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Register a new userOn the Moore-Gibson-Thompson equation with memory with nonconvex kernels
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We consider the MGT equation with memory $$partial_{ttt} u + alpha partial_{tt} u - beta Delta partial_{t} u - gammaDelta u + int_{0}^{t}g(s) Delta u(t-s) ds = 0.$$ We prove an existence and uniqueness result removing the convexity assumption on the convolution kernel $g$, usually adopted in the literature. In the subcritical case $alphabeta>gamma$, we establish the exponential decay of the energy, without leaning on the classical differential inequality involving $g$ and its derivative $g$, namely, $$g+delta gleq 0,quaddelta>0,$$ but only asking that $g$ vanishes exponentially fast.
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We are interested in the Moore-Gibson-Thompson(MGT) equation with memory begin{equation} onumber tau u_{ttt}+ alpha u_{tt}+c^2A u+bA u_t -int_0^tg(t-s)A w(s)ds=0. end{equation} We first classify the memory into three types. Then we study how a memory term creates damping mechanism and how the memory causes energy decay.
We study a temporally third order (Moore-Gibson-Thompson) equation with a memory term. Previously it is known that, in non-critical regime, the global solutions exist and the energy functionals decay to zero. More precisely, it is known that the energy has exponential decay if the memory kernel decays exponentially. The current work is a generalization of the previous one (Part I) in that it allows the memory kernel to be more general and shows that the energy decays the same way as the memory kernel does, exponentially or not.
P. Galenko et al. proposed a modified Cahn-Hilliard equation to model rapid spinodal decomposition in non-equilibrium phase separation processes. This equation contains an inertial term which causes the loss of any regularizing effect on the solutions. Here we consider an initial and boundary value problem for this equation in a two-dimensional bounded domain. We prove a number of results related to well-posedness and large time behavior of solutions. In particular, we analyze the existence of bounded absorbing sets in two different phase spaces and, correspondingly, we establish the existence of the global attractor. We also demonstrate the existence of an exponential attractor.
The paper is concerned with the exponential attractors for the viscoelastic wave model in $Omegasubset mathbb R^3$: $$u_{tt}-h_t(0)Delta u-int_0^inftypartial_sh_t(s)Delta u(t-s)mathrm ds+f(u)=h,$$ with time-dependent memory kernel $h_t(cdot)$ which is used to model aging phenomena of the material. Conti et al [Amer. J. Math., 2018] recently provided the correct mathematical setting for the model and a well-posedness result within the novel theory of dynamical systems acting on. time-dependent spaces, recently established by Conti, Pata and Temam [J. Differential Equations, 2013], and proved the existence and the regularity of the time-dependent global attractor. In this work, we further study the existence of the time-dependent exponential attractors as well as their regularity. We establish an abstract existence criterion via quasi-stability method introduced originally by Chueshov and Lasiecka [J. Dynam. Diff.Eqs.,2004], and on the basis of the theory and technique developed in [Amer. J. Math., 2018] we further provide a new method to overcome the difficulty of the lack of further regularity to show the existence of the time-dependent exponential attractor. And these techniques can be used to tackle other hyperbolic models.
In Cao, Du, Li and Li [8], a nonlocal diffusion model with free boundaries extending the local diffusion model of Du and Lin [12] was introduced and studied. For Fisher-KPP type nonlinearities, its long-time dynamical behaviour is shown to follow a spreading-vanishing dichotomy. However, when spreading happens, the question of spreading speed was left open in [8]. In this paper we obtain a rather complete answer to this question. We find a condition on the kernel function such that spreading grows linearly in time exactly when this condition holds, which is achieved by completely solving the associated semi-wave problem that determines this linear speed; when the kernel function violates this condition, we show that accelerating spreading happens.
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