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October 2021 Blow-up at space infinity for a quasilinear parabolic equation with space-dependent reaction
Ryuichi SUZUKI, Noriaki UMEDA
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Hokkaido Math. J. 50(3): 345-408 (October 2021). DOI: 10.14492/hokmj/2019-195

Abstract

We consider a nonnegative solution $u$ of the Cauchy problem for a quasilinear parabolic equation $u_t=\Delta u^m+\mu(x)u^p$ with the initial data $u_0(x)\,(\not\equiv 0)$ satisfying $\| \tilde\mu u_0\|_{L^{\infty}(\mathbf R^N)}<\infty$, where nonnegative function $\mu(x)$ satisfies some condition and $\tilde\mu=\mu^{1/(p-1)}$. We give sufficient conditions on $u_0$ for a weighted solution $\tilde\mu u$ to blow up at space infinity and for a direction $\psi \in \mathbf S^{N-1}$ to be a blow-up direction of $\tilde\mu u$. We also show that such a weighted solution $\tilde\mu u$ blows up completely at the blow-up time of $\tilde\mu u$.

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Ryuichi SUZUKI. Noriaki UMEDA. "Blow-up at space infinity for a quasilinear parabolic equation with space-dependent reaction." Hokkaido Math. J. 50 (3) 345 - 408, October 2021. https://doi.org/10.14492/hokmj/2019-195

Information

Received: 25 October 2019; Revised: 20 May 2020; Published: October 2021
First available in Project Euclid: 17 December 2021

Digital Object Identifier: 10.14492/hokmj/2019-195

Subjects:
Primary: 35B40 , 35B44 , 35B60 , 35K15 , 35K59 , 35K65

Keywords: Blow-up , Cauchy problem , minimal blow-up time , quasilinear parabolic equation , space infinity , space-dependent reaction

Rights: Copyright c 2021 Hokkaido University, Department of Mathematics

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Vol.50 • No. 3 • October 2021
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