Existence of a Stable Blow-up profile for the nonlinear heat equation\n with a critical power nonlinear gradient term

2015/06/27 by Slim Tayachi, Tayachi, Slim, Hatem Zaag +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1506.08306

openalex publication_date 2015/06/27 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We consider the nonlinear heat equation with a nonlinear gradient term:\n\∂_t u =\Δ u+\μ|\∇ u|^q+|u|^p-1u, ; \μ>0, ; q=2p/(p+1), ;\np>3, ; t\∈ (0,T), ; x\∈ R^N. We construct a solution which blows up in\nfinite time T>0. We also give a sharp description of its blow-up profile and\nshow that it is stable with respect to perturbations in initial data. The proof\nrelies on the reduction of the problem to a finite dimensional one, and uses\nthe index theory to conclude. The blow-up profile does not scale as\n(T-t)^1/2|\log(T-t)|^1/2, like in the standard nonlinear heat equation,\ni.e. \μ=0, but as (T-t)^1/2|\log(T-t)|^\β with\n\β=(p+1)/[2(p-1)]>1/2. We also show that u and \∇ u blow up\nsimultaneously and at a single point, and give the final profile. In\nparticular, the final profile is more singular than the case of the standard\nnonlinear heat equation.\n

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