Decay Of Two Types Of Wave Equations

In this paper, we study the decay of two types of wave equations.The one is Here,t∈R+,χ∈Ω(?) Rn are the time and space variables,respectively.AndΩis an open set with the smooth boundaryΓ.The coupling constantsβ≥0 and l＞0 are damping and spring coefficients,respectively.The funcions u(χ,t),v(χ,t) and w(x, t) are the displacements of three vibrating objects measured from their equilibrium position. C1,c2 and C3 are wave propagation speeds. The distributed springs and dampers linking the three vibrating objects are the coupling terms;i.e.,±l(u+v-2w) and±β(ut+vt-2wt).We reformulate the above equation into an evolution system and discuss the properties of the operator.Then,we prove the property of uniformly exponentially decay holds for it under certain conditions.The other is whereμ(t) satisfyμ'(t)≤0 andμ(t)≥μ(0)＞0.Ωis an open bounded domain of Rn with the smooth boundary ??Ω=Γ0∩Γ1,νis the unit outer normal vector,and f∈C1(R) is a function satisfying with for some constant d＞0 and p≥1 such that (n-2)p≤n. The partitionΓo andΓ1 are closed,disjoint, with meas(Γ0)＞0 and satisfying where m(χ)=χ-χ0,χ0∈Rn.First,we show the existence of the solutions of the above system,then,we prove general decay of solutions by introducing the Lyapunov function.