Nonexistence results for systems of elliptic and parabolic differential inequalities in exterior domains of \(\mathbb{R}^n\)
type: Item
instance of: scholarly article
title: Nonexistence results for systems of elliptic and parabolic differential inequalities in exterior domains of \(\mathbb{R}^n\)
zbMATH Open document ID: 1379.35163
author: Yuhua Sun
published in: Pacific Journal of Mathematics
publication date: 2017-11-22T00:00:00Z
review text: The author studies the nonexistence of nonnegative solutions of systems of the following parabolic differential inequalities \[ \begin{aligned} &\Delta u - \partial_t u + x^a |v|^p \leq 0 \\ &\Delta v - \partial_t v + x^b |u|^q \leq 0 , \quad (x,t) \in D \times (0,\infty) \\ &u(x,t) \geq f(x), \;v(x,t) \geq g(x) \\ &u(x,0)=u_0(x), \;v(x,0)=v_0(x), \;\;x \in {\bar{D}}^c \end{aligned} \] where \(D \subset \mathbb{R}^n\) is a bounded Lipschitz domain with \(n \geq 3\) containing the origin and \({\bar{D}}^c=\mathbb{R}^n \setminus \bar{D}\). The exponents satisfy \(a,b > -2\) and \(p,q >1\) , \(f,g \in L^1(\partial D)\) are nonnegative and positive somewhere functions and \(u_0(x), v_0(x)\) are nonnegative functions. The main goal is to introduce a unified approach for the investigation of nonexistence results of systems of parabolic differential inequalities. As the author notes in the elliptic case his results accord with those on the elliptic differential inequalities \[ \begin{aligned} &\Delta u + x^a |v|^p \leq 0 \\ &\Delta v + x^b |u|^q \leq 0 , \quad x \in {\bar{D}}^c \\ &u(x) \geq f(x), \;v(x) \geq g(x), \;\;x \in \partial D \end{aligned} \] given by \textit{M.-F. Bidaut-Véron} and \textit{S. Pohozaev} [J. Anal. Math. 84, 1--49 (2001; Zbl 1018.35040)].
reviewed by: Andrey Zahariev
Mathematics Subject Classification ID: 35K51
Mathematics Subject Classification ID: 35M87
zbMATH DE Number: 6810930
zbMATH Keywords: system of differential inequalities
zbMATH Keywords: critical exponents
zbMATH Keywords: exterior domain
MaRDI profile type: Publication
full work available at URL: pjm.2018.293.245
OpenAlex ID: W2768720495
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DOI: 10.2140/PJM.2018.293.245
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description: scientific article; zbMATH DE number 6810930