The blow-up rate for a non-scaling invariant semilinear heat equation

Abstract

We consider the semilinear heat equation $${\partial }_{t}u-\Delta u=f(u),(x,t)\in {\mathbb{R}}^N\times [0,T),(1)$$ with $f(u)=|u{|}^{p-1}u{\log }^a(2+u^2)$, where $p>1$ is Sobolev subcritical and $a\in \mathbb{R}$. We first show an upper bound for any blow-up solution of (1). Then, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), namely $u^{\prime }=|u{|}^{p-1}u{\log }^a(2+u^2)$. In other terms, all blow-up solutions in the Sobolev subcritical range are Type I solutions. Up to our knowledge, this is the first determination of the blow-up rate for a semilinear heat equation where the main nonlinear term is not homogeneous.

arXiv commentarXiv admin note: text overlap with arXiv:2012.05374

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Cited by in corpus (1)

• Universal estimates and Liouville theorems for superlinear problems without scale invariance