A reaction-diffusion problem with an obstacle potential is considered in a bounded domain of $\R^N$. Under the assumption that the obstacle $\K$ is a closed convex and bounded subset of $\mathbb{R}^n$ with smooth boundary or it is a closed $n$-dimensional simplex, we prove that the long-time behavior of the solution semigroup associated with this problem can be described in terms of an exponential attractor. In particular, the latter means that the fractal dimension of the associated global attractor is also finite.
In this paper, a standard about the existence and upper semi-continuity of pullback attractors in the non-initial space is established for some classes of non-autonomous SPDE. This pullback attractor, which is the omega-limit set of the absorbing set constructed in the initial space, is completely determined by the asymptotic compactness of solutions in both the initial and non-initial spaces. As applications, the existences and upper semi-continuity of pullback attractors in $H^1(\mathbb{R}^N)$ are proved for stochastic non-autonomous reaction-diffusion equation driven by a multiplicative noi…
This work is devoted to the study of the asymptotic behavior of nonautonomous reaction-diffusion equations in Dumbbell domains $Ω_{\varepsilon} \subset \mathbb{R}^{N}$. Each $Ω_{\varepsilon}$ is the union of a fixed open set $Ω$ and a channel $R_{\varepsilon}$ that collapses to a line segment $R_0$ as $\varepsilon \rightarrow 0^{+}$. We first establish the global existence of solution for each problem by using two properties of the parabolic equation considered, which are the positivity of the solutions and comparison results for them. We prove the existence of pullback and uniform attractors …
We establish the local input-to-state stability of a large class of disturbed nonlinear reaction-diffusion equations w.r.t. the global attractor of the respective undisturbed system.
On the robustness of pullback attractors for a nonlocal reaction-diffusion equation under perturbation
Centro de Investigación Operativa, Universidad Miguel Hernández de Elche（埃尔切米格尔·埃尔南德斯大学运营研究中心）；Universidad Miguel Hernández de Elche（埃尔切米格尔·埃尔南德斯大学）；Departamento de Ecuaciones Diferenciales y Análisis Numérico Universidad de Sevilla（塞维利亚大学微分方程与数值分析系）；Universidad de Sevilla（塞维利亚大学）
In this paper we study the structure of the global attractor for a reaction- di§usion equation in which uniqueness of the Cauchy problem is not guarantied. We prove that the global attractor can be characterized using either the unstable manifold of the set of stationary points or the stable one but considering in this last case only solutions in the set of bounded complete trajectories.
We prove the existence of a compact L^2-H^1 attractor for a reaction-diffusion equation in R^n. This improves a previous result of B. Wang concerning the existence of a compact L^2-L^2 attractor for the same equation.
We prove existence of global attractors for parabolic equations of the form $$u_t+β(x)u-\sum_{ij}\partial_i(a_{ij}(x)\partial_j u)=f(x,u)$$ with Dirichlet boundary condition on an arbitrary unbounded domain $Ω$ in $\R^3$, without smoothness assumptions on $a_{ij}(\cdot)$ and $\partialΩ$.
We study the long time behavior of solutions of the non-autonomous Reaction-Diffusion equation defined on the entire space R^n when external terms are unbounded in a phase space. The existence of a pullback global attractor for the equation is established in L^2(R^n) and H^1(R^n), respectively. The pullback asymptotic compactness of solutions is proved by using uniform a priori estimates on the tails of solutions outside bounded domains.
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 the non-autonomous equations converge to the solutions of the autonomous averaged equation. If the nonlinearity is dissipative, we prove existence of compact attractors and their upper-semicontinuity at infinity.
We consider a nonlinear reaction-diffusion equation settled on the whole euclidean space. We prove the well-posedness of the corresponding Cauchy problem in a general functional setting, namely, when the initial datum is uniformly locally bounded in L^2. Then we adapt the short trajectory method to establish the existence of the global attractor and, if the space dimension is at most 3, we also find an upper bound of its Kolmogorov's entropy.
In [Adv. Math., 267(2014), 277-306], Cheskidov and Lu develop a new framework of the evolutionary system that deals directly with the notion of a uniform global attractor due to Haraux, and by which a trajectory attractor is able to be defined for the original system under consideration. The notion of a trajectory attractor was previously established for a system without uniqueness by considering a family of auxiliary systems including the original one. In this paper, we further prove the existence of a notion of a strongly compact strong trajectory attractor if the system is asymptotically co…
Global existence of strong solutions and the existence of global and atrractors are established for generalized Shigesada-Kawasaki-Teramoto models on planar domains. The cross diffusion and reaction can have polynomial growth of any order.
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+β(x)u-Δu&=f(x,u),&&(t,x)\in[0,+\infty[\timesΩ, u&=0,&&(t,x)\in[0,+\infty[\times\partialΩ} {equation*} in $H^1_0(Ω)$ has finite Hausdorff dimension. Here $Ω$ is an arbitrary, possibly unbounded, domain in $\R^3$ and $f(x,u)$ is a nonlinearity of subcritical growth. The nonlinearity $f(x,u)$ needs not to satisfy any dissipativeness assumption and the invariant subset $\mathcal I$ needs not to be an an attractor. If $Ω$ is regular, $f…
In this work we consider a dissipative reaction-diffusion equation in a $d$-dimensional thin domain shrinking to a one dimensional segment and obtain good rates for the convergence of the attractors. To accomplish this, we use estimates on the convergence of inertial manifolds as developed previously in \cite{Arrieta-Santamaria-C0} and Shadowing theory.
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 constants involved in our asymptotic estimates, and to compare these bounds to previously known estimates for the dimension of the global attractor AK; K \in {D;N; P}, of reactiondiffusion equations subject to Dirichlet, Neumann and periodic boundary conditions. The explicit estimates we obtain show that the dimension of the global attractor AW is of differ…
The existence of an exponential attractor for the diffusive Hindmarsh-Rose equations on a three-dimensional bounded domain originated in the study of neurodynamics is proved through uniform estimates together with a new theorem on the squeezing property of an abstract reaction-diffusion equation also proved in this paper. The results infer that the global attractor whose existence has been established in [23] for the Hindmarsh-Rose semiflow has a finite fractal dimension.
The existence of a global attractor for the solution semiflow of the extended Brusselator system in the $L^2$ phase space is proved, which is a cubic-autocatalytic and partially reversible reaction-diffusion system with linear coupling between two compartments. The method of grouping and re-scaling estimation is developed to deal with the challenge in proving the absorbing property and the asymptotic compactness of this typical multi-component reaction-diffusion systems. It is also proved that the global attractor is an $(H, E)$ global attractor with the $L^\infty$ regularity and that the Haus…
Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction
Vienna University of Technology, Institute for Analysis and Scientific Computing（维也纳工业大学，分析与科学计算研究所）；University of Vienna, Faculty of Mathematics（维也纳大学，数学学院）
Under consideration is the hyperbolic relaxation of a semilinear reaction-diffusion equation on a bounded domain, subject to a dynamic boundary condition. We also consider the limit parabolic problem with the same dynamic boundary condition. Each problem is well-posed in a suitable phase space where the global weak solutions generate a Lipschitz continuous semiflow which admits a bounded absorbing set. We prove the existence of a family of global attractors of optimal regularity. After fitting both problems into a common framework, a proof of the upper-semicontinuity of the family of global at…
We consider a 2D system that models the nematic liquid crystal flow through the Navier--Stokes equations suitably coupled with a transport-reaction-diffusion equation for the averaged molecular orientations. This system has been proposed as a reasonable approximation of the well-known Ericksen--Leslie system. Taking advantage of previous well-posedness results and proving suitable dissipative estimates, here we show that the system endowed with periodic boundary conditions is a dissipative dynamical system with a smooth global attractor of finite fractal dimension.
We analyze the dynamics of the flow generated by a nonlinear parabolic problem when some reaction and potential terms are concentrated in a neighborhood of the boundary. We assume that this neighborhood shrinks to the boundary as a parameter εgoes to zero. Also, we suppose that the "inner boundary" of this neighborhood presents a highly oscillatory behavior. Our main goal here is to show the continuity of the family of attractors with respect to ε. Indeed, we prove upper semicontinuity under the usual properties of regularity and dissipativeness and, assuming hyperbolicity of the equilibria, w…
A method is proposed to deal with some multivalued semiflows with weak continuity properties. An application to the reaction-diffusion problems with nonmonotone multivalued semilinear boundary condition and nonmonotone multivalued semilinear source term is presented.
The paper is devoted to constructing a random exponential attractor for some classes of stochastic PDE's. We first prove the existence of an exponential attractor for abstract random dynamical systems and study its dependence on a parameter and then apply these results to a nonlinear reaction-diffusion system with a random perturbation. We show, in particular, that the attractors can be constructed in such a way that the symmetric distance between the attractors for stochastic and deterministic problems goes to zero with the amplitude of the random perturbation.
A horizontal $N$-dimensional plane, having a diffusion of its own, exchanges with the lower half space. There, a reaction-diffusion process, modelled by a free boundary problem, takes place. We wish to understand whether, and how, the free boundary meets the plane. The origin of this problem is a two-dimensional reaction diffusion model proposed some time ago by the second author, in collaboration with H. Berestycki and L. Rossi, to model how biological invasions can be enhanced by a line of fast diffusion. Some counter-intuitive numerical simulations of this model, due to A.-C. Coulon, have b…
We study the global existence and uniform-in-time bounds of classical solutions in all dimensions to reaction-diffusion systems dissipating mass. By utilizing the duality method and the regularization of the heat operator, we show that if the diffusion coefficients are close to each other, or if the diffusion coefficients are large enough compared to initial data, then the local classical solution exists globally and is bounded uniformly in time. Applications of the results include the validity of the Global Attractor Conjecture for complex balanced reaction systems with large diffusion.
In this work the existence of a global attractor for the solution semiflow of the coupled two-cell Brusselator model equations is proved. A grouping estimation method and a new decomposition approach are introduced to deal with the challenge in proving the absorbing property and the asymptotic compactness of this type of four-variable reaction-diffusion systems with cubic autocatalytic nonlinearity and with linear coupling. It is also proved that the Hausdorff dimension and the fractal dimension of the global attractor are finite.
This paper is concerned with a fully nonlinear variant of the Allen-Cahn equation with strong irreversibility, where each solution is constrained to be non-decreasing in time. Main purposes of the paper are to prove the well-posedness, smoothing effect and comparison principle, to provide an equivalent reformulation of the equation as a parabolic obstacle problem and to reveal long-time behaviors of solutions. More precisely, by deriving \emph{partial} energy-dissipation estimates, a global attractor is constructed in a metric setting, and it is also proved that each solution $u(x,t)$ converge…
Uniform $L^{\infty}$-Boundedness of Global Attractors for Reaction-Diffusion Equations with Neumann boundary condition in Uniformly Perturbed Non-Smooth Domains
Instituto de Matemática e Estatística, Universidade de São Paulo（圣保罗大学数学与统计学院）
In this paper we consider a mixed boundary value problem with a nonhomogeneous, nonlinear differential operator (called double phase operator), a nonlinear convection term (a reaction term depending on the gradient), three multivalued terms and an implicit obstacle constraint. Under very general assumptions on the data, we prove that the solution set of such implicit obstacle problem is nonempty (so there is at least one solution) and weakly compact. The proof of our main result uses the Kakutani-Ky Fan fixed point theorem for multivalued operators along with the theory of nonsmooth analysis a…
We study a reaction-diffusion-convection problem with nonlinear drift posed in a domain with periodically arranged obstacles. The non-linearity in the drift is linked to the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. Because of the imposed large drift scaling, this nonlinearity is expected to explode in the limit of a vanishing scaling parameter. As main working techniques, we employ two-scale formal homogenization asymptotics with drift to derive the corresponding upscaled model e…
This paper is concerned with qualitative properties of solutions to nonlocal reaction-diffusion equations of the form$$ \int\_{\mathbb{R}^N\setminus K} J(x-y)\,\big( u(y)-u(x) \big)\,\D y+f(u(x))=0, \quad x\in\R^N\setminus K,$$set in a perforated open set $\mathbb{R}^N\setminus K$, where $K\subset\mathbb{R}^N$ is a bounded compact "obstacle" and $f$ is a bistable nonlinearity. When $K$ is convex, we prove some Liouville-type results for solutions satisfying some asymptotic limiting conditions at infinity. We also establish a robustness result, assuming slightly relaxed conditions on $K$.
We solve and characterize the Lagrange multipliers of a reaction-diffusion system in the Gibbs simplex of R^{N+1} by considering strong solutions of a system of parabolic variational inequalities in R^N. Exploring properties of the two obstacles evolution problem, we obtain and approximate a N-system involving the characteristic functions of the saturated and/or degenerated phases in the nonlinear reaction terms. We also show continuous dependence results and we establish sufficient conditions of non-degeneracy for the stability of those phase subregions.
We give a comprehensive study of the analytic properties and long-time behavior of solutions of a reaction-diffusion system in a bounded domain in the case where the nonlinearity satisfies the standard monotonicity assumption. We pay the main attention to the supercritical case, where the nonlinearity is not subordinated to the linear part of the equation trying to put as small as possible amount of extra restrictions on this nonlinearity. The properties of such systems in the supercritical case may be very different in comparison with the standard case of subordinated nonlinearities. We exami…
We consider networks of reaction-diffusion systems of Hodgkin-Huxley type. We give a general mathematical framework, in which we prove existence and unicity of solutions as well as existence of invariant regions and of the attractor. Then, we illustrate some relevant numerical examples and exhibit bifurcation phenomena and propagation of bursting oscillations through one and two coupled systems.
The existence of a global attractor for wave equations in unbounded domains is a challenging problem due to the non-compactness of the Sobolev embeddings. To overcome this difficulty, some authors have worked with weighted Sobolev spaces which restrict the choice of the initial data. Using the "tail estimation method" introduced by B. Wang for reaction diffusion equations, we establish in this paper the existence of a global attractor for two wave equations in the traditional Hilbert spaces $\displaystyle H^1(Ω)\times L^2(Ω)$ where $Ω$ is an unbounded domain of $\R^N$. The first equation, with…
We investigate the long term behavior in terms of finite dimensional global and exponential attractors, as time goes to infinity, of solutions to a semilinear reaction-diffusion equation on non-smooth domains subject to nonlocal Robin boundary conditions, characterized by the presence of fractional diffusion on the boundary. Our results are of general character and apply to a large class of irregular domains, including domains whose boundary is Holder continuous and domains which have fractal-like geometry. In addition to recovering most of the existing results on existence, regularity, unique…
This note is a synthesis of my reflexions on some questions that have emerged during the MATRIX event "Recent Trends on Nonlinear PDEs of Elliptic and Parabolic Type" concerning the qualitative properties of solutions to some non local reaction-diffusion equations of the form L[u](x) + f (u(x)) = 0, for x $\in$ R n \ K, where K $\subset$ R N is a bounded smooth compact "obstacle", L is non local operator and f is a bistable nonlinearity. When K is convex and the nonlocal operator L is a continuous operator of convolution type then some Liouville-type results for solutions satisfying some asymp…
