Initial Boundary Value Problem For Some Quasilinear Wave Equations

In this paper we study the existence and asympotic behaviour of global solutions of the initial boundary value problems for some quasilinear wave equations:where h > 0 is a constant, which has a smooth boundary is a bounded domain of RN.In Chapter 2,by constructing the potential well and applying Galerkin method,we obtain a unique global generalized solution for the problem (l)-(2).The main result is the following:Theorem 1 Suppose that(A1) f C(R2), f(s,0)s 0, s R, f(s, ) is a local Lipschitz continuous function.where 0 3, B(s) (s 0) is a nonnegative continuous function.then T > 0,the problem(l) - (2) has a unique global generalized solutionEspecially , if(A4)0< 3, |f(s,0)| B2(|s| + l) ( >, B2 >0 are real numbers),then,the generalized solution is of asymptotic propertyC* is a insertion constant from H1 to L +1,then VT > 0, the problem (l)-(2) has a unique global generalized solutionEspecially,if (A4) holds,then the generalized solution is of asymptotic property(8).In Chapter 3,by applying Galerkin method,we obtain the global generalized solution for the problem (3)-(4) under the mixed boundary conditions.By using Nakao inequality, we prove the generalized solution is of asymptotic property.The main result is the following:Theorem 3 Suppose thatEspecially , ifwhere, 1?if a > 0 , then h(t) = t1+2/; 2 if a = 0 , then h(t) = e-t, > 0,thenwhere, 1 if a > 0 , then h1(t) = t-2/; 2 if a = 0 , then h1(t) = e , 1 > 0.In Chapter 4,by constructing the potential well and applying Galerkin method,we obtain a global generalized solution for the problem (5)-(7).By using Nakao inequality, we prove the generalized solution is of asymptotic property.The main result is the following:Theorem 4 Suppose thatTheorem 5 Suppose thatwhere k = log...