Quenching versus blow-up. (English) Zbl 0965.35086
The authors study the nonlinear initial-boundary value problem \(u_t=\Delta u - u^{-q},\) \(u(x,0)=u_0(x)\) on the bounded domain \(\Omega \subset {\mathbb R}^n\) with a smooth boundary and the Neumann boundary condition \(\partial u / \partial \nu=u^p,\) where \(p\) and \(q\) are positive parameters, \(0\leq u_0(x) \leq M,\) and \(\partial u_0 / \partial \nu =u_0^p\) on \(\partial \Omega.\) They derive criteria for finite time quenching (i.e. convergence of the solution to zero) and blow-up, estimate the rates of quenching and blow-up, and determine the sets of points in which quenching or blow-up can occur. The results parallel earlier analyses of this system with zero flux or zero reaction part.
Reviewer: David S.Boukal (České Budějovice)
MSC:
35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |
35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
35B40 | Asymptotic behavior of solutions to PDEs |
35K55 | Nonlinear parabolic equations |
35K65 | Degenerate parabolic equations |