article

Title: Boundedness of global solutions for a supercritical semilinear heat equation and its application

Authors: Noriko Mizoguchi

Issue: Volume 54 (2005), Issue 4, 1047-1060

Abstract:

$We consider a Cauchy problem and a Cauchy-Dirichlet problem in a ball in $\R^N$ for a semilinear heat equation \[ u_t=\Delta u+u^p \] with radially symmetric nonnegative initial data. Let \[ p>\frac{N-2\sqrt{N-1}}{N-4-2\sqrt{N-1}}\quad\mbox{and}\quad N\geq11. \] It is proved that if $u$ is a global solution of the Cauchy problem with initial data satisfying a condition near infinity, then there exists a positive constant $C$ such that $|u(t)|_{\infty}\leq C$ for all $t\geq0$. The uniform boundedness of solutions for the Cauchy-Dirichlet problem in a ball is also given. The result is used to show the existence of a solution for the Cauchy-Dirichlet problem in a ball which exhibits the type II blowup.$