EXISTENCE AND DECAY OF SOLUTIONS OF A NONLINEAR VISCOELASTIC PROBLEM WITH A MIXED NONHOMOGENEOUS CONDITION

Abstract : We study the initial-boundary value problem for a nonlinear wave equation given by u_{tt}-u_{xx}+\int_{0}^{t}k(t-s)u_{xx}(s)ds+ u_{t}^{q-2}u_{t}=f(x,t,u) , 0 < x < 1, 0 < t < T, u_{x}(0,t)=u(0,t), u_{x}(1,t)+\eta u(1,t)=g(t), u(x,0)=û_{0}(x), u_{t}(x,0)={û}_{1}(x), where \eta \geq 0, q\geq 2 are given constants {û}_{0}, {û}_{1}, g, k, f are given functions. In part I under a certain local Lipschitzian condition on f, a global existence and uniqueness theorem is proved. The proof is based on the paper [10] associated to a contraction mapping theorem and standard arguments of density. In Part} 2, under more restrictive conditions it is proved that the solution u(t) and its derivative u_{x}(t) decay exponentially to 0 as t tends to infinity.
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Numerical Functional Analysis and Optimization / Numerical Functional Analysis and Optimization An International Journal, 2008, 29 (11-12), pp.1363-1393
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Dernière modification le : jeudi 3 mai 2018 - 15:32:06
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  • HAL Id : hal-00136410, version 3
  • ARXIV : 0705.3959

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Long Nguyen Thanh, Alain Pham Ngoc Dinh, Le Xuan Truong. EXISTENCE AND DECAY OF SOLUTIONS OF A NONLINEAR VISCOELASTIC PROBLEM WITH A MIXED NONHOMOGENEOUS CONDITION. Numerical Functional Analysis and Optimization / Numerical Functional Analysis and Optimization An International Journal, 2008, 29 (11-12), pp.1363-1393. 〈hal-00136410v3〉

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