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


Authors: Catherine Bandle and Howard A. Levine
Journal: Trans. Amer. Math. Soc. 316 (1989), 595-622
MSC: Primary 35K57
DOI: https://doi.org/10.1090/S0002-9947-1989-0937878-9
MathSciNet review: 937878
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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 + \vert x{\vert^\sigma }{u^p}$ in the cone.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0937878-9
Keywords: Reaction-diffusion equations, unbounded regions
Article copyright: © Copyright 1989 American Mathematical Society

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