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We study a quite general family of nonlinear evolution equations of diffusive type with nonlocal effects. More precisely, we study porous medium equations with a fractional Laplacian pressure, and the problem is posed on a bounded space…

In this paper we prove a higher differentiability result for the solutions to a class of obstacle problems in the form \begin{equation*} \label{obst-def0} \min\left\{\int_\Omega F(x,Dw) dx : w\in \mathcal{K}_{\psi}(\Omega)\right\}…

This paper is concerned with the problem of scattering of time-harmonic acoustic waves from an impenetrable obstacle in a piecewise homogeneous medium. The well-posedness of the direct problem is established, employing the integral equation…

This is a study of a class of nonlocal nonlinear diffusion equations. We present a strong maximum principle for nonlocal time-dependent Dirichlet problems. Results are for bounded functions of space, rather than (semi)-continuous functions.…

We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion, $$ \{ll} \dfrac{\partial u}{\partial t} + (-\Delta)^{\sigma/2} (|u|^{m-1}u)=0, & \qquad x\in\mathbb{R}^N,\; t>0, [8pt]…

We show the existence of self-similar solutions with constant finite mass to the time-fractional Porous-Medium Equation for all spatial dimensions $d \ge 1$ and all exponents $m>m_c=(d-2)_+/d$. This range is optimal. We find two types of…

We show local higher integrability of derivative of a suitable weak solution to the surface growth model, provided a scale-invariant quantity is locally bounded. If additionally our scale-invariant quantity is small, we prove local…

We formulate a numerical method to solve the porous medium type equation with fractional diffusion \[ \frac{\partial u}{\partial t}+(-\Delta)^{\sigma/2} (u^m)=0 \] posed for $x\in \mathbb{R}^N$, $t>0$, with $m\geq 1$, $\sigma \in (0,2)$,…

In the paper, we derive Li-Yau gradient estimates and Souplet Zhang type estimates of the following equation \begin{equation*} \begin{split} u_t= \Delta_\xi p+\lambda u+A(u) , \end{split} \end{equation*} on complete noncompact metric…

In this paper we study a singularly modified version of the incompressible porous media equation. We investigate the implications for the local well-posedness of the equations by modifying, with a fractional derivative, the constitutive…

We study a fractional diffusion problem in the divergence form in one space dimension. We define a notion of the viscosity solution. We prove existence of viscosity solutions to the fractional diffusion problem with the Dirichlet boundary…

We prove sharp estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations on a bounded domain subject to a homogeneous Dirichlet boundary condition. Important special cases are the…

We prove optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains. This solves a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical…

This paper considers the existence of local and global-in-time strong solutions to the advection-diffusion equation with variable coefficients on an evolving surface with a boundary. We apply both the maximal $L^p$-in-time regularity for…

The problem of diffusion in a porous medium with a spatially varying porosity is considered. The particular microstructure analyzed comprises a collection of impenetrable spheres, though the methods developed are general. Two different…

This work extends Perron's method for the porous medium equation in the slow diffusion case. The main result shows that nonnegative continuous boundary functions are resolutive in a general cylindrical domain.

We construct non-negative weak solutions of fast diffusion equations with a divergence type of drift term satisfying the $L^q$-energy inequality and speed estimate in Wasserstein spaces under some integrability conditions on the drift term.…

We investigate qualitative properties of local solutions $u(t,x)\ge 0$ to the fast diffusion equation, $\partial_t u =\Delta (u^m)/m$ with $m<1$, corresponding to general nonnegative initial data. Our main results are quantitative…

Existence of specific \emph{eternal solutions} in exponential self-similar form to the following quasilinear diffusion equation with strong absorption$$\partial_t u=\Delta u^m-|x|^{\sigma}u^q,$$posed for…

We describe acceleration of the front propagation for solutions to a class of monostable nonlinear equations with a nonlocal diffusion in $\mathbb{R}^d$, $d\geq1$. We show that the acceleration takes place if either the diffusion kernel or…