Abstract
We prove the existence and uniqueness of a global solution of a damped quasilinear hyperbolic equation. Key point to our proof is the use of the Yosida approximation. Furthermore, we apply a method based on a specic integral inequality to prove that the solution decays exponentially to zero when the time t goes to innit y.
Highlights
Let Ω ⊂ RN be a bounded domain with smooth boundary Γ
In this paper we are concerned with the global existence and asymptotic behavior of solutions to the mixed problem
The author in [1] has been successful in proving the global existence and establishing the precise decay rate of solutions when Γ1 = ∅, g is nonlinear without any smallness conditions on the initial data and without the assumption g (x) ≥ τ > 0
Summary
Let Ω ⊂ RN be a bounded domain with smooth boundary Γ. In this paper we are concerned with the global existence and asymptotic behavior of solutions to the mixed problem u. The problem (P) with h = 0, g(x) = δx (δ > 0) and Γ1 = ∅ was studied by De Brito [3] She has shown the existence and uniqueness of global solutions for sufficiently small initial data by using a Galerkin method. The author in [1] has been successful in proving the global existence and establishing the precise decay rate of solutions when Γ1 = ∅, g is nonlinear without any smallness conditions on the initial data and without the assumption g (x) ≥ τ > 0. Our study is motivated by Ikehata and Okazawa’s work [5] where global existence was proved when g(x) = δx (δ > 0) and Dirichlet or Neumann boundary condition by using the Yosida approximation method together with compactness arguments.
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