Related papers: Nonlocal and nonlinear evolution equations in perf…
We consider the inviscid Leray-$\alpha$ equations - an inviscid nonlocal regularisation of the Euler equations. In the first part, we prove the convergence of strong solutions of the Leray-$\alpha$ equations to strong solutions of the Euler…
We study well-posedness of degenerate mixed-type parabolic-hyperbolic equations $$ \partial_tu+\textrm{div}\big(f(u)\big)=\mathcal{L}[b(u)] $$ on bounded domains with general Dirichlet boundary/exterior conditions. The nonlocal diffusion…
We consider a perforated domain $\Omega(\epsilon)$ of $\mathbb{R}^2$ with a small hole of size $\epsilon$ and we study the behavior of the solution of a mixed Neumann-Robin problem in $\Omega(\epsilon)$ as the size $\epsilon$ of the small…
In this paper, we study the solvability of the nonlinear Dirichlet problem with sum of the operators of independent non standard growths in a bounded domain $\Omega \subset \mathbb{R}^{n}$. We obtain sufficient conditions and show the…
In this paper, we investigate the existence of weak solution for a Kirchhoff type problem driven by a nonlocal operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions {\small$$…
In this paper we study the dependence of the H\"older estimates on the geometry of a domain with holes for the Neumann problem. For this, we study the H\"older regularity of the solutions to the Dirichlet and Neumann problems in the disk…
In the present paper, we consider the parabolic and hyperbolic inequalities with a singular potentials and with a critical nonlinearities in the annulus domain. The problems are studied with Neumann-type and Dirichlet-type boundary…
In this paper we study homogenization of quasi-linear partial differential equations of the form $-\mbox{div}\left( a\left( x,x/\varepsilon _h,Du_h\right) \right) =f_h$ on $\Omega $ with Dirichlet boundary conditions. Here the sequence…
In the first part of this article we deal with the existence of at least three non-trivial weak solutions of a nonlocal problem with nonstandard growth involving a nonlocal Robin type boundary condition. The second part of the article is…
In this work we consider the non local evolution problem \[ \begin{cases} \partial_t u(x,t)=-u(x,t)+g(\beta K(f\circ u)(x,t)+\beta h), ~x \in\Omega, ~t\in[0,\infty[;\\ u(x,t)=0, ~x\in\mathbb{R}^N\setminus\Omega, ~t\in[0,\infty[;\\…
Results on the peridynamics equilibrium and evolution equations over the space of periodic vector-distributions in multi-spatial dimensions are presented. The associated operator considered is the linear state-based peridynamic operator for…
Within the framework of Hilbert spaces, we solve nonlocal problems in bounded domains with prescribed conditions on the complement of the domain. Our main focus is on the inhomogeneous Neumann problem in a rather general setting. We also…
We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form \begin{equation}\label{abeqn} (\partial_t+X\cdot\nabla_Y)u=\nabla_X\cdot(A(\nabla_X u,X,Y,t)). \end{equation} The function…
We are concerned with Dirichlet problems of the form $${\mathop{\rm div}\nolimits} (|D u|^{p-2}Du)+f(u)=0\ \mbox{ in }\Omega,\qquad u=0\ \mbox{ on }\partial\Omega, $$ where $\Omega$ is a bounded domain of $\mathbb{R}^n$, $n\ge 2$, $1<p<n$…
This paper deals with the approximation of non-autonomous evolution equations of the form \begin{equation*}\label{Abstract equation} \dot u(t)+A(t)u(t)=f(t)\ \ t\in[0,T],\ \ u(0)=u_0. \end{equation*} where $A(t),\ t\in [0,T]$ arise from a…
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…
We study the non-autonomously forced Burgers equation $$ u_t(x,t) + u(x,t)u_x(x,t) - u_{xx}(x,t) = f(x,t) $$ on the space interval $(0,1)$ with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we…
We consider a linear non-autonomous evolutionary Cauchy problem \begin{equation} \dot{u} (t)+A(t)u(t)=f(t) \hbox{ for }\ \hbox{a.e. t}\in [0,T],\quad u(0)=u_0, \end{equation} where the operator $A(t)$ arises from a time depending…
We establish the asymptotic behaviour of the least energy solutions of the following nonlocal Neumann problem: \begin{align*} \left\{\begin{array}{l l} { d(-\Delta)^{s}u+ u= \abs{u}^{p-1}u } \text{ in $\Omega,$ } { \mathcal{N}_{s}u=0 }…
We state and prove estimates for the local boundedness of subsolutions of non-local, possibly degenerate, parabolic integro-differential equations of the form \begin{equation*} \partial_tu(x,t)+\mbox{P.V.}\int\limits_{\mathbb R^n}K(x,y,t)…