The Cauchy-Dirichlet problem for singular nonlocal diffusions on bounded domains

Abstract

<p style='text-indent:20px;'>We study the Cauchy-Dirichlet Problem (CDP) for a nonlinear and nonlocal diffusion equation of singular type of the form <inline-formula><tex-math id="M1">\begin{document}$ \partial_t u = - \mathcal{L} u^m $\end{document}</tex-math></inline-formula> posed on a bounded Euclidean domain <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset \mathbb{R}^N $\end{document}</tex-math></inline-formula> with smooth boundary and <inline-formula><tex-math id="M3">\begin{document}$ N\ge 1 $\end{document}</tex-math></inline-formula>. The linear diffusion operator <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{L} $\end{document}</tex-math></inline-formula> is a sub-Markovian operator, allowed to be of nonlocal type, while the nonlinearity is of singular type, namely <inline-formula><tex-math id="M5">\begin{document}$ u^m = |u|^{m-1}u $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M6">\begin{document}$ 0<m<1 $\end{document}</tex-math></inline-formula>. The prototype equation is the Fractional Fast Diffusion Equation (FFDE), when <inline-formula><tex-math id="M7">\begin{document}$ \mathcal{L} $\end{document}</tex-math></inline-formula> is one of the three possible Dirichlet Fractional Laplacians on <inline-formula><tex-math id="M8">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We provide a complete basic theory for solutions to (CDP): existence and uniqueness in the biggest class of data known so far, both for nonnegative and signed solutions; sharp smoothing estimates: classical <inline-formula><tex-math id="M9">\begin{document}$ L^p-L^\infty $\end{document}</tex-math></inline-formula> smoothing effects, and new weighted estimates, which represent a novelty also in local case, i.e. <inline-formula><tex-math id="M10">\begin{document}$ u_t = \Delta u^m $\end{document}</tex-math></inline-formula>. We compare two strategies to prove smoothing effects: Moser iteration VS Green function method.</p><p style='text-indent:20px;'>Due to the singular nonlinearity and to presence of nonlocal diffusion operators, the question of how solutions satisfy the lateral boundary conditions is delicate and we answer it by quantitative upper boundary estimates.</p><p style='text-indent:20px;'>Finally, we show that solutions extinguish in finite time and we provide upper and lower estimates for the extinction time, together with explicit sharp extinction rates in different norms.</p>