Uniform boundedness and blow-up rate of solutions in non-scale-invariant superlinear heat equations

TL;DR

This paper investigates the boundedness and blow-up behavior of radially symmetric solutions to superlinear heat equations with Dirichlet boundary conditions, providing new results on solution boundedness and blow-up rates in various dimensions.

Contribution

It establishes uniform boundedness of global solutions and rules out type II blow-up for certain superlinear heat equations, extending results to high dimensions and rapidly growing nonlinearities.

Findings

Global solutions are uniformly bounded under specified conditions.

Type II blow-up solutions do not exist for the studied equations.

Results apply to high dimensions and nonlinearities with super-exponential growth.

Abstract

For superlinear heat equations with the Dirichlet boundary condition, the $L^\infty$ estimates of radially symmetric solutions are studied. In particular, the uniform boundedness of global solutions and the non-existence of solutions with type II blow-up are proved. For the space dimension greater than $9$, our results are shown under the condition that an exponent representing the growth rate of a nonlinear term is between the Sobolev exponent and the Joseph-Lundgren exponent. In the case where the space dimension is greater than $2$ and smaller than $10$, our results are applicable for nonlinear terms growing extremely faster than the exponential function.