Global existence and nonexistence for a parabolic system with nonlinear boundary conditions
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.
scientific article; zbMATH DE number 1452496