Type: Article
Publication Date: 2001-06-01
Citations: 6
DOI: https://doi.org/10.21099/tkbjm/1496164222
In this note we obtain estimates in terms of the size of the initial data for the blow-up time of positive solutions of the heat equation in $R_{+}$ with a nonlinear boundary condition $-u_{X}(0, t)=$ $u^{P}(0, t)$ .where $p>1$ is fixed and $\lambda>0$ is a parameter.Throughout this note we assume that the initial datum $\phi$ is continuous, positive and bounded.Existence, uniqueness, regularity and continuous dependence on the initial data for this problem were proved, for instance, in [2].For problem (1), it is well known that if $\lambda$ is large enough the solution blows up in finite time $T_{\lambda}$ ( $T_{\lambda}$ depends on $\lambda$ ) if and only if $p>1$ , see for example, [1], [3], [4], [8], [10].This means that there exists a finite time $T_{\lambda}$ with $\lim_{t\nearrow T_{\lambda}}\Vert u(\cdot, t)\Vert_{\infty}=+\infty$ .