Uniform Attractors of Nonclassical Diffusion Equations Lacking Instantaneous Damping on R N $\mathbb{R}^{N}$ with Memo
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Uniform Attractors of Nonclassical Diffusion Equations Lacking Instantaneous Damping on RN with Memory Nguyen Duong Toan1
Received: 5 April 2020 / Accepted: 19 September 2020 / Published online: 6 October 2020 © Springer Nature B.V. 2020
Abstract In this paper, we consider the non-autonomous nonclassical diffusion equations on RN with hereditary memory ∞ ut − ut − κ(s)u(t − s)ds + f (x, u) = g(x, t). 0
The main characteristics of the model is that the equation does not contain a term of the form −u, which contributes to an instantaneous damping. We first investigate the existence and uniqueness of weak solutions to the initial-boundary-value problem for above-mentioned equation. Next, we study the long-time dynamical behavior of the solutions in the weak topological space H 1 (RN ) × L2μ (R+ , H 1 (RN )), where the nonlinearity is critical and the time-dependent forcing term is only translation bounded instead of translation compact. The results in this paper will extend and improve some results in (Conti et al. in Commun. Pure Appl. Anal. 19:2035–2050, 2020) in the non-autonomous and unbouded domain cases which have not been studied before. Mathematics Subject Classification (2010) 35B41 · 45K05 · 76R50 · 35D30 Keywords Nonclassical diffusion equation · Hereditary memory · Uniform attractor · Unbounded domain
1 Introduction The main goal of this paper is to discuss the long-time behavior of the solutions for the following equation ∞ ut − ut − κ(s)u(t − s)ds + f (x, u) = g(x, t), (x, t) ∈ RN × (τ, +∞), (1.1) 0
B N.D. Toan
[email protected]; [email protected]
1
Department of Mathematics, Haiphong University, 171 Phan Dang Luu, Kien An, Haiphong, Vietnam
790
N.D. Toan
with the initial data u(x, τ ) = uτ (x),
u(x, τ − s)|s>0 = qτ (x, s),
(1.2)
where N ≥ 3, the nonlinearity f and the external force g satisfy some certain conditions specified later. In the case of κ ≡ 0, equation (1.1) is an integro-differential relaxation of the nonclassical diffusion equation ut − ut − u + f (u) = g.
(1.3)
In 1980, the nonclassical diffusion equation was first introduced by Aifantis in [1]. It arises as a model to describe physical phenomena, such as non-Newtonian flows, soil mechanics and heat conduction theory (see, e.g., [1, 22, 26, 28]). In the last two decades, the existence and long-time behavior of solutions to nonclassical diffusion equations has extensively been studied by many authors, for both in bounded domain case (see [2, 10, 19, 23, 24, 29, 32, 35, 36, 38]) and in unbouded domain case (see [3, 5, 6, 34, 40]), and even in non-cylindrical domains (see [4, 27]). Now if we consider nonclassical diffusion equation containing viscoelasticity of the conductive medium, that is to say, we add a fading memory term to this equation, ∞ κ(s)u(t − s)ds + f (u) = g. (1.4) ut − ut − u − 0
The speed of energy dissipation for equation (1.4) is faster than for the usual nonclassical diffusion equation (1.3). The conduction of energy is not only affected by present external forces but also by historic
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