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

Long Nguyen Thanh (1) , Alain Pham Ngoc Dinh (2) , Le Xuan Truong (3)

1 Department of Mathematics and Computer Science
2 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
3 Department of Mathematics

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.

Mots clés: Faedo-Galerkin method; global existence; contraction; nonlinear wave equation; viscoelastic; exponential decay

Domaines: Equations aux dérivées partielles [math.AP]

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. 2007. ⟨hal-00136410v2⟩