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Abstract
A Dirichlet problem for a nonlinear wave equation is investigated. Under suitable assumptions, we prove the solvability and the uniqueness of a weak solution of the above problem. On the other hand, a high-order asymptotic expansion of a weak solution in many small parameters is studied. Our approach is based on the Faedo-Galerkin method, the compact imbedding theorems, and the Taylor expansion of a function.
Highlights
In the special cases, when the function μ x, t, u is independent of u, μ x, t, u ≡ 1, or μ x, t, u μ x, t, and the nonlinear term f has the simple forms, the problem 1.1, with various initial-boundary conditions, has been studied by many authors, for example, Ortiz and Dinh 1, Dinh and Long 2, 3, Long and Diem 4, Long et al 5, Long and Truong 6, 7, Long et al 8, Ngoc et al 9, and the references therein
Ficken and Fleishman and Rabinowitz studied the periodic-Dirichlet problem for hyperbolic equations containing a small parameter ε, in particular, the differential equation utt − uxx 2αut εf t, x, u, ut, ux
It is shown that the same results are valid for the equation utt − uxx g t, x, u g1 u ut εq t, x, u, ut, ux, 1.6 with sufficiently small ε and continuously differentiable q
Summary
∂ ∂x μ x, t, u ux f x, t, u, ux, ut , 0 < x < 1, 0 < t < T, 1.1 u 0, t u 1, t 0, 1.2 u x, 0 u0 x , ut x, 0 u1 x , 1.3 where u0, u1, μ, and f are given functions satisfying conditions specified later. In 12 , Kiguradze has established the existence and uniqueness of a classical solution u ∈ C2 0, a × Rn of the periodic-Dirichlet problem for the following nonlinear wave equation: utt − uxx g t, x, u g1 u ut, 1.5 under the assumption that g and g1 are continuously differentiable functions these conditions are sharp and cannot be weakened. The result obtained here is a relative generalization of 5–7, 14 , where asymptotic expansion of a weak solution in two or three small parameters is given
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