Integro-differential diffusion equations on graded Lie groups

We first study the existence, uniqueness and well-posedness of a general class of integro-differential diffusion equation on $L^p(\mathbb{G})$ $(1<p<+\infty$, $\mathbb{G}$ is a graded Lie group). We show the explicit solution of the considered equation by using the Fourier analysis of the group. The equation involves a nonlocal in time operator (with a general kernel) and a positive Rockland operator acting on $\mathbb{G}.$ Also, we provide $L^p(\mathbb{G})-L^q(\mathbb{G})$ $(1<p\leqslant 2\leqslant q<+\infty)$ norm estimates and time decay rate for the solutions. In fact, by using some contemporary results, one can translate the latter regularity problem to the study of boundedness of its propagator which strongly depends on the traces of the spectral projections of the Rockland operator. Moreover, in many cases, we can obtain time asymptotic decay for the solutions which depends intrinsically on the considered kernel. As a complement, we give some norm estimates for the solutions in terms of a homogeneous Sobolev space in $L^{2}(\mathbb{G})$ that involves the Rockland operator. We also give a counterpart of our results in the setting of compact Lie groups. Illustrative examples are also given.