Quasilinear Wave Equation, Initial Boundary Value Problems

In this paper we study the existence and asymptotic behaviour of global solutions of the initial boundary value problems for some quasilinear wave equations:Ω with smooth bounary (?)Ω 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(0.1)-(0.3).The main results are the followingTheorem 1 Suppose thatThen(0.1)-(0.3)admits a unique global generalized solutionTheorem 2 Under the conditions of TheoremThenWhere C is a constant.In Chapter 3, we get the global generalized solution of (0.4)-(0.6) by using Galerkin method; also, we obtain the uniqueness and decay of the solution by using Gronwall inequality and Nakao inequality respectively.Theorem 3 Suppose thatThen (0.4)-(0.6)admits a global generalized solutionTheorem 4 If the condition in theorem 3 holds, then we have the following decay estimateHereinTheorem 5 If the condition in Theorem 3 holds, and p ≤ N/N-2(N ≥ 3); u is the local solution of (0.4)-(0.6) satisfyingthenandHereinTheorem 6 If the condition in theorem3 holds, and p ≤N/N-2(N ≥ 3); then there exists a set S (?) [W ∩ H2(Ω)] × H10(Ω)containing(0,0) , such that (u0,u1) ∈ 5, where S is as shown in the proof of Theorem 6); and (0.4)-(0.6)admits a global generalized solutionMoreover, the decay estimate in Theorem 4 holds also.