Related papers: Upper semicontinuity for a class of nonlocal evolu…
In this work we consider the nonlocal evolution equation $$ \frac{\partial u(w,t)}{\partial t}=-u(w,t)+ \int_{S^{1}}J(wz^{-1})f(u(z,t))dz+ h, \,\,\, h > 0 $$ which arises in models of neuronal activity, in $L^{2}(S^{1})$, where $S^{1}$…
In this work we consider the non local evolution equation with time-dependent terms which arises in models of phase separation in $\mathbb{R}^N$ \[ \partial_t u=- u + g \left(\beta(J*u) +\beta h(t,u)\right) \] under some restrictions on…
In this paper we consider the non local non autonomous evolution problem \[ \begin{cases} \partial_t u =- u + g \left(\beta(t)(Ku) \right)\ \ \mbox{in}\ \ \Omega,\\ u = 0\ \ \mbox{in}\ \ \mathbb{R}^N\backslash\Omega, \end{cases} \] where…
The paper investigates the existence and upper semicontinuity of uniform attractors of the perturbed non-autonomous Kirchhoff wave equations with strong damping and supercritical nonlinearity: $u_{tt}-\Delta u_{t}-(1+\epsilon\|\nabla…
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[;\\…
In this paper we study several aspects related with solutions of nonlocal problems whose prototype is $$ u_t =\displaystyle \int_{\mathbb{R}^N} J(x-y) \big( u(y,t) -u(x,t) \big) \mathcal G\big( u(y,t) -u(x,t) \big) dy \qquad \mbox{ in } \,…
We consider a family of semilinear parabolic problems with nonlinear boundary conditions \[ \left\{ \begin{aligned} u_t(x,t) &=\Delta u(x,t) -au(x,t) + f(u(x,t)),\ x \in \Omega_\epsilon \mbox{ and } t>0\,,\\ \displaystyle\frac{\partial…
The aim of this paper is to study the robustness of the family of pullback attractors associated to a non-autonomous coupled system of strongly damped wave equations, given by the following evolution system $$\left\{ \begin{array}{lr}…
The paper investigates the existence of global attractors and their upper semicontinuity for a structural damped wave equation on $\mathbb{R}^{N}: u_{tt}-\Delta u+(-\Delta)^\alpha u_{t}+u_{t}+u+g(u)=f(x)$, where $\alpha\in (1/2, 1)$ is…
Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated by the semilinear damped wave equation u_{tt}+\alpha u_t+\beta(x)u-\Deltau = f(x,u), (t,x)\in[0,+\infty[\times\Omega, u = 0,…
The aim of this work is to examine the upper-semicontinuity properties of the family of global attractors admitted by a non-isothermal viscous relaxation of some nonlocal Cahn-Hilliard equations. We prove that the family of global…
In this work we analyze the behavior of the solutions to nonlocal evolution equations of the form $u_t(x,t) = \int J(x-y) u(y,t) \, dy - h_\epsilon(x) u(x,t) + f(x,u(x,t))$ with $x$ in a perturbed domain $\Omega^\epsilon \subset \Omega$…
If the semigroup is slowly non-dissipative, i.e., its solutions can diverge to infinity as time tends to infinity, one still can study its dynamics via the approach by the unbounded attractors - the counterpart of the classical notion of…
This paper is devoted to initial-boundary value problem of an extensible beam equation with degenerate nonlocal energy damping in $\Omega\subset\mathbb{R}^n$: $u_{tt}-\kappa\Delta u+\Delta^2u-\gamma(\Vert \Delta u\Vert^2+\Vert…
In this paper, we prove the existence of a compact global attractor for the flow generated by equation $$ \frac{\partial u}{\partial t}(x,t)+u(x,t)= \int_{\mathbb{R}^{N}}J(x-y)(f( u(y,t))dy+ h, \quad h > 0, \quad x\in \mathbb{R}^{N}, \quad…
We consider evolution equations of the form \begin{equation*}\label{Abstract equation} \dot u(t)+ A(t)u(t)=0,\ \ t\in[0,T],\ \ u(0)=u_0, \end{equation*} where $A(t),\ t\in [0,T],$ are associated with a non-autonomous sesquilinear form…
We address, in a three-dimensional spatial setting, both the viscous and the standard Cahn-Hilliard equation with a nonconstant mobility coefficient. As it was shown in J.W. Barrett and J.W. Blowey, Math. Comp., 68 (1999), 487-517, one…
Under consideration is the damped semilinear wave equation \[ u_{tt}+u_t-\Delta u+u+f(u)=0 \] in a bounded domain $\Omega$ in $\mathbb{R}^3$ subject to an acoustic boundary condition with a singular perturbation, which we term "massless…
This work investigates the semilinear wave equation featuring the displacement dependent term $\sigma(u)\partial_t u $ and nonlinearity $f(u)$. By developing refined space-time a priori estimates under extended ranges of the nonlinearity…
We carry out an analysis of the existence of solutions for a class of nonlinear partial differential equations of parabolic type. The equation is associated to a nonlocal initial condition, written in general form which includes, as…