关键词: 非线性波方程, 初边值问题, 局部解, 解的爆破

Abstract:

In this paper, the following initial boundary value problem  of the nonlinear wave equation involving the nonlinear damping term and the
nonlinear source term
u_tt -u_xxt -u_xx -(σ(u²_x)u_x)_x+δ|u_t|^p-1u_t=μ|u|^q-1u, 0 < x <1, 0 ≤ t ≤ T,      
u(0, t)=u(1, t)=0,  0 ≤ t ≤ T,                                                                     

u(x, 0)=u₀(x),  u_t(x, 0)=u₁(x), 0 ≤ x ≤1                                           
is discussed. This paper gives sufficient conditions of  blow-up of the solutions for the problem in finite time and proves the existence and uniqueness of the local generalized solution and classical solution of this problem.

Key words: Nonlinear wave equation, Initial boundary value problem, Local solution, Blow-up of solution