On well-posedness of the semilinear heat equation on the sphere

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?CBeltrami 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.