  We prove the $L^p-L^q$ $(1<p\leqslant 2\leqslant q<+\infty)$ 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.
