Journal
JOURNAL OF DIFFERENTIAL EQUATIONS
Volume 251, Issue 7, Pages 1972-1989Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2011.05.033
Keywords
Semilinear parabolic equations; Semilinear elliptic equations; Laplace-Beltrami operator; Semigroup theory; Singular solutions
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We investigate existence and uniqueness of solutions to semilinear parabolic and elliptic equations in bounded domains of the n-dimensional hyperbolic space (n >= 3). L-p -> L-q estimates for the semigroup generated by the Laplace-Beltrami operator are obtained and then used to prove existence and uniqueness results for parabolic problems. Moreover, under proper assumptions on the nonlinear function, we establish uniqueness of positive classical solutions and nonuniqueness of singular solutions of the elliptic problem; furthermore, for the corresponding semilinear parabolic problem, nonuniqueness of weak solutions is stated. (C) 2011 Elsevier Inc. All rights reserved.
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