On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains

Abstract:

In this paper we study the first initial-boundary value problem for ${u_t} = \Delta u + {u^p}$ in conical domains $D = (0,\infty ) \times \Omega \subset {R^N}$ where $\Omega \subset {S^{N - 1}}$ is an open connected manifold with boundary. We obtain some extensions of some old results of Fujita, who considered the case $D = {R^N}$. Let $\lambda = - {\gamma _ - }$ where ${\gamma _ - }$ is the negative root of $\gamma (\gamma + N - 2) = {\omega _1}$ and where ${\omega _1}$ is the smallest Dirichlet eigenvalue of the Laplace-Beltrami operator on $\Omega$. We prove: If $1 < p < 1 + 2/(2 + \lambda )$, there are no nontrivial global solutions. If $1 < p < 1 + 2/\lambda$, there are no stationary solutions in $D - \{ 0\}$ except $u \equiv 0$. If $1 + 2/\lambda < p < (N + 1)/(N - 3)$ (if $N > 3$, arbitrary otherwise) there are singular stationary solutions ${u_s}$. If $u(x,0) \leqslant {u_s}(x)$, the solutions are global. If $1 + 2/\lambda < p < (N + 2)/(N - 2)$ and $u(x,0) \leqslant {u_s}$, with $u(x,0) \in C(\overline D )$, the solutions decay to zero. If $1 + 2/N < p$, there are global solutions. For $1 < p < \infty$, there are ${L^\infty }$ data of arbitrarily small norm, decaying exponentially fast at $r = \infty$, for which the solution is not global. We show that if $D$ is the exterior of a bounded region, there are no global, nontrivial, positive solutions if $1 < p < 1 + 2/N$ and that there are such if $p > 1 + 2/N$. We obtain some related results for ${u_t} = \Delta u + |x{|^\sigma }{u^p}$ in the cone.