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

2020/08/03 by Mohamed Majdoub, Majdoub, Mohamed · 2 citations
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2008.01290

openalex publication_date 2020/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the large-time behavior of sign-changing solutions of the\ninhomogeneous equation u_t-\Δ u=|x|^\α |u|^p+\ζ(t) , mathbf\nw(x) in (0,\∞)\×\ℝ^N, where N\≥ 3, p>1, \α>-2,\n z, mathbf w are continuous functions such that \ζ(t)=t^\σ or\n\ζ(t)\∼ t^\σ as t\→ 0, \ζ(t)\∼ t^m as t\→\∞ . We\nobtain local existence for \σ>-1. We also show the following:\n beginitemize item If m\≤ 0, p<\(N-2m+\α)/(N-2m-2) and\n\∫_\ℝ^N mathbf w(x)dx>0, then all solutions blow up in finite\ntime; item If m> 0, p>1 and \∫_\ℝ^N mathbf w(x)dx>0, then\nall solutions blow up in finite time; item If \ζ(t)=t^\σ with\n-1<\σ<0, then for u₀:=u(t=0) and mathbf w sufficiently small the\nsolution exists globally. enditemize We also discuss lower dimensions. The\nmain novelty in this paper is that blow up depends on the behavior of \ζ\nat infinity.\n

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