Lᵖ-Lq estimates for non-local heat and wave type equations on locally compact groups

We prove the Lᵖ-Lq (1<p 2 q<+∞) norm estimates for the solutions of heat and wave type equations on a locally compact separable unimodular group G by using an integro-differential operator in time and any positive left invariant operator (maybe unbounded) on G. We complement our studies by giving asymptotic time estimates for the solutions, which in some cases are sharp.