Uniform boundedness and blow-up rate of solutions in non-scale-invariant superlinear heat equations

2024/12/29 by Yohei Fujishima, Fujishima, Yohei, Toru Kan +1 · 2 citations
Engineering · Mathematics · #Stability and Controllability of Differential Equations #Advanced Mathematical Physics Problems #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2412.20402

Abstract

For superlinear heat equations with the Dirichlet boundary condition, the L^∞ estimates of radially symmetric solutions are studied. In particular, the uniform boundedness of global solutions and the non-existence of solutions with type II blow-up are proved. For the space dimension greater than 9, our results are shown under the condition that an exponent representing the growth rate of a nonlinear term is between the Sobolev exponent and the Joseph-Lundgren exponent. In the case where the space dimension is greater than 2 and smaller than 10, our results are applicable for nonlinear terms growing extremely faster than the exponential function.

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