The Fujita exponent for the Cauchy problem in the hyperbolic space

It is well known that the heat kernel in the hyperbolic space has a different behavior for large times than the one in the Euclidean space. The main purpose of this paper is to study its effect on the positive solutions of Cauchy problems with power nonlinearities. Existence and non-existence result...

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Bibliographic Details
Published in:Journal of Differential Equations 2011-10, Vol.251 (8), p.2143-2163
Main Authors: Bandle, Catherine, Pozio, Maria Assunta, Tesei, Alberto
Format: Article
Language:English
Subjects:
Blow-up
Existence and non-existence of solutions
Fujita exponent
Hyperbolic space
Semilinear parabolic equations
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Summary:It is well known that the heat kernel in the hyperbolic space has a different behavior for large times than the one in the Euclidean space. The main purpose of this paper is to study its effect on the positive solutions of Cauchy problems with power nonlinearities. Existence and non-existence results for local solutions are derived. Emphasis is put on their long time behavior and on Fujitaʼs phenomenon. To have the same situation as for the Cauchy problem in R N , namely finite time blow up for all solutions if the exponent is smaller than a critical value and existence of global solutions only for powers above the critical exponent, we must introduce a weight depending exponentially on the time. In this respect the situation is similar to problems in bounded domains with Dirichlet boundary conditions. Important tools are estimates for the heat kernel in the hyperbolic space and comparison principles.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2011.06.001