Time-vanishing properties of solutions of some degenerate parabolic equations with strong absorption (Q2770158)

The author studies the time compact support property for \(\partial_t u-\Delta_p u+b(x)u^q=0,\) \(p>2\) and for \(\partial_t u-\Delta(u^m) +b(x) u^q=0,\) \(m>1,\) \(0\leq q<1\) in \(\Omega\times (0,\infty)\), where \(\Omega\) is an open bounded subset of \({\mathbb R}^N\) and \(b\) is a measurable nonnegative function in \(\Omega.\) Some criteria involving the asymptotics as \(h\to 0\) of the first eigenvalue of \(-\Delta_p +{b(x)\over h^p}\) are given which imply such a phenomenon. Applying these criteria it is obtained that the time vanishing property occurs when \(1/b\in L^s(\Omega)\) for some \(s=s(q,p,m).\)