We are concerned with existence, uniqueness and nonuniqueness of nonnegative solutions to the semilinear heat equation in open subsets of the n-dimensional sphere. Existence and uniqueness results are obtained using L p → L q estimates for the semigroup generated by the Laplace-Beltrami operator. Moreover, under proper assumptions on the nonlinear function, we establish nonuniqueness of weak solutions, when n ≥ 3; to do this, we shall prove uniqueness of positive classical solutions and nonuniqueness of singular solutions of the corresponding semilinear elliptic problem.
On well-posedness of the semilinear heat equation on the sphere / F. Punzo. - In: JOURNAL OF EVOLUTION EQUATIONS. - ISSN 1424-3199. - 12:3(2012), pp. 571-592.
On well-posedness of the semilinear heat equation on the sphere
F. Punzo
2012
Abstract
We are concerned with existence, uniqueness and nonuniqueness of nonnegative solutions to the semilinear heat equation in open subsets of the n-dimensional sphere. Existence and uniqueness results are obtained using L p → L q estimates for the semigroup generated by the Laplace-Beltrami operator. Moreover, under proper assumptions on the nonlinear function, we establish nonuniqueness of weak solutions, when n ≥ 3; to do this, we shall prove uniqueness of positive classical solutions and nonuniqueness of singular solutions of the corresponding semilinear elliptic problem.Pubblicazioni consigliate
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