On a nonlinear coupled system of thermoelastic type with acoustic boundary conditions
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On a nonlinear coupled system of thermoelastic type with acoustic boundary conditions P. Braz e Silva1 · H. R. Clark2 · C. L. Frota3
Received: 28 October 2014 / Accepted: 21 April 2015 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015
Abstract This paper deals with the existence, uniqueness, and asymptotic behavior of global solutions for a parabolic–hyperbolic coupled system with both local and nonlocal nonlinearities under mixed nonlinear acoustic boundary conditions. Keywords Acoustic boundary conditions · Nonlinear coupled system · Thermoelastic system Mathematics Subject Classification
35B40 · 35L05 · 35L20
1 Introduction Let ⊂ Rn be a bounded, open and connected set, situated locally on one side of its boundary , which is assumed to be a C 2 (n − 1)-dimensional compact manifold without boundary. Suppose also to be made up of two disjoint parts 0 and 1 ( = 0 ∪1 and 0 ∩1 = ∅), both connected and with positive measure. We look for functions u, θ : × (0, ∞) → R and δ : 1 × (0, ∞) → R solutions for the nonlinear coupled system of the thermoelastic type with acoustic boundary conditions
Communicated by Geraldo Diniz.
B
H. R. Clark [email protected] P. Braz e Silva [email protected] C. L. Frota [email protected]
1
DMAT, Universidade Federal de Pernambuco, Recife, PE, Brazil
2
IME, Universidade Federal Fluminense, Niterói, RJ, Brazil
3
DMA, Universidade Estadual de Maringá, Maringá, PR, Brazil
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P. Braz e Silva et al.
u − αu + λ|u|ρ u + (a · ∇)θ = 0
θ − β
in × (0, ∞),
(1.1)
in × (0, ∞),
(1.2)
θ dx θ + (a · ∇)u = 0
u = 0 on 0 × (0, ∞), u + f 1 δ + f 2 δ + f 3 δ = 0
(1.3)
on 0 × (0, ∞),
(1.4)
∂u − δ + η(·, u ) = 0 on 1 × (0, ∞), ∂ν
(1.5)
θ = 0 on × (0, ∞),
(1.6)
where α : [0, ∞) → R and β : R → R are given functions; λ, ρ are positive real constants n ai ∂∂xi and a = (a1 , . . . , an ) is a constant known vector of Rn ; (a · ∇) is the operator and =
n i=1
i=1
∂2 ∂ xi2
is the Laplace operator; f i , for i = 1, 2, 3, are given real-valued functions
defined on 1 ; ν denotes the unit outward normal vector on 1 ; and η : 1 × R → R is a given function. Here, denotes the partial derivative with respect to t and all these derivatives are in the sense of the distributions (cf. Schwartz 1966). We consider the system above under the initial conditions u(x, 0) = u 0 (x), u (x, 0) = u 1 (x) for x ∈ ,
(1.7)
θ (x, 0) = θ0 (x) for x ∈ ,
(1.8)
∂u 0 (1.9) (x) − η(x, u 1 (x)) for x ∈ 1 , ∂ν where u 0 , u 1 , θ0 : → R and δ0 : 1 → R are given functions. Precisely, this paper is concerned with the existence, uniqueness and asymptotic behavior of global solutions for the initial-boundary value problem (1.1)–(1.9). Since the Eqs. (1.1), (1.2) have time-dependent coefficients α = α(t), β = β(t), one cannot use semigroup theory to study the solvability of system (1.1)–(1.9). So, to prove the existence and uniqueness of solutions we employ Faedo–Galerkin–Lions’ method and the energy method, respectively, with no
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