Finite time extinction for a diffusion equation with spatially inhomogeneous strong absorption. (Q6133014)

The authors consider a diffusion equation with strong absoption \(\partial_t u - \Delta u^m + |x|^\sigma u^q=0\), where \(m\ge1\) and \(q\in(0,1)\). Two results on finite-time extinction of weak solutions are given: (i) for any nonnegative, bounded initial data, provided that \(\sigma < \sigma^* := 2(1-q)/(m-1)\), and (ii) for ``flat'' initial data \(u_0(x) \le A|x|^a\), with suitable \(A\) and \(a\), if \(\sigma \ge \sigma^*\). Apart from relying on the (long-established) property of instantaneous shrinking of supports, the proofs are short and simple.