Blow-up of solutions of nonlinear degenerate parabolic equations
The authors consider the global existence of solutions of the parabolic initial-boundary value problem \[ u_ t=u^ 2(\Delta u+u),\quad (x\in \Omega,\quad t>0);\quad u=0,\quad (x\in \partial \Omega,\quad t>0);\quad u(x,0)=\phi (x)\geq 0;\quad (x\in \Omega) \] where \(\Omega\) is a bounded domain in \({\mathbb{R}}^ n\). They prove rigorous results concerning the blow-up set and in particular, prove that the blow-up set is \(\{\) x: - \(\pi\) /2\(\leq x\leq \pi /2\}\) in the case n-1 and \(\phi\) even. The techniques employed extend to a wide variety of nonlinear parabolic equations.
parabolic initial-boundary value problem
scientific article; zbMATH DE number 4004752