Kobayashi

We studythe existence and the large time behavior of global-in-time solutions of a nonlinear integral equation with a generalized heat kernel\begin{eqnarray*} & & u(x,t)=\int_{{\mathbb R}^N}G(x-y,t)\varphi(y)dy\\ & & \qquad\quad+\int_0^t\int_{{\mathbb R}^N}G(x-y,t-s)F(y,s,u(y,s),\dots,\nabla^\ell u(y,s))dyds,\end{eqnarray*}where … We studythe existence and the large time behavior of global-in-time solutions of a nonlinear integral equation with a generalized heat kernel\begin{eqnarray*} & & u(x,t)=\int_{{\mathbb R}^N}G(x-y,t)\varphi(y)dy\\ & & \qquad\quad+\int_0^t\int_{{\mathbb R}^N}G(x-y,t-s)F(y,s,u(y,s),\dots,\nabla^\ell u(y,s))dyds,\end{eqnarray*}where $\varphi\in W^{\ell,\infty}({\mathbb R}^N)$ and $\ell\in\{0,1,\dots\}$.The arguments of this paper are applicable tothe Cauchy problem for various nonlinear parabolic equationssuch as fractional semilinear parabolic equations, higher order semilinear parabolic equationsand viscous Hamilton-Jacobi equations.

Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces

2015-10-01

Kazuhiro Ishige , Ryuichi Sato

We establish the local existence and the uniqueness of solutionsof the heat equation with a nonlinear boundary conditionfor the initial data in uniformly local $L^r$ spaces.Furthermore, we study the sharp … We establish the local existence and the uniqueness of solutionsof the heat equation with a nonlinear boundary conditionfor the initial data in uniformly local $L^r$ spaces.Furthermore, we study the sharp lower estimates of the blow-up timeof the solutions with the initial data $\lambda\psi$ as $\lambda\to 0$ or $\lambda\to\infty$and the lower blow-up estimates of the solutions.

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Blow-up problems for a semilinear heat equation with large diffusion

2004-12-12

Kazuhiro Ishige , Hiroki Yagisita

Asymptotics for a nonlinear integral equation with a generalized heat kernel

2014-06-08

Kazuhiro Ishige , Tatsuki Kawakami , Kanako Kobayashi

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Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition

2012-01-01

Марек Фила , Kazuhiro Ishige , Tatsuki Kawakami

We study the large time behavior of positive solutions forthe Laplace equation on the half-space with a nonlinear dynamical boundary condition.We show the convergence to the Poisson kernel in a … We study the large time behavior of positive solutions forthe Laplace equation on the half-space with a nonlinear dynamical boundary condition.We show the convergence to the Poisson kernel in a suitable sense provided initial dataare sufficiently small.

Refined asymptotic profiles for a semilinear heat equation

2011-05-25

Kazuhiro Ishige , Tatsuki Kawakami

On a new kind of convexity for solutions of parabolic problems

2010-12-08

Kazuhiro Ishige , Paolo Salani

We introduce the notion of $\alpha$-parabolic quasi-concavity for functions of space and time, which extends the usual notion of quasi-concavity and the notion of parabolic quasi-cocavity introduced in [18]. Then … We introduce the notion of $\alpha$-parabolic quasi-concavity for functions of space and time, which extends the usual notion of quasi-concavity and the notion of parabolic quasi-cocavity introduced in [18]. Then we investigate the $\alpha$-parabolic quasi-concavity of solutions to parabolic problems with vanishing initial datum. The results here obtained are generalizations of some of the results of [18].

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Existence of solutions for a higher-order semilinear parabolic equation with singular initial data

2020-04-23

Shinya Okabe , Kazuhiro Ishige , Tatsuki Kawakami

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Blow-up set for type I blowing up solutions for a semilinear heat equation

2013-03-14

Yohei Fujishima , Kazuhiro Ishige

Let u be a type I blowing up solution of the Cauchy–Dirichlet problem for a semilinear heat equation, \left\{\begin{matrix} \partial _{t}u = \mathrm{\Delta }u + u^{p}, & x \in \Omega … Let u be a type I blowing up solution of the Cauchy–Dirichlet problem for a semilinear heat equation, \left\{\begin{matrix} \partial _{t}u = \mathrm{\Delta }u + u^{p}, & x \in \Omega ,\:t > 0, \\ u(x,t) = 0, & x \in \partial \Omega ,\:t > 0, \\ u(x,0) = \varphi (x), & x \in \Omega , \\ \end{matrix}\right. where \Omega is a (possibly unbounded) domain in \mathbf{R}^{N} , N \geq 1 , and p > 1 . We prove that, if \varphi \in L^{\infty }(\Omega ) \cap L^{q}(\Omega ) for some q \in [1,\infty) , then the blow-up set of the solution u is bounded. Furthermore, we give a sufficient condition for type I blowing up solutions not to blow up on the boundary of the domain \Omega . This enables us to prove that, if \Omega is an annulus, then the radially symmetric solutions of ( P ) do not blow up on the boundary \partial\Omega .

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Global solutions of the heat equation with a nonlinear boundary condition

2010-02-03

Kazuhiro Ishige , Tatsuki Kawakami

Blow-up behavior for semilinear heat equations with boundary conditions

2003-01-01

Kazuhiro Ishige , Noriko Mizoguchi

We study the blow-up behavior of solutions of the semilinear heat equations under the Dirichlet boundary condition or the Neumann boundary condition. We prove the nondegeneracy of blow-up of the … We study the blow-up behavior of solutions of the semilinear heat equations under the Dirichlet boundary condition or the Neumann boundary condition. We prove the nondegeneracy of blow-up of the solutions. Furthermore, we give a result on the blow-up rate of the solutions under the Neumann boundary condition.

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Location of blow-up set for a semilinear parabolic equation with large diffusion

2003-11-01

Kazuhiro Ishige , Noriko Mizoguchi

The heat kernel of a Schrödinger operator with inverse square potential

2017-05-08

Kazuhiro Ishige , Yoshitsugu Kabeya , El Maati Ouhabaz

We consider the Schrödinger operator H = − Δ + V ( | x | ) with radial potential V which may have singularity at 0 and a quadratic decay … We consider the Schrödinger operator H = − Δ + V ( | x | ) with radial potential V which may have singularity at 0 and a quadratic decay at infinity. First, we study the structure of positive harmonic functions of H and give their precise behavior. Second, under quite general conditions we prove an upper bound for the correspond heat kernel p ( x , y , t ) of the type 0 0 , where U is a positive harmonic function of H. Third, if U 2 is an A 2 weight on R N , then we prove a lower bound of a similar type.

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An intrinsic metric approach to uniqueness of the positive Cauchy problem for parabolic equations

1998-02-01

Kazuhiro Ishige , Minoru Murata

Large time behaviors of hot spots for the heat equation with a potential

2008-04-03

Kazuhiro Ishige , Yoshitsugu Kabeya

On the Fujita exponent for a semilinear heat equation with a potential term

2008-03-10

Kazuhiro Ishige

Blow-up behavior of solutions to a degenerate parabolic–parabolic Keller–Segel system

2016-03-23

Kazuhiro Ishige , Philippe Laurençot , Noriko Mizoguchi

Solvability of the Heat Equation with a Nonlinear Boundary Condition

2019-01-01

Kotaro Hisa , Kazuhiro Ishige

In this paper we obtain necessary conditions and sufficient conditions for the solvability of the problem $({\rm P})\, \{ \partial_t u=\Delta u,\, x\in{\bf R}^N_+,\,t>0; \partial_\nu u=u^p,\, x\in\partial{\bf R}^N_+,\, t>0; u(x,0)=\mu(x)\ge … In this paper we obtain necessary conditions and sufficient conditions for the solvability of the problem $({\rm P})\, \{ \partial_t u=\Delta u,\, x\in{\bf R}^N_+,\,t>0; \partial_\nu u=u^p,\, x\in\partial{\bf R}^N_+,\, t>0; u(x,0)=\mu(x)\ge 0,\, x\in D:=\overline{{\bf R}^N_+} \}$, where $N\ge 1$, $p>1$, and $\mu$ is a nonnegative measurable function in ${\bf R}^N_+$ or a Radon measure in ${\bf R}^N$ with $\mbox{supp}\,\mu\subset D$. Our sufficient conditions and necessary conditions enable us to identify the strongest singularity of the initial data for the solvability for problem $({\rm P})$.

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Existence of positive solutions of a semilinear elliptic equation with a dynamical boundary condition

2015-04-03

Марек Фила , Kazuhiro Ishige , Tatsuki Kawakami

Movement of hot spots on the exterior domain of a ball under the Neumann boundary condition

2004-12-22

Kazuhiro Ishige

An Intrinsic Metric Approach to Uniqueness of the Positive Dirichlet Problem for Parabolic Equations in Cylinders

1999-11-01

Kazuhiro Ishige

Blow-up set for a semilinear heat equation with small diffusion

2010-04-12

Yohei Fujishima , Kazuhiro Ishige

Parabolic power concavity and parabolic boundary value problems

2013-12-04

Kazuhiro Ishige , Paolo Salani

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Sharp decay estimates in Lorentz spaces for nonnegative Schrödinger heat semigroups

2014-09-23

Norisuke Ioku , Kazuhiro Ishige , Eiji Yanagida

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<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msup></mml:math>norms of nonnegative Schrödinger heat semigroup and the large time behavior of hot spots

2012-01-09

Kazuhiro Ishige , Yoshitsugu Kabeya

Convexity breaking of the free boundary for porous medium equations

2010-02-25

Kazuhiro Ishige , Paolo Salani

We investigate the preservation of convexity of the free boundary by the solutions of the porous medium equation. We prove that starting with an initial datum with some kind of … We investigate the preservation of convexity of the free boundary by the solutions of the porous medium equation. We prove that starting with an initial datum with some kind of suboptimal αconcavity property, the convexity of the positivity set can be lost in a short time.

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Blowup for a Fourth-Order Parabolic Equation with Gradient Nonlinearity

2020-01-01

Kazuhiro Ishige , Nobuhito Miyake , Shinya Okabe

Let $u$ be a solution to the Cauchy problem for a fourth-order nonlinear parabolic equation $\partial_t u+(-\Delta)^2u=-\nabla\cdot(|\nabla u|^{p-2}\nabla u)$ on ${\bf R}^N$, where $p>2$ and $N\ge 1$. In this paper … Let $u$ be a solution to the Cauchy problem for a fourth-order nonlinear parabolic equation $\partial_t u+(-\Delta)^2u=-\nabla\cdot(|\nabla u|^{p-2}\nabla u)$ on ${\bf R}^N$, where $p>2$ and $N\ge 1$. In this paper we give a sufficient condition for the maximal existence time $T_M(u)$ of the solution $u$ to be finite. Furthermore, we show that if $T_M(u)<\infty$, then $\|\nabla u(t)\|_{L^\infty({\bf R}^N)}$ blows up at $t=T_M(u)$, and we obtain lower estimates on the blow-up rate. We also give a sufficient condition on the existence of global-in-time solutions to the Cauchy problem.

Nonlinear characterizations of stochastic completeness

2020-05-15

Gabriele Grillo , Kazuhiro Ishige , Matteo Muratori

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Asymptotic expansions of solutions of the Cauchy problem for nonlinear parabolic equations

2013-10-01

Kazuhiro Ishige , Tatsuki Kawakami

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Asymptotic behavior of solutions for some semilinear heat equations in $R^N$

2009-01-01

Kazuhiro Ishige , Tatsuki Kawakami

We consider the Cauchy problem of the semilinear heat equation,$\partial_t u = \Delta u +f(u)$ in $R^N \times (0,\infty),$$u (x,0) = \phi (x) \ge 0$ in $R^N,\quad\quad$where $N \geq 1$, … We consider the Cauchy problem of the semilinear heat equation,$\partial_t u = \Delta u +f(u)$ in $R^N \times (0,\infty),$$u (x,0) = \phi (x) \ge 0$ in $R^N,\quad\quad$where $N \geq 1$, $f \in C^1([0,\infty))$,and $\phi \in L^1(R^N) \cap L^{\infty}(R^N)$.We study the asymptoticbehavior of the solutions in the $L^q$ spaces with $q \in [1,\infty]$,by using the relative entropy methods.

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Initial trace for a doubly nonlinear parabolic equation

2011-08-11

Kazuhiro Ishige , Juha Kinnunen

An intrinsic metric approach to uniqueness of the positive Cauchy–Neumann problem for parabolic equations

2002-12-01

Kazuhiro Ishige

The Cauchy problem for the Finsler heat equation

2018-03-07

Goro Akagi , Kazuhiro Ishige , Ryuichi Sato

Abstract Let H be a norm of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:math> {\mathbb{R}^{N}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>H</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> {H_{0}} the dual norm of H . … Abstract Let H be a norm of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:math> {\mathbb{R}^{N}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>H</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> {H_{0}} the dual norm of H . Denote by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>H</m:mi> </m:msub> </m:math> {\Delta_{H}} the Finsler–Laplace operator defined by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>H</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>:=</m:mo> <m:mrow> <m:mi>div</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>H</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:msub> <m:mo>∇</m:mo> <m:mi>ξ</m:mi> </m:msub> <m:mo>⁡</m:mo> <m:mi>H</m:mi> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {\Delta_{H}u:=\operatorname{div}(H(\nabla u)\nabla_{\xi}H(\nabla u))} . In this paper we prove that the Finsler–Laplace operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>H</m:mi> </m:msub> </m:math> {\Delta_{H}} acts as a linear operator to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>H</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> {H_{0}} -radially symmetric smooth functions. Furthermore, we obtain an optimal sufficient condition for the existence of the solution to the Cauchy problem for the Finsler heat equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mo>∂</m:mo> <m:mi>t</m:mi> </m:msub> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>H</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo rspace="12.5pt">,</m:mo> <m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:mrow> <m:mo rspace="4.2pt">,</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> \partial_{t}u=\Delta_{H}u,\quad x\in\mathbb{R}^{N},\,t>0, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>N</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {N\geq 1} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mo>∂</m:mo> <m:mi>t</m:mi> </m:msub> <m:mo>:=</m:mo> <m:mfrac> <m:mo>∂</m:mo> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mi>t</m:mi> </m:mrow> </m:mfrac> </m:mrow> </m:math> {\partial_{t}:=\frac{\partial}{\partial t}} .

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Gradient estimates for the heat equation in the exterior domains under the Neumann boundary condition

2009-05-01

Kazuhiro Ishige

We consider the Cauchy-Neumann problem for the heat equation in the exterior domain $\Omega$ of a compact set in ${\bf R}^N$ ($N\ge 2$). In this paper we give an estimate … We consider the Cauchy-Neumann problem for the heat equation in the exterior domain $\Omega$ of a compact set in ${\bf R}^N$ ($N\ge 2$). In this paper we give an estimate of the $L^\infty$-norm of the gradient of the solutions.

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Large time behavior of solutions of a semilinear elliptic equation with a dynamical boundary condition

2013-01-01

Марек Фила , Kazuhiro Ishige , Tatsuki Kawakami

The main purpose of the paper is to study the large-time behavior of positive solutions of a semilinear elliptic equation with a dynamical boundary condition. We show that small solutions … The main purpose of the paper is to study the large-time behavior of positive solutions of a semilinear elliptic equation with a dynamical boundary condition. We show that small solutions behave asymptotically like suitable multiples of the Poisson kernel.

Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball

2007-07-01

Kazuhiro Ishige , Yoshitsugu Kabeya

We consider the initial-boundary value problem \begin{equation}\qquad\left\{ \begin{array}{lll} \frac{\partial}{\partial t}u = \Delta u-V(|x|)u & \mbox{in}& \Omega_L\times(0,\infty), \\ \mu u+(1-\mu)\frac{\partial}{\partial\nu}u = 0 & \mbox{on}& \partial\Omega_L\times(0,\infty), \\ u(\cdot,0) = \phi(\cdot)\in L^p(\Omega_L), & … We consider the initial-boundary value problem \begin{equation}\qquad\left\{ \begin{array}{lll} \frac{\partial}{\partial t}u = \Delta u-V(|x|)u & \mbox{in}& \Omega_L\times(0,\infty), \\ \mu u+(1-\mu)\frac{\partial}{\partial\nu}u = 0 & \mbox{on}& \partial\Omega_L\times(0,\infty), \\ u(\cdot,0) = \phi(\cdot)\in L^p(\Omega_L), & p \ge 1,& \end{array}\right. \tag{P}\end{equation} where $\Omega_L = \{ |x| \in \mathbf{R}^{N} : |x\ > L\}, N \geq 2, L > 0, 0 \leq \mu \leq 1, v$ is the outer unit normal vector to $\partial \Omega_L$, and $V$ is a nonnegative smooth function such that $V(r) = O(r^{-2})$ as $r \to \infty$. In this paper, we study the decay rates of the derivatives $\bigtriangledown^j_x u$ of the solution $u$ to $(P)$ as $t \to \infty$.

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Blow-up time and blow-up set of the solutions for semilinear heat equations with large diffusion

2002-01-01

Kazuhiro Ishige

We consider blow-up problems of the Cauchy-Neumann problem for semilinear heat equations with large diffusion. We prove that, in cylindrical domains, the solutions blow up only at the edge of … We consider blow-up problems of the Cauchy-Neumann problem for semilinear heat equations with large diffusion. We prove that, in cylindrical domains, the solutions blow up only at the edge of the domain for almost all initial data. Furthermore, we give an estimate of the blow-up time of the solutions.

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Geometric Properties for Parabolic and Elliptic PDE's

2016-01-01

Filippo Gazzola , Kazuhiro Ishige , Carlo Nitsch +1 more

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Blow-up for a semilinear parabolic equation with large diffusion on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mi mathvariant="bold">R</mml:mi><mml:mi>N</mml:mi></mml:msup></mml:math>

2010-12-22

Yohei Fujishima , Kazuhiro Ishige

Movement of hot spots on the exterior domain of a ball under the Dirichlet boundary condition

2007-01-01

Kazuhiro Ishige

We consider the Cauchy-Dirichlet problem of the heat equation in the exterior domain of a ball, and study the movement of hot spots $H(t)$ as $t\to\infty$. In particular, we give … We consider the Cauchy-Dirichlet problem of the heat equation in the exterior domain of a ball, and study the movement of hot spots $H(t)$ as $t\to\infty$. In particular, we give a rate for the hot spots to run away from the boundary of the domain as $t\to\infty$. Furthermore we give a sufficient condition for the hot spots to consist of only one point after a finite time.

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Minimal solutions of a semilinear elliptic equation with a dynamical boundary condition

2015-12-01

Марек Фила , Kazuhiro Ishige , Tatsuki Kawakami

Asymptotic Expansions of Solutions of Fractional Diffusion Equations

2017-01-01

Kazuhiro Ishige , Tatsuki Kawakami , Hironori Michihisa

In this paper we obtain the precise description of the asymptotic behavior of the solution $u$ of $$ \partial_t u+(-\Delta)^{\frac{\theta}{2}}u=0\quad\mbox{in}\quad{\bf R}^N\times(0,\infty), \qquad u(x,0)=\varphi(x)\quad\mbox{in}\quad{\bf R}^N, $$ where $0<\theta<2$ and $\varphi\in L_K:=L^1({\bf … In this paper we obtain the precise description of the asymptotic behavior of the solution $u$ of $$ \partial_t u+(-\Delta)^{\frac{\theta}{2}}u=0\quad\mbox{in}\quad{\bf R}^N\times(0,\infty), \qquad u(x,0)=\varphi(x)\quad\mbox{in}\quad{\bf R}^N, $$ where $0<\theta<2$ and $\varphi\in L_K:=L^1({\bf R}^N,\,(1+|x|)^K\,dx)$ with $K\ge 0$. Furthermore, we develop the arguments in [15] and [18] and establish a method to obtain the asymptotic expansions of the solutions to a nonlinear fractional diffusion equation $$ \partial_t u+(-\Delta)^{\frac{\theta}{2}}u=|u|^{p-1}u\quad\mbox{in}\quad{\bf R}^N\times(0,\infty), $$ where $0<\theta<2$ and $p>1+\theta/N$.

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Existence of solutions to a fractional semilinear heat equation in uniformly local weak Zygmund-type spaces

2025-05-29

Norisuke Ioku , Kazuhiro Ishige , Tatsuki Kawakami

Initial traces of solutions to a semilinear heat equation under the Dirichlet boundary condition

2025-03-24

Kotaro Hisa , Kazuhiro Ishige

Riemannian starshape and capacitary problems

2025-03-23

Kazuhiro Ishige , Paolo Salani , Asuka Takatsu

Fundamental solution to the heat equation in a half-space with a dynamical boundary condition

2025-02-17

Kazuhiro Ishige , Sho Katayama , Tatsuki Kawakami

Initial traces of solutions to a semilinear heat equation under the Dirichlet boundary condition

2024-12-08

Kotaro Hisa , Kazuhiro Ishige

We study qualitative properties of initial traces of nonnegative solutions to a semilinear heat equation in a smooth domain under the Dirichlet boundary condition. Furthermore, for the corresponding Cauchy--Dirichlet problem, … We study qualitative properties of initial traces of nonnegative solutions to a semilinear heat equation in a smooth domain under the Dirichlet boundary condition. Furthermore, for the corresponding Cauchy--Dirichlet problem, we obtain sharp necessary conditions and sufficient conditions on the existence of nonnegative solutions and identify optimal singularities of solvable nonnegative initial data.

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Fundamental solution to the heat equation with a dynamical boundary condition

2024-10-10

Kazuhiro Ishige , Sho Katayama , Tatsuki Kawakami

We give an explicit representation of the fundamental solution to the heat equation on a half-space of ${\mathbb R}^N$ with the homogeneous dynamical boundary condition, and obtain upper and lower … We give an explicit representation of the fundamental solution to the heat equation on a half-space of ${\mathbb R}^N$ with the homogeneous dynamical boundary condition, and obtain upper and lower estimates of the fundamental solution. These enable us to obtain sharp decay estimates of solutions to the heat equation with the homogeneous dynamical boundary condition. Furthermore, as an application of our decay estimates, we identify the so-called Fujita exponent for a semilinear heat equation on the half-space of ${\mathbb R}^N$ with the homogeneous dynamical boundary condition.

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Riemannian starshape and capacitary problems

2024-08-29

Kazuhiro Ishige , Paolo Salani , Asuka Takatsu

We prove the Riemannian version of a classical Euclidean result: every level set of the capacitary potential of a starshaped ring is starshaped. In the Riemannian setting, we restrict ourselves … We prove the Riemannian version of a classical Euclidean result: every level set of the capacitary potential of a starshaped ring is starshaped. In the Riemannian setting, we restrict ourselves to starshaped rings in a warped product of an open interval and the unit sphere. We also extend the result by replacing the Laplacian with the $q$-Laplacian.

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Existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term

2024-07-25

Kazuhiro Ishige , Tatsuki Kawakami , Ryo Takada

We study the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term. For this aim, we establish decay estimates of the fractional heat semigroup in … We study the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term. For this aim, we establish decay estimates of the fractional heat semigroup in several uniformly local Zygumnd spaces. Furthermore, we apply the real interpolation method in uniformly local Zygmund spaces to obtain sharp integral estimates on the inhomogeneous term and the nonlinear term. This enables us to find sharp sufficient conditions for the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term.

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Existence of solutions for a semilinear parabolic system with singular initial data

2024-07-03

Yohei Fujishima , Kazuhiro Ishige , Tatsuki Kawakami

Let $(u,v)$ be a solution to the Cauchy problem for a semilinear parabolic system \[ \mathrm{(P)} \qquad \cases{ \partial_t u=D_1\Delta u+v^p\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ \partial_t v=D_2\Delta v+u^q\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ (u(\cdot,0),v(\cdot,0))=(\mu,\nu) & … Let $(u,v)$ be a solution to the Cauchy problem for a semilinear parabolic system \[ \mathrm{(P)} \qquad \cases{ \partial_t u=D_1\Delta u+v^p\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ \partial_t v=D_2\Delta v+u^q\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ (u(\cdot,0),v(\cdot,0))=(\mu,\nu) & $\quad\mbox{in}\quad{\mathbb{R}}^N,$ } \] where $N\ge 1$, $T>0$, $D_1>0$, $D_2>0$, $0<p\le q$ with $pq>1$, and $(\mu,\nu)$ is a pair of nonnegative Radon measures or locally integrable nonnegative functions in ${\mathbb R}^N$. In this paper we establish sharp sufficient conditions on the initial data for the existence of solutions to problem~(P) using uniformly local Morrey spaces and uniformly local weak Zygmund type spaces.

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Eventual concavity properties of the heat flow

2024-05-29

Kazuhiro Ishige

Abstract The eventual concavity properties are useful to characterize geometric properties of the final state of solutions to parabolic equations. In this paper we give characterizations of the eventual concavity … Abstract The eventual concavity properties are useful to characterize geometric properties of the final state of solutions to parabolic equations. In this paper we give characterizations of the eventual concavity properties of the heat flow for nonnegative, bounded measurable initial functions with compact support.

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A parabolic PDE-based approach to Borell--Brascamp--Lieb inequality

2024-05-26

Kazuhiro Ishige , Qing Liu , Paolo Salani

In this paper, we provide a new PDE proof for the celebrated Borell--Brascamp--Lieb inequality. Our approach reveals a deep connection between the Borell--Brascamp--Lieb inequality and properties of diffusion equations of … In this paper, we provide a new PDE proof for the celebrated Borell--Brascamp--Lieb inequality. Our approach reveals a deep connection between the Borell--Brascamp--Lieb inequality and properties of diffusion equations of porous medium type pertaining to the large time asymptotics and preservation of a generalized concavity of the solutions. We also recover the equality condition in the special case of the Pr\'ekopa--Leindler inequality by further exploiting known properties of the heat equation including the eventual log-concavity and backward uniqueness of solutions.

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Non-preservation of concavity properties by the Dirichlet heat flow on Riemannian manifolds

2024-05-06

Kazuhiro Ishige , Asuka Takatsu , Haruto Tokunaga

We prove that no concavity properties are preserved by the Dirichlet heat flow in a totally convex domain of a Riemannian manifold unless the sectional curvature vanishes everywhere on the … We prove that no concavity properties are preserved by the Dirichlet heat flow in a totally convex domain of a Riemannian manifold unless the sectional curvature vanishes everywhere on the domain.

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Local solvability and dilation-critical singularities of supercritical fractional heat equations

2024-05-03

Yohei Fujishima , Kotaro Hisa , Kazuhiro Ishige +1 more

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Existence of solutions to a fractional semilinear heat equation in uniformly local weak Zygmund type spaces

2024-02-22

Norisuke Ioku , Kazuhiro Ishige , Tatsuki Kawakami

In this paper we introduce uniformly local weak Zygmund type spaces, and obtain an optimal sufficient condition for the existence of solutions to the critical fractional semilinear heat equation. In this paper we introduce uniformly local weak Zygmund type spaces, and obtain an optimal sufficient condition for the existence of solutions to the critical fractional semilinear heat equation.

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Supercritical Hénon-type equation with a forcing term

2024-01-01

Kazuhiro Ishige , Sho Katayama

Abstract This article is concerned with the structure of solutions to the elliptic problem for a Hénon-type equation with a forcing term: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mspace width="11.3em" /> <m:mo>−</m:mo> <m:mi … Abstract This article is concerned with the structure of solutions to the elliptic problem for a Hénon-type equation with a forcing term: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mspace width="11.3em" /> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>α</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mi>κ</m:mi> <m:mi>μ</m:mi> <m:mo>,</m:mo> <m:mspace width="1.0em" /> <m:mspace width="0.1em" /> <m:mtext>in</m:mtext> <m:mspace width="0.1em" /> <m:mspace width="0.33em" /> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width="1.0em" /> <m:mi>u</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mspace width="1.0em" /> <m:mspace width="0.1em" /> <m:mtext>in</m:mtext> <m:mspace width="0.1em" /> <m:mspace width="0.33em" /> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width="13.0em" /> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="normal">P</m:mi> </m:mrow> <m:mrow> <m:mi>κ</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \hspace{11.3em}-\Delta u=\alpha \left(x){u}^{p}+\kappa \mu ,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}u\gt 0,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{13.0em}\left({{\rm{P}}}_{\kappa }) where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>N</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:math> N\ge 3 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>1</m:mn> </m:math> p\gt 1 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>κ</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:math> \kappa \gt 0 , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha is a positive continuous function in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mo>\</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:math> {{\mathbb{R}}}^{N}\setminus \left\{0\right\} , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>μ</m:mi> </m:math> \mu is a nonnegative Radon measure in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:math> {{\mathbb{R}}}^{N} . Under suitable assumptions on the exponent <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p , the coefficient <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha , and the forcing term <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>μ</m:mi> </m:math> \mu , we give a complete classification of the existence/nonexistence of solutions to problem ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="normal">P</m:mi> </m:mrow> <m:mrow> <m:mi>κ</m:mi> </m:mrow> </m:msub> </m:math> {{\rm{P}}}_{\kappa } ) with respect to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>κ</m:mi> </m:math> \kappa .

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Solvability of the heat equation on a half-space with a dynamical boundary condition and unbounded initial data

2023-06-22

Марек Фила , Kazuhiro Ishige , Tatsuki Kawakami

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General framework to construct local-energy solutions of nonlinear diffusion equations for growing initial data

2023-02-15

Goro Akagi , Kazuhiro Ishige , Ryuichi Sato

This paper presents an integrated framework to construct local-energy solutions to fairly general nonlinear diffusion equations for initial data growing at infinity under suitable assumptions on local-energy estimates for approximate … This paper presents an integrated framework to construct local-energy solutions to fairly general nonlinear diffusion equations for initial data growing at infinity under suitable assumptions on local-energy estimates for approximate solutions. A delicate issue for constructing local-energy solutions resides in the identification of weak limits of nonlinear terms for approximate solutions in a limiting procedure. Indeed, such an identification process often needs the maximal monotonicity of nonlinear elliptic operators (involved in the doubly-nonlinear equations) as well as uniform estimates for approximate solutions; however, even the monotonicity is violated due to a localization of the equations, which is also necessary to derive local-energy estimates for approximate solutions. In the present paper, such an inconsistency is systematically overcome by reducing the original equation to a localized one, where a (no longer monotone) localized elliptic operator is decomposed into the sum of a maximal monotone operator and a perturbation, and by integrating all the other relevant processes. Furthermore, the general framework developed in the present paper is also applied to the Finsler porous medium and fast diffusion equations, which are variants of the classical PME and FDE and also classified as a doubly-nonlinear equation.

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Initial traces and solvability for a semilinear heat equation on a half space of ℝ^{ℕ}

2023-02-15

Kotaro Hisa , Kazuhiro Ishige , Jin Takahashi

We show the existence and the uniqueness of initial traces of nonnegative solutions to a semilinear heat equation on a half space of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R … We show the existence and the uniqueness of initial traces of nonnegative solutions to a semilinear heat equation on a half space of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript upper N"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> under the zero Dirichlet boundary condition. Furthermore, we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the corresponding Cauchy–Dirichlet problem. Our necessary conditions and sufficient conditions are sharp and enable us to find optimal singularities of initial data for the solvability of the Cauchy–Dirichlet problem.

A general nonlinear characterization of stochastic incompleteness

2023-01-01

Gabriele Grillo , Kazuhiro Ishige , Matteo Muratori +1 more

Stochastic incompleteness of a Riemannian manifold $M$ amounts to nonconservation of probability for the heat semigroup on $M$. We show that this property is equivalent to the existence of nonnegative, … Stochastic incompleteness of a Riemannian manifold $M$ amounts to nonconservation of probability for the heat semigroup on $M$. We show that this property is equivalent to the existence of nonnegative, nontrivial, bounded subsolutions to $\Delta W=\psi(W)$ for one, hence all, general nonlinearity $\psi$ which is only required to be continuous in $[0,+\infty)$, nondecreasing, with $\psi(0)=0$ and $\psi>0$ in $(0,+\infty)$. We also prove that it is equivalent to nonuniqueness of nonnegative bounded solutions to the nonlinear parabolic problems $u_t=\Delta\phi(u)$ with nonnegative, bounded initial data, for one, hence all, general nonlinearity $\phi$ which is required to be continuous in $[0,+\infty)$, locally Lipschitz in $(0, +\infty)$ and strictly increasing, including e.g.~equations of both fast diffusion and porous medium type. This strongly improves previous results of \cite{GIM}, that in the elliptic case were obtained for solutions only and assuming in addition that $\psi$ is \emph{convex}, strictly increasing and $C^1$ in $[0,+\infty)$, whereas in the parabolic case they were established assuming that $\phi$ is \emph{concave}, strictly increasing and $C^1$ in $(0,+\infty)$.

Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity

2023-01-01

Kazuhiro Ishige , Shinya Okabe , Tokushi Sato

Abstract In our previous paper [K. Ishige, S. Okabe, and T. Sato, A supercritical scalar field equation with a forcing term , J. Math. Pures Appl. 128 (2019), pp. 183–212], … Abstract In our previous paper [K. Ishige, S. Okabe, and T. Sato, A supercritical scalar field equation with a forcing term , J. Math. Pures Appl. 128 (2019), pp. 183–212], we proved the existence of a threshold <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>κ</m:mi> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> <m:mo>></m:mo> <m:mn>0</m:mn> </m:math> {\kappa }^{\ast }\gt 0 such that the elliptic problem for an inhomogeneous elliptic equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mi>κ</m:mi> <m:mi>μ</m:mi> </m:math> -\Delta u+u={u}^{p}+\kappa \mu in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="bold">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:math> {{\bf{R}}}^{N} possesses a positive minimal solution decaying at the space infinity if and only if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>κ</m:mi> <m:mo>≤</m:mo> <m:msup> <m:mrow> <m:mi>κ</m:mi> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> </m:math> 0\lt \kappa \le {\kappa }^{\ast } . Here, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>N</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:math> N\ge 2 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>μ</m:mi> </m:math> \mu is a nontrivial nonnegative Radon measure in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="bold">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:math> {{\bf{R}}}^{N} with a compact support, and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>1</m:mn> </m:math> p\gt 1 is in the Joseph-Lundgren subcritical case. In this article, we prove the existence of nonminimal positive solutions to the elliptic problem. Our arguments are also applicable to inhomogeneous semilinear elliptic equations with exponential nonlinearity.

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Local solvability and dilation-critical singularities of supercritical fractional heat equations

2023-01-01

Yohei Fujishima , Kotaro Hisa , Kazuhiro Ishige +1 more

We consider the Cauchy problem for fractional semilinear heat equations with supercritical nonlinearities and establish both necessary conditions and sufficient conditions for local-in-time solvability. We introduce the notion of a … We consider the Cauchy problem for fractional semilinear heat equations with supercritical nonlinearities and establish both necessary conditions and sufficient conditions for local-in-time solvability. We introduce the notion of a dilation-critical singularity (DCS) of the initial data and show that such singularities always exist for a large class of supercritical nonlinearities. Moreover, we provide exact formulae for such singularities.

Eventual concavity properties of the heat flow

2023-01-01

Kazuhiro Ishige

The eventual concavity properties are useful to characterize geometric properties of the final state of solutions to parabolic equations. In this paper we give characterizations of the eventual concavity properties … The eventual concavity properties are useful to characterize geometric properties of the final state of solutions to parabolic equations. In this paper we give characterizations of the eventual concavity properties of the heat flow for nonnegative, bounded measurable initial functions with compact support.

Hierarchy of deformations in concavity

2022-12-12

Kazuhiro Ishige , Paolo Salani , Asuka Takatsu

Solvability of superlinear fractional parabolic equations

2022-12-10

Yohei Fujishima , Kotaro Hisa , Kazuhiro Ishige +1 more

Abstract We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the … Abstract We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the solvability of the Cauchy problem and the strength of the singularities of the initial measure.

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Refined Asymptotic Expansions of Solutions to Fractional Diffusion Equations

2022-11-04

Kazuhiro Ishige , Tatsuki Kawakami

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Preface

2022-07-26

Kazuhiro Ishige , Tohru Ozawa , Senjo Shimizu +1 more

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Existence of solutions to nonlinear parabolic equations via majorant integral kernel

2022-06-19

Kazuhiro Ishige , Tatsuki Kawakami , Shinya Okabe

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Decay estimates for Schrödinger heat semigroup with inverse square potential in Lorentz spaces

2022-03-01

Kazuhiro Ishige , Yujiro Tateishi

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Thresholds for the existence of solutions to inhomogeneous elliptic equations with general exponential nonlinearity

2022-01-01

Kazuhiro Ishige , Shinya Okabe , Tokushi Sato

Abstract In this paper we study the existence and the nonexistence of solutions to an inhomogeneous non-linear elliptic problem (P) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"><m:mrow><m:mo>−</m:mo><m:mi>Δ</m:mi><m:mi>u</m:mi><m:mo>+</m:mo><m:mi>u</m:mi><m:mo>=</m:mo><m:mi>F</m:mi><m:mo stretchy="false">(</m:mo><m:mi>u</m:mi><m:mo stretchy="false">)</m:mo><m:mo>+</m:mo><m:mi>κ</m:mi><m:mi>μ</m:mi><m:mo> </m:mo><m:mo> </m:mo><m:mtext>in</m:mtext><m:mo> </m:mo><m:mo> </m:mo><m:msup><m:mi mathvariant="bold">R</m:mi><m:mi>N</m:mi></m:msup><m:mo>,</m:mo><m:mo> … Abstract In this paper we study the existence and the nonexistence of solutions to an inhomogeneous non-linear elliptic problem (P) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"><m:mrow><m:mo>−</m:mo><m:mi>Δ</m:mi><m:mi>u</m:mi><m:mo>+</m:mo><m:mi>u</m:mi><m:mo>=</m:mo><m:mi>F</m:mi><m:mo stretchy="false">(</m:mo><m:mi>u</m:mi><m:mo stretchy="false">)</m:mo><m:mo>+</m:mo><m:mi>κ</m:mi><m:mi>μ</m:mi><m:mo> </m:mo><m:mo> </m:mo><m:mtext>in</m:mtext><m:mo> </m:mo><m:mo> </m:mo><m:msup><m:mi mathvariant="bold">R</m:mi><m:mi>N</m:mi></m:msup><m:mo>,</m:mo><m:mo> </m:mo><m:mi>u</m:mi><m:mo>></m:mo><m:mn>0</m:mn><m:mo> </m:mo><m:mo> </m:mo><m:mtext>in</m:mtext><m:mo> </m:mo><m:mo> </m:mo><m:msup><m:mi mathvariant="bold">R</m:mi><m:mi>N</m:mi></m:msup><m:mo>,</m:mo><m:mo> </m:mo><m:mi>u</m:mi><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo stretchy="false">)</m:mo><m:mo>→</m:mo><m:mn>0</m:mn><m:mo> </m:mo><m:mo> </m:mo><m:mtext>as</m:mtext><m:mo> </m:mo><m:mo> </m:mo><m:mo>|</m:mo><m:mi>x</m:mi><m:mo>|</m:mo><m:mo>→</m:mo><m:mi>∞</m:mi><m:mo>,</m:mo></m:mrow></m:math> - \Delta u + u = F(u) + \kappa \mu \quad {\kern 1pt} {\rm in}{\kern 1pt} \quad {{\bf R}^N},\quad u > 0\quad {\kern 1pt} {\rm in}{\kern 1pt} \quad {{\bf R}^N},\quad u(x) \to 0\quad {\kern 1pt} {\rm as}{\kern 1pt} \quad |x| \to \infty , where F = F ( t ) grows up (at least) exponentially as t → ∞. Here N ≥ 2, κ > 0, and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mrow><m:mi>μ</m:mi><m:mo>∈</m:mo><m:msubsup><m:mi>L</m:mi><m:mtext>c</m:mtext><m:mn>1</m:mn></m:msubsup><m:mo stretchy="false">(</m:mo><m:msup><m:mi mathvariant="bold">R</m:mi><m:mi>N</m:mi></m:msup><m:mo stretchy="false">)</m:mo><m:mo>\</m:mo><m:mo>{</m:mo><m:mn>0</m:mn><m:mo>}</m:mo></m:mrow></m:math> \mu \in L_{\rm{c}}^1({{\bf R}^N})\backslash \{ 0\} is nonnegative. Then, under a suitable integrability condition on μ , there exists a threshold parameter κ * > 0 such that problem (P) possesses a solution if 0 < κ < κ * and it does not possess no solutions if κ > κ * . Furthermore, in the case of 2 ≤ N ≤ 9, problem (P) possesses a unique solution if κ = κ * .

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Solvability of Superlinear Fractional Parabolic Equations

2022-01-01

Yohei Fujishima , Kotaro Hisa , Kazuhiro Ishige +1 more

We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship … We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the solvability of the Cauchy problem and the strength of the singularities of the initial measure.

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Characterization of $F$-concavity preserved by the Dirichlet heat flow

2022-01-01

Kazuhiro Ishige , Paolo Salani , Asuka Takatsu

$F$-concavity is a generalization of power concavity and, actually, the largest available generalization of the notion of concavity. We characterize the $F$-concavities preserved by the Dirichlet heat flow in convex … $F$-concavity is a generalization of power concavity and, actually, the largest available generalization of the notion of concavity. We characterize the $F$-concavities preserved by the Dirichlet heat flow in convex domains on ${\mathbb R}^n$, and complete the study of preservation of concavity properties by the Dirichlet heat flow, started by Brascamp and Lieb in 1976 and developed in some recent papers. More precisely: (1) we discover hot-concavity, which is the strongest $F$-concavity preserved by the Dirichlet heat flow; (2) we show that log-concavity is the weakest $F$-concavity preserved by the Dirichlet heat flow; quasi-concavity is also preserved only for $n=1$; (3) we prove that if $F$-concavity does not coincide with log-concavity and it is not stronger than log-concavity and $n\ge 2$, then there exists an $F$-concave initial datum such that the corresponding solution to the Dirichlet heat flow is not even quasi-concave, hence losing any reminiscence of concavity. Furthermore, we find a sufficient and necessary condition for $F$-concavity to be preserved by the Dirichlet heat flow. We also study the preservation of concavity properties by solutions of the Cauchy--Dirichlet problem for linear parabolic equations with variable coefficients and for nonlinear parabolic equations such as semilinear heat equations, the porous medium equation, and the parabolic $p$-Laplace equation.

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Initial traces and solvability for a semilinear heat equation on a half space of ${\mathbb R}^N$

2022-01-01

Kotaro Hisa , Kazuhiro Ishige , Jin Takahashi

We show the existence and the uniqueness of initial traces of nonnegative solutions to a semilinear heat equation on a half space of ${\mathbb R}^N$ under the zero Dirichlet boundary … We show the existence and the uniqueness of initial traces of nonnegative solutions to a semilinear heat equation on a half space of ${\mathbb R}^N$ under the zero Dirichlet boundary condition. Furthermore, we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the corresponding Cauchy--Dirichlet problem. Our necessary conditions and sufficient conditions are sharp and enable us to find optimal singularities of initial data for the solvability of the Cauchy--Dirichlet problem.

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Solvability of the heat equation on a half-space with a dynamical boundary condition and unbounded initial data

2022-01-01

Марек Фила , Kazuhiro Ishige , Tatsuki Kawakami

We study the linear heat equation on a halfspace with a linear dynamical boundary condition. We are interested in an appropriate choice of the function space of initial functions such … We study the linear heat equation on a halfspace with a linear dynamical boundary condition. We are interested in an appropriate choice of the function space of initial functions such that the problem possesses a solution. It was known before that bounded initial data guarantee solvability. Here we extend that result by showing that data from a weighted Lebesgue space will also do so.

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Supercritical Henon type equation with a forcing term

2022-01-01

Kazuhiro Ishige , Sho Katayama

This paper is concerned with the structure of solutions to the elliptic problem for an Henon type equation with a forcing term. Under suitable assumptions on the forcing term, we … This paper is concerned with the structure of solutions to the elliptic problem for an Henon type equation with a forcing term. Under suitable assumptions on the forcing term, we give a complete classification of the existence/nonexistence of solutions to the problem.

Supercritical H\'Enon Type Equation with a Forcing Term

2022-01-01

Kazuhiro Ishige , Sho Katayama

This paper is concerned with the structure of solutions to the elliptic problem for an H\'enon type equation with a forcing term\begin{equation}\tag{$\mbox{P}_\kappa$}-\Delta u=\alpha(x)u^p+\kappa\mu \quad\mbox{in}\quad{\mathbb R}^N,\quadu>0\quad\mbox{in}\quad{\mathbb R}^N,\quadu(x)\to 0\quad\mbox{as}\quad |x|\to\infty, \end{equation}where $N\ge … This paper is concerned with the structure of solutions to the elliptic problem for an H\'enon type equation with a forcing term\begin{equation}\tag{$\mbox{P}_\kappa$}-\Delta u=\alpha(x)u^p+\kappa\mu \quad\mbox{in}\quad{\mathbb R}^N,\quadu>0\quad\mbox{in}\quad{\mathbb R}^N,\quadu(x)\to 0\quad\mbox{as}\quad |x|\to\infty, \end{equation}where $N\ge 3$, $p>1$, $\kappa>0$, $\alpha$ is a positive continuous function in ${\mathbb R}^N\setminus\{0\}$, and $\mu$ is a nonnegative Radon measure in ${\mathbb R}^N$. Under suitable assumptions on the exponent~$p$, the coefficient~$\alpha$, and the forcing term~$\mu$, we give a complete classification of the existence/nonexistence of solutions to problem~($\mbox{P}_\kappa$).

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New characterizations of log-concavity via Dirichlet heat flow

2021-11-22

Kazuhiro Ishige , Paolo Salani , Asuka Takatsu

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Optimal singularities of initial functions for solvability of a semilinear parabolic system

2021-11-11

Yohei Fujishima , Kazuhiro Ishige

Let $(u, v)$ be a nonnegative solution to the semilinear parabolic system $$ \mbox{(P)} \qquad \left\{ \begin{array}{ll} \partial_{t} u = D_{1} \Delta u + v^{p}, \quad x \in \mathbf{R}^{N}, \ … Let $(u, v)$ be a nonnegative solution to the semilinear parabolic system $$ \mbox{(P)} \qquad \left\{ \begin{array}{ll} \partial_{t} u = D_{1} \Delta u + v^{p}, \quad x \in \mathbf{R}^{N}, \ t > 0,\\ \partial_{t} v = D_{2} \Delta v + u^{q}, \quad x \in \mathbf{R}^{N}, \ t > 0,\\ (u(\cdot,0), v(\cdot,0)) = (\mu, \nu), \quad x \in \mathbf{R}^{N}, \end{array} \right. $$ where $D_{1}$, $D_{2} > 0$, $0 1$ and $(\mu, \nu)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in $\mathbf{R}^{N}$. In this paper we study sufficient conditions on the initial data for the solvability of problem (P) and clarify optimal singularities of the initial functions for the solvability.

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Power concavity for elliptic and parabolic boundary value problems on rotationally symmetric domains

2021-10-25

Kazuhiro Ishige , Paolo Salani , Asuka Takatsu

We study power concavity of rotationally symmetric solutions to elliptic and parabolic boundary value problems on rotationally symmetric domains in Riemannian manifolds. As applications of our results to the hyperbolic … We study power concavity of rotationally symmetric solutions to elliptic and parabolic boundary value problems on rotationally symmetric domains in Riemannian manifolds. As applications of our results to the hyperbolic space [Formula: see text], we have The first (positive) Dirichlet eigenfunction of the Laplacian on a ball in [Formula: see text] raised to some power [Formula: see text] is strictly concave; Let [Formula: see text] be the heat kernel on [Formula: see text]. Then [Formula: see text] is strictly log-concave in [Formula: see text] for [Formula: see text] and [Formula: see text].

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Decay estimates for Schrödinger heat semigroup with inverse square potential in Lorentz spaces II

2021-09-08

Kazuhiro Ishige , Yujiro Tateishi

Let <inline-formula><tex-math id="M1">\begin{document}$ H: = -\Delta+V $\end{document}</tex-math></inline-formula> be a nonnegative Schrödinger operator on <inline-formula><tex-math id="M2">\begin{document}$ L^2({\bf R}^N) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ N\ge 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ V $\end{document}</tex-math></inline-formula> … Let <inline-formula><tex-math id="M1">\begin{document}$ H: = -\Delta+V $\end{document}</tex-math></inline-formula> be a nonnegative Schrödinger operator on <inline-formula><tex-math id="M2">\begin{document}$ L^2({\bf R}^N) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ N\ge 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ V $\end{document}</tex-math></inline-formula> is a radially symmetric inverse square potential. Let <inline-formula><tex-math id="M5">\begin{document}$ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $\end{document}</tex-math></inline-formula> be the operator norm of <inline-formula><tex-math id="M6">\begin{document}$ \nabla^\alpha e^{-tH} $\end{document}</tex-math></inline-formula> from the Lorentz space <inline-formula><tex-math id="M7">\begin{document}$ L^{p, \sigma}({\bf R}^N) $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M8">\begin{document}$ L^{q, \theta}({\bf R}^N) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M9">\begin{document}$ \alpha\in\{0, 1, 2, \dots\} $\end{document}</tex-math></inline-formula>. We establish both of upper and lower decay estimates of <inline-formula><tex-math id="M10">\begin{document}$ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $\end{document}</tex-math></inline-formula> and study sharp decay estimates of <inline-formula><tex-math id="M11">\begin{document}$ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $\end{document}</tex-math></inline-formula>. Furthermore, we characterize the Laplace operator <inline-formula><tex-math id="M12">\begin{document}$ -\Delta $\end{document}</tex-math></inline-formula> from the view point of the decay of <inline-formula><tex-math id="M13">\begin{document}$ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $\end{document}</tex-math></inline-formula>.

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Initial traces and solvability of Cauchy problem to a semilinear parabolic system

2021-05-04

Yohei Fujishima , Kazuhiro Ishige

Let $(u, v)$ be a solution to a semilinear parabolic system $$ \mbox{(P)} \qquad \left\{ \begin{array}{ll} \partial_{t} u = D_1 \Delta u+v^{p} \quad \mbox{in} \quad \mathbf{R}^{N} \times (0, T),\\ \partial_{t} … Let $(u, v)$ be a solution to a semilinear parabolic system $$ \mbox{(P)} \qquad \left\{ \begin{array}{ll} \partial_{t} u = D_1 \Delta u+v^{p} \quad \mbox{in} \quad \mathbf{R}^{N} \times (0, T),\\ \partial_{t} v = D_2 \Delta v+u^{q} \quad \mbox{in} \quad \mathbf{R}^{N} \times (0, T),\\ u, v \geq 0 \quad \mbox{in} \quad \mathbf{R}^{N} \times (0, T),\\ (u(\cdot, 0), v(\cdot, 0)) = (\mu, \nu) \quad \mbox{in} \quad \mathbf{R}^N, \end{array} \right. $$ where $N \geq 1$, $T > 0$, $D_1 > 0$, $D_2 > 0$, $0 1$ and $(\mu, \nu)$ is a pair of Radon measures or nonnegative measurable functions in $\mathbf{R}^{N}$. In this paper we study qualitative properties of the initial trace of the solution $(u, v)$ and obtain necessary conditions on the initial data $(\mu, \nu)$ for the existence of solutions to problem (P).

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Power concavity and Dirichlet heat flow

2021-01-01

Kazuhiro Ishige , Paolo Salani , Asuka Takatsu

We show that log-concavity is the weakest power concavity preserved by the Dirichlet heat flow in $N$-dimensional convex domains, where $N\ge 2$ (indeed, we prove that starting with a negative … We show that log-concavity is the weakest power concavity preserved by the Dirichlet heat flow in $N$-dimensional convex domains, where $N\ge 2$ (indeed, we prove that starting with a negative power concave initial datum may result in losing immediately any reminiscence of concavity). Jointly with what we already know, i.e. that log-concavity is the strongest power concavity preserved by the Dirichlet heat flow, we see that log-concavity is indeed the only power concavity preserved by the Dirichlet heat flow.

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General framework to construct local-energy solutions of nonlinear diffusion equations for growing initial data

2021-01-01

Goro Akagi , Kazuhiro Ishige , Ryuichi Sato

This paper presents an integrated framework to construct local-energy solutions to fairly general nonlinear diffusion equations for initial data growing at infinity under suitable assumptions on local-energy estimates for approximate … This paper presents an integrated framework to construct local-energy solutions to fairly general nonlinear diffusion equations for initial data growing at infinity under suitable assumptions on local-energy estimates for approximate solutions. A delicate issue for constructing local-energy solutions resides in the identification of weak limits of nonlinear terms for approximate solutions in a limiting procedure. Indeed, such an identification process often needs the maximal monotonicity of nonlinear elliptic operators (involved in the doubly-nonlinear equations) as well as uniform estimates for approximate solutions; however, even the monotonicity is violated due to a localization of the equations, which is also necessary to derive local-energy estimates for approximate solutions. In the present paper, such an inconsistency will be systematically overcome by reducing the original equation to a localized one, where a (no longer monotone) localized elliptic operator will be decomposed into the sum of a maximal monotone one and a perturbation, and by integrating all the other relevant processes. Furthermore, the general framework developed in the present paper will also be applied to the Finsler porous medium and fast diffusion equations, which are variants of the classical PME and FDE and also classified as a doubly-nonlinear equation.

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Refined asymptotic expansions of solutions to fractional diffusion equations

2021-01-01

Kazuhiro Ishige , Tatsuki Kawakami

In this paper, as an improvement of the paper [K. Ishige, T. Kawakami and H. Michihisa, SIAM J. Math. Anal. 49 (2017) pp. 2167--2190], we obtain the higher order asymptotic … In this paper, as an improvement of the paper [K. Ishige, T. Kawakami and H. Michihisa, SIAM J. Math. Anal. 49 (2017) pp. 2167--2190], we obtain the higher order asymptotic expansions of the large time behavior of the solution to the Cauchy problem for inhomogeneous fractional diffusion equations and nonlinear fractional diffusion equations.

Existence of solutions to nonlinear parabolic equations via majorant integral kernel

2021-01-01

Kazuhiro Ishige , Tatsuki Kawakami , Shinya Okabe

We establish the existence of solutions to the Cauchy problem for a large class of nonlinear parabolic equations including fractional semilinear parabolic equations, higher-order semilinear parabolic equations, and viscous Hamilton-Jacobi … We establish the existence of solutions to the Cauchy problem for a large class of nonlinear parabolic equations including fractional semilinear parabolic equations, higher-order semilinear parabolic equations, and viscous Hamilton-Jacobi equations by using the majorant kernel introduced in [K. Ishige, T. Kawakami, and S. Okabe, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire 37 (2020), 1185--1209].

Nonlinear characterizations of stochastic completeness

2020-05-15

Gabriele Grillo , Kazuhiro Ishige , Matteo Muratori

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Existence of solutions for an inhomogeneous fractional semilinear heat equation

2020-05-15

Kotaro Hisa , Kazuhiro Ishige , Jin Takahashi

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Existence of solutions for a higher-order semilinear parabolic equation with singular initial data

2020-04-23

Shinya Okabe , Kazuhiro Ishige , Tatsuki Kawakami

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The large diffusion limit for the heat equation in the exterior of the unit ball with a dynamical boundary condition

2020-04-08

Марек Фила , Kazuhiro Ishige , Tatsuki Kawakami +1 more

We study the heat equation in the exterior of the unit ball with a linear dynamical boundary condition. Our main aim is to find upper and lower bounds for the … We study the heat equation in the exterior of the unit ball with a linear dynamical boundary condition. Our main aim is to find upper and lower bounds for the rate of convergence to solutions of the Laplace equation with the same dynamical boundary condition as the diffusion coefficient tends to infinity.

Power concavity for elliptic and parabolic boundary value problems on rotationally symmetric domains

2020-02-24

Kazuhiro Ishige , Paolo Salani , Asuka Takatsu

We study power concavity of rotationally symmetric solutions to elliptic and parabolic boundary value problems on rotationally symmetric domains in Riemannian manifolds. As applications of our results to the hyperbolic … We study power concavity of rotationally symmetric solutions to elliptic and parabolic boundary value problems on rotationally symmetric domains in Riemannian manifolds. As applications of our results to the hyperbolic space ${\bf H}^N$ we have: $\bullet$ The first Dirichlet eigenfunction on a ball in ${\bf H}^N$ is strictly positive power concave; $\bullet$ Let $\Gamma$ be the heat kernel on ${\bf H}^N$. Then $\Gamma(\cdot,y,t)$ is strictly log-concave on ${\bf H}^N$ for $y\in {\bf H}^N$ and $t>0$.

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The large diffusion limit for the heat equation with a dynamical boundary condition

2020-01-24

Марек Фила , Kazuhiro Ishige , Tatsuki Kawakami

We study the heat equation on a half-space with a linear dynamical boundary condition. Our main aim is to show that, if the diffusion coefficient tends to infinity, then the … We study the heat equation on a half-space with a linear dynamical boundary condition. Our main aim is to show that, if the diffusion coefficient tends to infinity, then the solutions converge (in a suitable sense) to solutions of the Laplace equation with the same dynamical boundary condition.

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Coauthor	Papers Together
Tatsuki Kawakami	40
Paolo Salani	24
Марек Фила	17
Kotaro Hisa	15
Yohei Fujishima	15
Yoshitsugu Kabeya	13
Asuka Takatsu	12
Shinya Okabe	9
Ryuichi Sato	7
Johannes Lankeit	6
Michinori Ishiwata	5
Norisuke Ioku	5
Robert Laister	4
Goro Akagi	4
Yujiro Tateishi	4
Sho Katayama	4
Gabriele Grillo	4
Tokushi Sato	4
Noriko Mizoguchi	4
Asato Mukai	4
Matteo Muratori	4
Junyong Eom	3
Kazushige Nakagawa	3
Jin Takahashi	3
Eiji Yanagida	3
Qing Liu	2
Kanako Kobayashi	2
Minoru Murata	2
El Maati Ouhabaz	2
Hiroki Yagisita	2
Jin Takahashi	2
Haruto Tokunaga	1
Philippe Laurençot	1
Kanako Kobayashi	1
Daniele Andreucci	1
Ryo Takada	1
Senjo Shimizu	1
Hironori Michihisa	1
Filippo Gazzola	1
Tohru Ozawa	1
Fabio Punzo	1
Carlo Nitsch	1
Mikołaj Sierżęga	1
Yasushi Taniuchi	1
Nobuhito Miyake	1
Juha Kinnunen	1
Hiroki Maekawa	1
Qing Liu	1
Paolo Salani	1
Sho Katayama	1