Blow-up versus global existence of solutions for reaction–diffusion equations on classes of Riemannian manifolds

Abstract

AbstractIt is well known from the work of Bandle et al. (J Differ Equ 251:2143–2163, 2011) that the Fujita phenomenon for reaction–diffusion evolution equations with power nonlinearities does not occur on the hyperbolic space $${\mathbb {H}}^N$$ H N , thus marking a striking difference with the Euclidean situation. We show that, on classes of manifolds in which the bottom $$\Lambda $$ Λ of the $$L^2$$ L 2 spectrum of $$-\Delta $$ - Δ is strictly positive (the hyperbolic space being thus included), a different version of the Fujita phenomenon occurs for other kinds of nonlinearities, in which the role of the critical Fujita exponent in the Euclidean case is taken by $$\Lambda $$ Λ . Such nonlinearities are time-independent, in contrast to the ones studied in Bandle et al. (2011). As a consequence of our results we show that, on a class of manifolds much larger than the case $$M={\mathbb {H}}^N$$ M = H N considered in Bandle et al. (2011), solutions to (1.1) with power nonlinearity $$f(u)=u^p$$ f ( u ) = u p , $$p>1$$ p > 1 , and corresponding to sufficiently small data, are global in time. Though qualitative similarities with similar problems in bounded, Euclidean domains can be seen in the results, the methods are significantly different because of noncompact setting dealt with.