Global existence and blow up results for a heat equation with nonlinear nonlocal term. (Q454549)

The author studies global existence and blow up phenomena of solutions to the nonlocal parabolic equation of the form NEWLINE\[NEWLINE u_t-\Delta u =\int _0^t k(t,s)| u| ^{p-1}u(s) \text{d}s,\; \; p>1 NEWLINE\]NEWLINE both in \(R^n\) and in a bounded smooth domain \(\Omega \) with the homogeneous Dirichlet boundary condition. The kernel \(k\) is assumed to satisfy \(k(\lambda t,\lambda s) =\lambda ^{-\gamma } k(t,s)\) for all \(0<s<t, \lambda >0\). In addition, if \(\gamma <2\), and \(\int _0^1 k(1,\eta )\text{d}\eta <\infty \), it is shown that there is a relation between \(\gamma , \;p,\) and the behavior of \(k(1,\eta )\) at \(\{0,1\}\), such that solutions blow up in a finite time. Relations that assure global existence for sufficiently small initial data are also given.