Time-Fractional Porous Medium Type Equations. Sharp Time Decay and Regularization

Abstract

We consider a class of porous medium type of equations with Caputo time derivative. The prototype problem reads as $\Dc u=-\A u^m$ and is posed on a bounded Euclidean domain $\Omega\subset\mathbb{R}^N$ with zero Dirichlet boundary conditions. The operator $\A$ falls within a wide class of either local or nonlocal operators, and the nonlinearity is allowed to be of degenerate or singular type, namely, $0<m<1$ and $m>1$. This equation is the most general form of a variety of models used to describe anomalous diffusion processes with memory effects, and finds application in various fields, including visco-elastic materials, signal processing, biological systems and geophysical science. We show existence of unique solution and new $L^p-L^\infty$ smoothing effects. The comparison principle, which we provide in the most general setting, serves as a crucial tool in the proof and provides a novel monotonicity formula. Consequently, we establish that the regularizing effects from the diffusion are stronger than the memory effects introduced by the fractional time derivative. Moreover, the solution attains the boundary conditions pointwise. Finally, we prove that the solution does not vanish in finite time if $0<m<1$, unlike the case with the classical time derivative. Indeed, we provide a sharp rate of decay for any $L^p$-norm of the solution for any $m>0$. Our findings indicate that memory effects weaken the spatial diffusion and mitigate the difference between slow and fast diffusion.