Grow-up for a quasilinear heat equation with a localized reaction in\n higher dimensions

2018/01/29 by Raúl Ferreira, Ferreira, Raul, Arturo de Pablo +1
Computer Science · Mathematics · #35K55 #35K57 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1801.09538

openalex publication_date 2018/01/29 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

We study the behaviour of nonnegative solutions to the quasilinear heat\nequation with a reaction localized in a ball u_t=
Delta u^m+a(x)u^p, for\nm>0, 0<p\≤\max 1,m , a(x)= mathds1_BL(x), 0<L<\∞ and\nN\≥2. We study when solutions, which are global in time, are bounded or\nunbounded. In particular we show that the precise value of the length L plays\na crucial role in the critical case p=m for N\≥3. We also obtain the\nasymptotic behaviour of unbounded solutions and prove that the grow-up rate is\ndifferent in most of the cases to the one obtained when L=\∞.\n

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