Related papers: The Dirichlet problem for some nonlocal diffusion …
In this article, we are interested in the Dirichlet problem for parabolic viscous Hamilton-Jacobi Equations. It is well-known that the gradient of the solution may blow up in finite time on the boundary of the domain, preventing a classical…
This paper provides sharp quantitative and constructive estimates of nonnegative solutions $u(t,x)\geq 0$ to the nonlinear fractional diffusion equation, $$\partial_t u +{\mathcal L} F(u)=0,$$ also known as filtration equation, posed in a…
In this paper we obtain interior regularity estimates for viscosity solutions of nonlocal Dirichlet problems that degenerate when the gradient of the solution vanishes. Interior H\"older estimates are obtained when the order of the…
We study the long-time behavior of the unique viscosity solution $u$ of the viscous Hamilton-Jacobi Equation $u_t-\Delta u + |Du|^m = f\hbox{in }\Omega\times (0,+\infty)$ with inhomogeneous Dirichlet boundary conditions, where $\Omega$ is a…
We study the fully nonlinear heat equation $b(\partial_tu)\partial_tu=\Delta u$ posed in a bounded domain with Dirichlet boundary conditions. Here $b(s)=b^-$ if $s<0$, $b(s)=b^+$ if $s>0$, $b^-\neq b^+$ being two positive constants. This…
We study well-posedness of degenerate mixed-type parabolic-hyperbolic equations $$ \partial_tu+\text{div}\big(f(u)\big)=\mathcal{L}[b(u)] $$ on bounded domains with general Dirichlet boundary/exterior conditions. The nonlocal diffusion…
We investigate bubbling solutions for the nonlocal equation \[ A_\Omega^s u =u^p,\ u >0 \quad \mbox{in } \Omega, \] under homogeneous Dirichlet conditions, where $\Omega$ is a bounded and smooth domain. The operator $A_\Omega^s$ stands for…
One-dimensional free boundary problem for a nonlinear diffusion - convection equation with a Dirichlet condition at fixed face $x=0$, variable in time, is considered. Throught several transformations the problem is reduced to a free…
We study the solutions to a nonlocal 1-Laplacian equation given by $$ 2\text{P.V.}\int_{\mathbb{R}^N}\frac{u(x)-u(y)}{|u(x)-u(y)|} \frac{dy}{|x-y|^{N+s}}=f(x) \quad \textmd{for } x\in \Omega, $$ with Dirichlet boundary condition $u(x)=0$ in…
We establish the interior $C^{1,\alpha}$-estimate for viscosity solutions of degenerate/singular fully nonlinear parabolic equations $$u_t = |Du|^{\gamma}F(D^2u) + f.$$ For this purpose, we prove the well-posedness of the regularized…
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 study two identification problems in relation with a strongly degenerate parabolic diffusion equation characterized by a vanishing diffusion coefficient $u\in W^{1,\infty},$ with the property $\frac{1}{u}\notin L^{1}. $ The aim is to…
We consider the partial differential equation $$ u-f={\rm div}\left(u^m\frac{\nabla u}{|\nabla u|}\right) $$ with $f$ nonnegative and bounded and $m\in\mathbb{R}$. We prove existence and uniqueness of solutions for both the Dirichlet…
The paper deals with the asymptotic behavior of solutions to a non-local diffusion equation, $u_t=J*u-u:=Lu$, in an exterior domain, $\Omega$, which excludes one or several holes, and with zero Dirichlet data on…
We study whether the solutions of a parabolic equation with diffusion given by the fractional Laplacian and a dominating gradient term satisfy Dirichlet boundary data in the classical sense or in the generalized sense of viscosity…
We present a numerical approximation method for linear diffusion-reaction problems with possibly discontinuous Dirichlet boundary conditions. The solution of such problems can be represented as a linear combination of explicitly known…
In this paper, by variational and topological arguments based on linking and $\nabla$-theorems, we prove the existence of multiple solutions for the following nonlocal problem with mixed Dirichlet-Neumann boundary data, $$ \left\{…
We study the viscosity solutions to the first eigenvalue equation. We consider $\Omega$ a bounded B-regular domain in $\mathbb{C}^n$ and we prove that the Dirichlet problem $\Lambda_{1}(D_{\mathbb{C}}^2 u)=f$ in $\Omega$ and $u=\varphi$ on…
We use a diffuse interface method for solving Poisson's equation with a Dirichlet condition on an embedded curved interface. The resulting diffuse interface problem is identified as a standard Dirichlet problem on approximating regular…
We study the boundary regularity of solutions to the porous medium equation $u_t = \Delta u^m$ in the degenerate range $m>1$. In particular, we show that in cylinders the Dirichlet problem with positive continuous boundary data on the…