Well-posedness and blow-up for an inhomogeneous semilinear parabolic equation

TL;DR

This paper studies the long-term behavior of solutions to an inhomogeneous semilinear parabolic equation, revealing how blow-up or global existence depends on the asymptotic behavior of the inhomogeneous term.

Contribution

It introduces new results on the conditions for blow-up or global existence based on the growth of the inhomogeneous term at infinity.

Findings

Solutions blow up in finite time under certain conditions on p and the inhomogeneous term.

Global existence occurs when the inhomogeneous term grows like a negative power of t with small initial data.

Blow-up behavior is influenced by the asymptotic behavior of the inhomogeneous forcing term.

Abstract

We consider the large-time behavior of sign-changing solutions of the inhomogeneous equation $u_t-\Delta u=|x|^\alpha |u|^{p}+\zeta(t)\,{\mathbf w}(x)$ in $(0,\infty)\times\mathbb{R}^N$, where $N\geq 3$, $p>1$, $\alpha>-2$, $\z, {\mathbf w}$ are continuous functions such that $\zeta(t)=t^\sigma$ or $\zeta(t)\sim t^\sigma$ as $t\to 0$, $\zeta(t)\sim t^m$ as $t\to\infty$ . We obtain local existence for $\sigma>-1$. We also show the following: \begin{itemize} \item If $m\leq 0$, $p<\frac{N-2m+\alpha}{N-2m-2}$ and $\int_{\mathbb{R}^N}{\mathbf w}(x)dx>0$, then all solutions blow up in finite time; \item If $m> 0$, $p>1$ and $\int_{\mathbb{R}^N}{\mathbf w}(x)dx>0$, then all solutions blow up in finite time; \item If $\zeta(t)=t^\sigma$ with $-1<\sigma<0$, then for $u_0:=u(t=0)$ and ${\mathbf w}$ sufficiently small the solution exists globally. \end{itemize} We also discuss lower dimensions. The main novelty in this paper is that blow up depends on the behavior of $\zeta$ at infinity.