Heat- and Wave-Type Equations with Nonlocal Operators, I. Compact Lie Groups

We prove existence and uniqueness and give the analytical solution of heat and wave type equations on a compact Lie group G by using a nonlocal (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 Lp(G) with 1 < p <= 2 q < +infinity. 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.