Abstract
In this paper, we study the existence of Young measure solutions to a fourth-order wave equation with variable exponent nonlinearity on a bounded domain. The asymptotic behavior of the Young measure solutions is also investigated by applying a lemma developed by Nakao.
Highlights
In this paper, we consider the initial boundary value problem of the following model: ∂ u ∂t +| u|p(x)– u + |u|q(x)– u + a |u|q(x) dx ∂u = f (x, t),∂t (x, t) ∈ QT, u = u =, (x, t) ∈ ∂ × (, T), ( . ) u(x, ) = u (x),∂u(x, ) ∂t = u (x), x∈, where ⊂ RN (N ≥ ) is a bounded domain with smooth boundary ∂, < T < ∞ is a given constant, and QT = × (, T)
After Kováčik and Rákosník first discussed the variable exponent Lebesgue space Lp(x) and Sobolev space W k,p(x) in [ ], a lot of research has been done concerning these kinds of variable exponent spaces; see for example [, ] for the properties of such spaces and [ – ] for the applications of variable exponent spaces on partial differential equations
In [ ] Růžička presented the mathematical theory for the application of variable exponent spaces in electro-rheological fluids
Summary
In Section , we obtain the existence of weak solutions of problem ) and some a priori estimates, we get the existence of Young measure solutions by letting ε → . If < p– ≤ p+ < ∞, the space W k,p(x)( ) is separable and reflexive; If p : → ( , ∞) is a bounded log-Hölder continuous function, C ∞( ) is dense in W k,p(x)( ).
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