On the Cauchy problem for evolution $p(x)$-Laplace equation
On the Cauchy problem for evolution $p(x)$-Laplace equation
We consider the Cauchy problem for the equation \text{$u_{t}-\operatorname{div} \left( a(x,t) |\nabla u|^{p(x)-2}\nabla u\right) =f(x,t)$ in $S_{T}=\mathbb{R}^{n}\times(0,T)$} with measurable but possibly discontinuous variable exponent p(x):\,\mathbb{R}^{n}\mapsto [p^-,p^+]\subset (1,\infty) . It is shown that for every u(x,0)\in L^{2}(\mathbb{R}^{n}) and f\in L^{2}(S_T) the problem has at least one weak solution u\in C^{0}([0,T];L^{2} _{loc}(\mathbb{R}^{n}))\cap …