scholarly journals Global Existence and Global Nonexistence of Solutions of the Cauchy Problem for a Nonlinearly Damped Wave Equation

1998 ◽  
Vol 228 (1) ◽  
pp. 181-205 ◽  
Author(s):  
Howard A Levine ◽  
Sang Ro Park ◽  
James Serrin
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Global Existence ◽  
Damped Wave Equation ◽  
Global Nonexistence ◽  
The Cauchy Problem ◽  
Nonexistence Of Solutions

BLOW-UP OF THE SOLUTION FOR SEMILINEAR DAMPED WAVE EQUATION WITH TIME-DEPENDENT DAMPING

2012 ◽  
Vol 14 (05) ◽  
pp. 1250034
Author(s):  
JIAYUN LIN ◽  
JIAN ZHAI
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Initial Data ◽  
Upper Bound ◽  
Blow Up ◽  
Time Dependent ◽  
Damped Wave Equation ◽  
Large Initial Data ◽  
Power Type ◽  
The Cauchy Problem

We consider the Cauchy problem for the damped wave equation with time-dependent damping and a power-type nonlinearity |u|ρ. For some large initial data, we will show that the solution to the damped wave equation will blow up within a finite time. Moreover, we can show the upper bound of the life-span of the solution.

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”.

Sign in / Sign up

Export Citation Format

Share Document