International Journal of Mathematics and Mathematical Sciences
Volume 17 (1994), Issue 3, Pages 561-570
doi:10.1155/S0161171294000815

Decay of solutions of a nonlinear hyperbolic system in noncylindrical domain

Abstract

In this paper we study the existence of solutions of the following nonlinear hyperbolic svstem|u″+A(t)u+b(x)G(u)=f   in   Qu=0   on   Σu(0)=uο   u1(0)=u1where Q is a noncylindrical domain of ℝn+1 with lateral boundary Σ, u−(u1,u2) a vector defined on Q, {A(t),   0≤t≤+∞} is a family of operators in ℒ(Hο1(Ω),H−1(Ω)), where A(t)u=(A(t)u1,A(t)u2) and G:ℝ2→ℝ2 a continuous function such that x.G(x)≥0, for x∈ℝ2.Moreover, we obtain that the solutions of the above system with dissipative term u′ have exponential decay.