Weak solution for a fourth-order nonlinear wave equation

摘要

The existence and the nonexistence, the uniqueness and the energy decay estimate of solution for the fourth-order nonlinear wave equation u_(tt) + alpha DELTA~2 u-b DELTA u_t -beta DELTA u + u_t|u_t|~r + g(u) =0 in SIGMA x (0, infinity ) are studied with the boundary condition u = partial deriv u/partial deriv v = 0 on partial deriv SIGMA and the initial condition u(X,0) = u_0(x), u_t(x, 0) =u,(x,0) in bounded domain SIGMA is contained in R~n, n > = 1. The energy decay rate of the global solution is estimated by the multiplier method. The blow-up result of the solution in finite time is established by the ideal of a potential well theory, and the existence of the solution is gotten by the Galekin approximation method.