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

Source abstract

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