A Fujita type result for a degenerate Neumann problem in domains with noncompact boundary
The authors consider the following Neumann problem \[ \begin{aligned} (w^{\beta})_{t} -\text{ div } (| Dw| ^{m-1}Dw) =w^{\nu}, &\quad \text{in} Q_{T},\\ | Dw| ^{m-1}Dw\cdot \mathbf{n}=0, &\quad \text{on} \partial \Omega\times (0,T),\\ w^{\beta}(x,0)= w_{0}^{\beta}(x), &\quad \text{in} \Omega,\end{aligned} \] where \(\Omega\subset {\mathbb R}^{N}\), \(N\geq 2\), is a unbounded domain with noncompact boundary \(\partial \Omega\), \(Q_{T}=\Omega\times(0,T)\), \(T\leq \infty\), \(0<\beta\leq 1\), \(m>1\), \(\nu>1\), \(w_{0}(x)\geq 0\) with \(w^{\beta}_{0}\in L^{1}_{loc}(\bar \Omega)\) and \(\mathbf{n}\) is the unit outward normal to \(\partial \Omega\). The authors study existence or nonexistence of global in time solutions and also derive estimates of the finite speed of propagation of the support of \(w\). It is shown how these properties depend on the geometry of \(\Omega\).
scientific article; zbMATH DE number 1274906