Lyapunov functionals for reaction–diffusion equations with memory

We consider a reaction‐diffusion equation in which the usual diffusion term also depends on the past history of the diffusion itself. This equation has been analysed by several authors, with an emphasis on the longtime behaviour of the solutions. In this respect, the first results have been obtained by using the past history approach. They show that the equation, subject to a suitable boundary condition, defines a dissipative dynamical system which possesses a global attractor. A similar theorem has been recently proved by Chepyzhov and Miranville, using a different method based on the notion of trajectory attractors. In addition, those authors provide sufficient conditions that ensure the existence of a Lyapunov functional. Here we show that a similar result can be demonstrated within the past history approach, with less restrictive conditions. Copyright © 2005 John Wiley & Sons, Ltd.     

Random Attractor for the Nonclassical Diffusion Equation with Fading Memory

In this paper,we consider the stochastic nonclassical diffusion equationwith fading memory on a bounded domain. By decomposition of the solution operator, we give the necessary condition of asymptotic smoothness of the solution to the initial boundary value problem, and then we prove the existence of a random attractor in the space $M_1=D(A^{\frac{1}{2}}) × L^2_μ(R^+, D(A^{\frac{1}{2}}))$, where A=-Δ with Dirichlet boundary condition.     

Attractors for the non-autonomous nonclassical diffusion equations with fading memory

The long-time dynamical behavior of the non-autonomous nonclassical diffusion equation with fading memory, when nonlinearity is critical, is discussed for in the weak topological space . First, the asymptotic regularity of solutions is proven, and then the existence of a compact uniform attractor together with its structure and regularity is obtained, while the time-dependent forcing term is only translation bounded instead of translation compact. The result extends and improves some results given in [Y. Xiao, Attractors for a nonclassical diffusion equation, Acta Math. Appl. Sin. Engl. Ser. 18 (2002) 273–276; C. Sun, M. Yang, Dynamics of the nonclassical diffusion equations, Asympt. Anal. 59 (2008) 51–81].     

Global Attractors for a Nonclassical Diffusion Equation

We prove the existence of global attractors in H0^1 （Ω） for a nonclassical diffusion equation. Two types of nonlinearity f are considered： one is the critical exponent, and the other is the polynomial growth of arbitrary order.     

Trajectory attractors for nonclassical diffusion equations with fading memory

In this article, we consider the existence of trajectory and global attractors for nonclassical diffusion equations with linear fading memory. For this purpose, we will apply the method presented by Chepyzhov and Miranville [7,8], in which the authors provide some new ideas in describing the trajectory attractors for evolution equations with memory.     

Global behavior of solutions to the fast diffusion equation with boundary flux governed by memory

We study global existence and blow up in finite time for a one‐dimensional fast diffusion equation with memory boundary condition. The problem arises out of a corresponding model formulated from tumor‐induced angiogenesis. We obtain necessary and sufficient conditions for global existence of solutions to the problem. Copyright © 2016 John Wiley & Sons, Ltd.     

Upper semicontinuity of global attractors for a viscoelastic equations with nonlinear density and memory effects

This paper is devoted to showing the upper semicontinuity of global attractors associated with the family of nonlinear viscoelastic equations in a three‐dimensional space, for f growing up to the critical exponent and dependent on ρ ∈ [0,4), as ρ→0⁺. This equation models extensional vibrations of thin rods with nonlinear material density ?(?_tu) = |?_tu|^ρ and presence of memory effects. This type of problems has been extensively studied by several authors; the existence of a global attractor with optimal regularity for each ρ ∈ [0,4) were established only recently. The proof involves the optimal regularity of the attractors combined with Hausdorff's measure.     

Attractors for the nonclassical diffusion equations with fading memory

We discuss long-time dynamical behavior of the nonclassical diffusion equation with fading memory when nonlinearity is critical. The existence and regularity of global attractors in weak topological space and strong topological space are obtained, while the forcing term only belongs to H⁻¹(Ω) and L²(Ω) respectively. The results in this part are new and appear to be optimal corresponding to the forcing term.     

Identification of the relaxation kernel in diffusion processes and viscoelasticity with memory via deconvolution

We present an algorithm for the identification of the relaxation kernel in the theory of diffusion systems with memory (or of viscoelasticity), which is linear, in the sense that we propose a linear Volterra integral equation of convolution type whose solution is the relaxation kernel. The algorithm is based on the observation of the flux through a part of the boundary of a body. The identification of the relaxation kernel is ill posed, as we should expect from an inverse problem. In fact, we shall see that it is mildly ill posed, precisely as the deconvolution problem which has to be solved in the algorithm we propose. Copyright © 2016 John Wiley & Sons, Ltd.     

Robust exponential attractors for a class of non-autonomous semi-linear second-order evolution equation with memory and critical nonlinearity

In this article, we investigate a class of non-autonomous semi-linear second-order evolution with memory terms, expressed by the convolution integrals, which account for the past history of one or more variables. First, the asymptotic regularity of solutions is proved, while the nonlinearity is critical and the time-dependent external forcing term is assumed to be only translation-bounded (instead of translation-compact), and then the existence of compact uniform attractors together with its structure and regularity is established. Finally, the existence of robust family of exponential attractors is constructed.     

Stability of global and exponential attractors for a three‐dimensional conserved phase‐field system with memory

We consider a conserved phase‐field system on a tri‐dimensional bounded domain. The heat conduction is characterized by memory effects depending on the past history of the (relative) temperature ?, which is represented through a convolution integral whose relaxation kernel k is a summable and decreasing function. Therefore, the system consists of a linear integrodifferential equation for ?, which is coupled with a viscous Cahn–Hilliard type equation governing the order parameter χ. The latter equation contains a nonmonotone nonlinearity ? and the viscosity effects are taken into account by a term ?αΔ?_tχ, for some α?0. Rescaling the kernel k with a relaxation time ε>0, we formulate a Cauchy–Neumann problem depending on ε and α. Assuming a suitable decay of k, we prove the existence of a family of exponential attractors {?_α,ε} for our problem, whose basin of attraction can be extended to the whole phase–space in the viscous case (i.e. when α>0). Moreover, we prove that the symmetric Hausdorff distance of ?_α,ε from a proper lifting of ?_α,0 tends to 0 in an explicitly controlled way, for any fixed α?0. In addition, the upper semicontinuity of the family of global attractors {??_α,ε} as ε→0 is achieved for any fixed α>0. Copyright © 2009 John Wiley & Sons, Ltd.     

The dependence on fractional orders of mild solutions to the fractional diffusion equation with memory

In this paper we investigate the Cauchy problem for a fractional diffusion equation and the time-fractional derivative is taken in the Caputo type sense. We give a representation of solutions under Fourier series and analyze initial value problems for the semi-linear fractional diffusion equation with a memory term. We also discuss the stability of the fractional derivative order for the time under some assumptions on the input data. Our key idea is to use Mittag-Leffler functions, the Banach fixed point theorem, and some Sobolev embeddings.     

记忆型梁方程强解的全局吸引子

记忆型梁方程出现于—般的Kirchhoff粘弹性梁模型中．本文在记忆核满足指数衰退的条件下证明了系统的能量也是指数衰退的．进一步,通过对条件(C)的验证获得了系统强解的全局吸引子．     

OPTIMAL HARVESTING POLICY FOR INSHORE-OFFSHORE FISHERY MODEL WITH IMPULSIVE DIFFUSION

This article studies the inshore-offshore fishery model with impulsive diffusion. The existence and global asymptotic stability of both the trivial periodic solution and the positive periodic solution are obtained. The complexity of this system is also analyzed. Moreover, the optimal harvesting policy are given for the inshore subpopulation, which includes the maximum sustainable yield and the corresponding harvesting effort.     

Nonclassical symmetry solutions for reaction–diffusion equations with explicit spatial dependence

Nonclassical symmetry methods are used to study the linear diffusion equation with a nonlinear source term which includes explicit spatial dependence. Mathematical forms for the spatial dependence are found which enable strictly nonclassical symmetries to be admitted when the nonlinearity is cubic. A number of new exact solutions are constructed, and an application of one of these solutions to diploid population genetics is discussed.