Heat and wave type equations with non-local operators, I. Compact Lie groups

Abstract

We prove existence, uniqueness and give the analytical solution of heat and wave type equations on a compact Lie group $G$ by using a non-local (in time) differential operator and a positive left invariant operator (maybe unbounded) acting on the group. For heat type equations, solutions are given in $L^q(G)$ for data in $L^p(G)$ with $1<p⩽2⩽q<+\infty$. We also provide some asymptotic estimates (large-time behavior) for the solutions. Some examples are given. Also, for wave type equations, we give the solution on some suitable Sobolev spaces over $L^2(G)$. We complement our results, by studying a multi-term heat type equation as well.

arXiv commentThe paper has been accepted for publication in International Mathematics Research Notices IMRN

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Cited by in corpus (1)

• Integro-differential diffusion equations on graded Lie groups