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We study the positivity and regularity of solutions to the fractional porous medium equations $u_t+(-Delta)^su^m=0$ in $(0,infty)timesOmega$, for $m>1$ and $sin (0,1)$ and with Dirichlet boundary data $u=0$ in $(0,infty)times({mathbb R}^NsetminusOmega)$, and nonnegative initial condition $u(0,cdot)=u_0geq0$. Our first result is a quantitative lower bound for solutions which holds for all positive times $t>0$. As a consequence, we find a global Harnack principle stating that for any $t>0$ solutions are comparable to $d^{s/m}$, where $d$ is the distance to $partialOmega$. This is in sharp contrast with the local case $s=1$, in which the equation has finite speed of propagation. After this, we study the regularity of solutions. We prove that solutions are classical in the interior ($C^infty$ in $x$ and $C^{1,alpha}$ in $t$) and establish a sharp $C^{s/m}_x$ regularity estimate up to the boundary. Our methods are quite general, and can be applied to a wider class of nonlocal parabolic equations of the form $u_t-mathcal L F(u)=0$ in $Omega$, both in bounded or unbounded domains.
We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,mu_X)$ satisfying a $2$-Poincare inequality. Given a bounded doma
We give a detailed study of the infinite-energy solutions of the Cahn-Hilliard equation in the 3D cylindrical domains in uniformly local phase space. In particular, we establish the well-posedness and dissipativity for the case of regular potentials
We prove that Gevrey regularity is propagated by the Boltzmann equation with Maxwellian molecules, with or without angular cut-off. The proof relies on the Wild expansion of the solution to the equation and on the characterization of Gevrey regularity by the Fourier transform.
We study existence of global solutions and finite time blow-up of solutions to the Cauchy problem for the porous medium equation with a variable density $rho(x)$ and a power-like reaction term $rho(x) u^p$ with $p>1$; this is a mathematical model of
We are concerned with nonnegative solutions to the Cauchy problem for the porous medium equation with a variable density $rho(x)$ and a power-like reaction term $u^p$ with $p>1$. The density decays {it fast} at infinity, in the sense that $rho(x)sim