Existence and Long-Time Behavior of Variational Solutions to a Class of Nonclassical Diffusion Equations in Noncylindric
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Existence and Long-Time Behavior of Variational Solutions to a Class of Nonclassical Diffusion Equations in Noncylindrical Domains Nguyen Duong Toan
Received: 9 October 2013 / Revised: 20 January 2014 / Accepted: 23 January 2014 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015
Abstract We prove the existence and uniqueness of variational solutions to the following non-autonomous nonclassical diffusion equation ut − ut − u + f (u) = g(x, t) in a noncylindrical domain with the homogeneous Dirichlet boundary condition, under assumptions that the spatial domains are bounded and increase with time, the nonlinearity f satisfies growth and dissipativity conditions of Sobolev type, and the external force g is time-dependent. Moreover, the nonautonomous dynamical system generated by this class of solutions is shown to have a pullback attractor Aˆ = {A(t) : t ∈ R}. Keywords Nonclassical diffusion equation · Variational solution · Pullback attractor · Penalty method Mathematics Subject Classification (2010) 35B41 · 35D99
1 Introduction Nonclassical diffusion equation arises as a model to describe physical phenomena, such as non-Newtonian flows, soil mechanics, and heat conduction (see, e.g., [1, 13, 18, 20]). In the last few years, the existence and long-time behavior of solutions to nonclassical diffusion equations have attracted the attention of many authors, for both autonomous case [16, 19, 21, 22] and nonautonomous case [2–5, 17]. However, most of the existing results are in the case of cylindrical domains, except the recent work [4] in which the nonclassical diffusion equations in noncylindrical domains with nonlinearity of polynomial type were investigated. In this paper, we will study the existence and long-time behavior of variational solutions to this equation in the case of non-cylindrical domains, the nonlinearity f is of Sobolev type, and the external force g that may depend on time t. N. D. Toan () Department of Mathematics, Haiphong University, 171 Phan Dang Luu, Kien An, Haiphong, Vietnam e-mail: [email protected]
N. D. Toan
Let {t }t∈R be a family of nonempty bounded open subsets of RN such that s < t ⇒ s ⊂ t . We will frequently use the notations t × {t}, Qτ := Qτ,T := t∈(τ,T )
τ,T :=
t × {t},
t∈(τ,∞)
∂t × {t}, τ :=
t∈(τ,T )
∂t × {t}.
t∈(τ,∞)
We are interested in the nonautonomous nonclassical diffusion equation ⎧ ⎨ ut − ut − u + f (u) = g(t, x) in Qτ , u=0 on τ , ⎩ u|t=τ = uτ on τ ,
(1.1)
where uτ ∈ H01 (τ ) is given, and the nonlinearity f and the external force g satisfy the following conditions: (H1)
f ∈ C 1 (R; R) satisfies f (u) ≥ −, |f (u)| ≤ C 1 + |u|ρ , uf (u) − κF (u) lim inf ≥ 0, |u|→∞ u2 F (u) lim inf 2 ≥ 0, |u|→∞ u
where 0 < ρ ≤ primitive of f ; (H2)
N+2 N−2 ,
and κ are two positive constants, and F (u) =
g ∈ L2 (Qτ ), where Qτ :=
t∈(τ,∞) t
(1.2) (1.3) (1.4) (1.5) u 0
f (s)ds is a
× {t}.
Since the open set t changes with time t, problem (1.1) i
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