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The well-posedness of a generalized Coleman--Gurtin equation equipped with dynamic boundary conditions with memory was recently established by the author with C.G. Gal. In this article we report advances concerning the asymptotic behavior…

In this paper, we consider the asymptotic behavior of weak solutions for non-autonomous diffusion equations with delay in time-dependent spaces when the nonlinear function $f$ is critical growth, the delay term $g(t, u_t)$ contains some…

In this paper, we derive optimal upper and lower bounds on the dimension of the attractor AW for scalar reaction-diffusion equations with a Wentzell (dynamic) boundary condition. We are also interested in obtaining explicit bounds about the…

This paper proposes an abstract theory concerned with dynamical systems generated by doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations in a reflexive Banach space setting. In order to…

We consider the initial value problem for the semilinear plate equation with nonlocal nonlinearity. We prove the existence of global attractor and then establish the regularity and finite dimensionality of this attractor.

We consider a linear implicit-explicit (IMEX) time discretization of the Cahn-Hilliard equation with a source term, endowed with Dirichlet boundary conditions. For every time step small enough, we build an exponential attractor of the…

We consider a family of non-autonomous reaction-diffusion equations with almost periodic, rapidly oscillating principal part and nonlinear interactions. As the frequency of the oscillations tends to infinity, we prove that the solutions of…

This paper presents estimates of the convergence of asymptotic dynamics of reaction-diffusion equations with nonlinear boundary conditions. We show how the convergence of the global attractors can be affected by the variations of diffusion…

In this paper dynamic von Karman equations with localized interior damping supported in a boundary collar are considered. Hadamard well-posedness for von Karman plates with various types of nonlinear damping are well-known, and the…

Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated by the semilinear reaction diffusion equation u_t+\beta(x)u-\Delta u&=f(x,u),&&(t,x)\in[0,+\infty[\times\Omega,…

We study the long-time behavior of solutions of the one dimensional wave equation with nonlinear damping coefficient. We prove that if the damping coefficient function is strictly positive near the origin then this equation possesses a…

This paper is concerned with long-time dynamics of semilinear wave equations defined on bounded domains of $\mathbb{R}^3$ with cubic nonlinear terms and locally distributed damping. The existence of regular finite-dimensional global…

A non--linear diffusion equation is derived by taking into account hopping rates depending on the occupation of next neighbouring sites. There appears additonal repulsive and attractive forces leading to a changed local mobiltiy. The…

This paper is concerned with the asymptotic behavior of solutions of the two-dimensional Navier-Stokes equations with both non-autonomous deterministic and stochastic terms defined on unbounded domains. We first introduce a continuous…

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…

We consider a diffusion on a bounded domain, assuming that the system is irreducible inside the domain and that the diffusion has varying degree of degeneracy on the domain's boundary. The long-term statistical properties of typical…

We deal with a class of parabolic nonlinear evolution equations with state-dependent delay. This class covers several important PDE models arising in biology. We first prove well-posedness in a certain space of functions which are Lipschitz…

This work is devoted to the study of the asymptotic behavior of nonautonomous reaction-diffusion equations in Dumbbell domains $\Omega_{\varepsilon} \subset \mathbb{R}^{N}$. Each $\Omega_{\varepsilon}$ is the union of a fixed open set…

The global asymptotic behavior of a stochastic Hopfield neural network model (HNNM) with delays is explored by studying the existence and structure of random attractors. It is first proved that the trajectory field of the stochastic delayed…

We prove the existence and uniqueness of tempered random attractors for stochastic Reaction-Diffusion equations on unbounded domains with multiplicative noise and deterministic non-autonomous forcing. We establish the periodicity of the…