scholarly journals Global Existence of Strong Solutions to a Class of Fully Nonlinear Wave Equations with Strongly Damped Terms

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Zhigang Pan ◽  
Hong Luo ◽  
Tian Ma
Keyword(s):  
Global Existence ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Galerkin Approximation ◽  
Strong Solutions ◽  
Nonlinear Wave Equations ◽  
Existence Of A Solution ◽  
Fully Nonlinear ◽  
Sequence Techniques ◽  
Strongly Damped

We consider the global existence of strong solutionu, corresponding to a class of fully nonlinear wave equations with strongly damped termsutt-kΔut=f(x,Δu)+g(x,u,Du,D2u)in a bounded and smooth domainΩinRn, wheref(x,Δu)is a given monotone inΔunonlinearity satisfying some dissipativity and growth restrictions andg(x,u,Du,D2u)is in a sense subordinated tof(x,Δu). By using spatial sequence techniques, the Galerkin approximation method, and some monotonicity arguments, we obtained the global existence of a solutionu∈Lloc∞((0,∞),W2,p(Ω)∩W01,p(Ω)).

scholarly journals Global existence and decay estimates for nonlinear wave equations with space-time dependent dissipative term

2015 ◽  
Vol 12 (02) ◽  
pp. 249-276
Author(s):  
Tomonari Watanabe
Keyword(s):  
Global Existence ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Space Time ◽  
Time Dependent ◽  
Nonlinear Wave Equations ◽  
Energy Estimates ◽  
Space Region ◽  
Decay Estimates ◽  
Dissipative Term

We study the global existence and the derivation of decay estimates for nonlinear wave equations with a space-time dependent dissipative term posed in an exterior domain. The linear dissipative effect may vanish in a compact space region and, moreover, the nonlinear terms need not be in a divergence form. In order to establish higher-order energy estimates, we introduce an argument based on a suitable rescaling. The proposed method is useful to control certain derivatives of the dissipation coefficient.

Global Existence and Asymptotic Behavior of Solutions for Nonlinear Wave Equations

2014 ◽  
Vol 490-491 ◽  
pp. 327-330
Author(s):  
Ji Bing Zhang ◽  
Yun Zhu Gao
Keyword(s):  
Asymptotic Behavior ◽  
Global Existence ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Nonlinear Damping ◽  
Nonlinear Wave Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Potential Well ◽  
Source Terms ◽  
Damping And Source Terms

In this paper, we concern with the nonlinear wave equations with nonlinear damping and source terms. By using the potential well method, we obtain a result for the global existence and asymptotic behavior of the solutions.

scholarly journals On the existence of solutions of strongly damped nonlinear wave equations

2000 ◽  
Vol 23 (6) ◽  
pp. 369-382 ◽  
Author(s):  
Jong Yeoul Park ◽  
Jeong Ja Bae
Keyword(s):  
Nonlinear Wave ◽  
Existence And Uniqueness ◽  
Wave Equations ◽  
Existence Of Solutions ◽  
Hyperbolic Type ◽  
Nonlinear Wave Equations ◽  
Gamma Delta ◽  
Uniqueness Of Solutions ◽  
Alpha Beta ◽  
Strongly Damped

We investigate the existence and uniqueness of solutions of the following equation of hyperbolic type with a strong dissipation:utt(t,x)−(α+β(∫Ω|∇u(t,y)|2dy)γ)Δu(t,x)                                −λΔut(t,x)+μ|u(t,x)|q−1u(t,x)=0,     x∈Ω,t≥0            u(0,x)=u0(x),          ut(0,x)=u1(x),      x∈Ω,  u|∂Ω=0, whereq>1,λ>0,μ∈ℝ,α,β≥0,α+β>0, andΔis the Laplacian inℝN.

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