Instantaneous compactification of supports of solutions to semilinear parabolic equations and systems thereof (Q914951)

The main topic of this paper is a phenomenon discovered by \textit{L. C. Evans} and \textit{B. E. Knerr} [Ill. J. Math. 23, 153-166 (1979; Zbl 0403.35052)] for a heat equation with a nonlinear damping term, of the type: \[ Au=\partial u/\partial t-\Delta u+g(x,t)| u|^ p sign u=0,\;u(x,0)=f(x),\;x\in R^ n. \] It is assumed that \(g\geq 0\), \(f\geq 0\). This phenomenon consists of spatial support of the solutions (called instantaneous compactification of the support). Theorem 1 states that if \(p<1\), \[ g(x,t)\geq g_ 0(1+| x|)^{- \beta}\;(g_ 0\geq 0,\;\beta \geq 0), \] \[ f(x)\leq M(1+| x|)^{- \alpha}\;(M>0,\;\alpha>0) \] \(\alpha(1-p)>\beta\), then \(u(x,t)\) has a bounded support in x. Theorem 2 offers nonexistence of compact support condition. Similar statements are proved regarding a specific component of a solution.