Global existence and nonexistence for a parabolic system with nonlinear boundary conditions. (English) Zbl 1004.35012

Global existence and blow-up results for positive solutions to the weakly coupled system \(u_t = \triangle u + h(u,v),\;v_t = \triangle v + r(u, v)\) with nonlinear boundary conditions \(u_\nu = f(v),\;v_\nu = g(u)\) are established. The functions \(h\) and \(r\) are supposed to be nonnegative, smooth and such that \(h(u,v)/u,\;r(u,v)/v\) are globally bounded. The general results are then applied to the special choice of \(f\) and \(g\).
Theorem 4.1. Let \(p \geq q > 0\) and \(f(z) = (p+1)z^p, g(z) = (q+1)z^q\). If \(pq \leq 1\), every positive solution is global, whereas if \(pq > 1\), every positive solution blows up in finite time.
Theorem 4.2. Let \(f(z) = z \log z\) and \(g(z) = ((p+1)/2)^2z^p \log z\) with \(p > 0\) and \(u_0 \geq 1,\;v_0 \geq 1\) (\(u_0\) and \(v_0\) are initial values of \(u\) and \(v\)). If \(p<1\) every solution is global, whereas if \(p \geq 1\) every solution blows up in finite time.


35B33 Critical exponents in context of PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B60 Continuation and prolongation of solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
35K50 Systems of parabolic equations, boundary value problems (MSC2000)