On the critical behavior for the semilinear biharmonic heat equation with forcing term in an exterior domain

TL;DR

This paper studies the critical exponents and blow-up behavior of solutions to the semilinear biharmonic heat equation with forcing in exterior domains, revealing how the forcing term influences solution lifespan and critical thresholds.

Contribution

It introduces new critical exponents for the biharmonic heat equation with forcing in exterior domains and analyzes their dependence on the forcing term and domain dimension.

Findings

Critical exponent p_{Crit} equals infinity for N=2,3,4.

Critical exponent p_{Crit} equals N/(N-4) for N≥5.

Forcing term introduces a second critical exponent affecting solution behavior.

Abstract

In this paper, we investigate the critical behavior of solutions to the semilinear biharmonic heat equation with forcing term $f(x),$ under six homogeneous boundary conditions. This paper is the first since the seminal work by Bandle, Levine, and Zhang [J. Math. Anal. Appl. 251 (2000) 624-648], to focus on the study of critical exponents in exterior problems for semilinear parabolic equations with a forcing term. By employing a method of test functions and comparison principle, we derive the critical exponents $p_{Crit}$ in the sense of Fujita. Moreover, we show that $p_{Crit}=\infty$ if $N=2,3,4$ and $p_{Crit}=\frac{N}{N-4}$ if $N \geq 5$. The impact of the forcing term on the critical behavior of the problem is also of interest, and thus a second critical exponent in the sense of Lee-Ni, depending on the forcing term is introduced. We also discuss the case $f\equiv 0$, and present the finite-time blow-up results and lifespan estimates of solutions for the subcritical and critical cases. The lifespan estimates of solutions are obtained by employing the method proposed by Ikeda and Sobajama in [Nonlinear Anal. 182 (2019) 57-74].