Extinction and non-extinction for a \(p\)-Laplacian equation with nonlinear source
The phenomenon of extinction in finite time is known for the \(p\)-Laplace equation \[ {\partial u\over\partial t}= \text{div}(|\nabla u|^{p-2}\nabla u) \] in the singular case \(1< p< 2\). The authors add the source term \(\lambda u^q\) to the right-hand member; \(\lambda> 0\), \(q> 0\) and \(u\) is nonnegative. The critical exponent for extinction is J\(q= p-1\). The authors show that in the critical case the situation is determined by a nonlinear eigenvalue problem, apparently first studied by \textit{E. H. Lieb} [Invent. Math. 74, 441--448 (1983; Zbl 0538.35058)]. Lemma 2.4 has previously been proved by \textit{P. Lindqvist} [Proc. Am. Math. Soc. 109, No. 1, 157--164 (1990; Zbl 0714.35029)].
evolutionary \(p\)-Laplace equation' nonlinear source term
scientific article; zbMATH DE number 5354992