Báo Cáo Báo cáo hóa học ON THE EXISTENCE OR THE ABSENCE OF GLOBAL SOLUTIONS OF THE CAUCHY CHARACTERISTIC PRO

ON THE EXISTENCE OR THE ABSENCE OF GLOBAL SOLUTIONS OF THE CAUCHY CHARACTERISTIC PROBLEM FOR SOME NONLINEAR HYPERBOLIC EQUATIONS S. KHARIBEGASHVILI Received 20 October 2004 For wave equations with power nonlinearity we investigate the problem of the existence or nonexistence of global solutions of the Cauchy characteristic problem in the light cone of the future. 1. Statement of the problem Consider a nonlinear wave equation of the type u := ∂2 u ư ∆u = f (u) + F, ∂t 2 (1.1) where f and F are the given real functions; note that f is a nonlinear and u is an unknown real function, ∆ = n=1 ∂2 /∂xi2 . i For (1.1), we consider the Cauchy characteristic problem on finding in a truncated light cone of the future DT : |x| 1, T = const 0, a solution u(x,t) of that equation by the boundary condition u|ST = g, (1.2) where g is the given real function on the characteristic conic surface ST : t = |x|, t ≤ T. When considering the case T = +∞ we