非线性黏性波动方程强解的存在性

Existence for the Strong Solution of Nonliner Viscoelastic Wave Equations

文章我们着重讨论以下具有边界阻尼的非线性黏性波动方程强解的存在性.设Ω是Rn的具有光滑边界Γ=Γ0∪Γ1的星形有界区域,这里Γ0与Γ1是不相交闭集,ν为外向单位法向量.在Ω上研究了具有边界阻尼项的非线性黏性波动方程ytt-Δy+∫0th(t-τ)Δy(τ)dτ+F(x,t,y,Δy)=0,(x,t)∈Ω×(0,∞);y=0,(x,t)∈Γ1×(0,∞);y /ν-∫0th(t-τ)y/ν(τ)dτ+byt=0,(x,t)∈Γ0×(0,∞);y(x,0)=y0(x),yt(x,0)=y1(x),x∈Ω.这里b>0.我们利用Faedo-Galerkin方法证明上述问题强解的存在性.

We study existence for the strong solution of nonliner viscoelastic wave equations with boundary damping.Let Ω be a bounded star-shaped domain of Rn,n≥1,with a smooth boundary Γ=Γ0∪Γ1.Here,Γ0 and Γ1 are closed and disjoint anνrepresents the unit outward normal toΓ.In this article,we study the nonlinear viscoelastic equation ytt-Δy+∫0th(t-τ)Δy(τ)dτ+F(x,t,y,Δy)=0,(x,t)∈Ω×(0,∞);y=0,(x,t)∈Γ1×(0,∞);y /ν-∫0th(t-τ)y/ν(τ)dτ+byt=0,(x,t)∈Γ0×(0,∞);y(x,0)=y0(x),yt(x,0)=y1(x),x∈Ω.Where b>0.The existence is proved by means of Faedo-Galerkin method.