Đang chuẩn bị nút TẢI XUỐNG, xin hãy chờ
Tải xuống
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 − ∆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. | 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 __ d2 u _ Du u - Au f u F 1.1 dt2 where f and F are the given real functions note that f is a nonlinear and u is an unknown real function A y 1 d2 dx2. For 1.1 we consider the Cauchy characteristic problem on finding in a truncated light cone of the future Dt x t T x x1 . xn n 1 T const 0 a solution u x t of that equation by the boundary condition uIst 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 TO we assume that D- t x and s _ dl D t x . Note that the questions on the existence or nonexistence of a global solution of the Cauchy problem for semilinear equations of type 1.1 with initial conditions u t o u0 du dtlt 0 u1 have been considered in 1 2 6 7 8 10 13 14 15 16 17 18 22 23 26 30 31 . As for the characteristic problem in a linear case that is for problem 1.1 - 1.2 when the right-hand side of 1.1 does not involve the nonlinear summand f u this problem is as is known formulated correctly and the global solvability in the corresponding spaces of functions takes place 3 4 5 11 25 . Below we will distinguish the particular cases of the nonlinear function f f u when problem 1.1 - 1.2 is globally solvable in one case and unsolvable in the other one. Copyright 2006 Hindawi Publishing Corporation BoundaryValue Problems 2005 3 2005 359-376 DOI 10.1155 BVP.2005.359 360 The Cauchy characteristic problem 2. Global solvability of the problem Consider the case for f u -A u pu where A 0 and p 0 are the given real numbers. In this case 1.1 takes the form d2u Lu 4-pT - ku -A
