Nonexistence results for systems of elliptic and parabolic differential inequalities in exterior domains of \(\mathbb{R}^n\)
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)].
scientific article; zbMATH DE number 6810930