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arXiv · 2411.03983

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

Nurdaulet N. Tobakhanov·Berikbol T. Torebek
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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].

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Nurdaulet N. Tobakhanov, Berikbol T. Torebek. 2024-11-06. On the critical behavior for the semilinear biharmonic heat equation with forcing term in an exterior domain. https://doi.org/10.1016/j.jde.2025.113758

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