Some remarks on global existence to the Cauchy problem of the wave equation with nonlinear dissipation

2008 ◽  
Vol 281 (12) ◽  
pp. 1694-1707 ◽  
Author(s):  
Nour-Eddine Amroun ◽  
Abbès Benaissa
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Global Existence ◽  
Nonlinear Dissipation ◽  
The Cauchy Problem

scholarly journals Global existence and energy decay of solutions to the Cauchy problem for a wave equation with a weakly nonlinear dissipation

2004 ◽  
Vol 2004 (11) ◽  
pp. 935-955 ◽  
Author(s):  
Abbès Benaissa ◽  
Soufiane Mokeddem
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Global Existence ◽  
Stable Set ◽  
Nonlinear Dissipation ◽  
Weakly Nonlinear ◽  
Dissipative Term ◽  
Decay Properties ◽  
Decay Of Solutions ◽  
The Cauchy Problem

We prove the global existence and study decay properties of the solutions to the wave equation with a weak nonlinear dissipative term by constructing a stable set inH1(ℝn).

A global existence theorem for the Cauchy problem of nonlinear wave equations

1988 ◽  
Vol 109 (3-4) ◽  
pp. 261-269 ◽  
Author(s):  
Jianmin Gao ◽  
Lichen Xu
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Global Existence ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Lorentz Invariance ◽  
Nonlinear Wave Equations ◽  
Homogeneous Wave ◽  
The Cauchy Problem ◽  
Global Existence Theorem

SynopsisIn this paper we consider the global existence (in time) of the Cauchy problem of the semilinear wave equation utt – Δu = F(u, Du), x ∊ Rn, t > 0. When the smooth function F(u, Du) = O((|u| + |Du|)k+1) in a small neighbourhood of the origin and the space dimension n > ½ + 2/k + (1 + (4/k)2)½/2, a unique global solution is obtained under suitable assumptions on initial data. The method used here is associated with the Lorentz invariance of the wave equation and an improved Lp–Lq decay estimate for solutions of the homogeneous wave equation. Similar results can be extended to the case of “fully nonlinear wave equations”.

scholarly journals Global existence of solution for multidimensional generalized double dispersion equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xiaoqiang Dai
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Dispersion Equation ◽  
Existence Of Solutions ◽  
Existence Of Solution ◽  
New Methods ◽  
Global Existence Of Solutions ◽  
The Cauchy Problem ◽  
Double Dispersion

Abstract In this paper, we study the Cauchy problem of multidimensional generalized double dispersion equation. To prove the global existence of solutions, we introduce some new methods and ideas, and fill some gaps in the established results.

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