Initial Boundary Value Problems For Two Classes Of Nonlinear Wave Equations

In this paper we study the existence and asymptic behaviour of global solution of the initial boundary value problems for two classes of nonlinear wave equations:where Ω (?) R^N(1 ≤ N ≤ 3 for problem (1)-(4) and N ≥ 1 for problem (5)-(8)) is a bounded domain which has a smooth boundary (?)Ω, (?)Ω = Γ₀ ∪ Γ₁,Γ₀ ∩ Γ₁ = (?), where Γ₀ and Γ₁ are measurable over (?)Ω, endowed with the (N-1)-dimensional Lebesgue measure. m,q ≥ 2,p > 3 are reals. v is the outward normal to the boundary (?)Ω.In Chapter 2, by applying Galerkin method, we obtain the global generalized solution for the problem (1)-(4). By using Nakao inequality, we prove the generalized solution is of asymptotic property. The main result is the following:Theorem 1 Suppose that 2 < m < r, and one of the following conditions is satisfied: (i) q = 2,3 0, problem (l)-(4)has global generalized solution u = u{x,t)satisfyingut G Lm(0,T;Lm(ri))nL?(0,T;L?(n)) Furthermore, we have the decay estimates:(1) if Condition (i) is satisfied, thenUMt)f + {h ï¿¡)I|VU(*)II2 < E{Q)exp-ti1[t - 1]+, t > 0;(2) if Condition (ii) is satisfied, thenkht(t)\\2 + (i - i)||Vu(t)||2 < (^(o)1^ +m2-1(! - i)[t-1]+)^, t > o.where//i,/Z2 are positive constants , [t-l]+ = max{t-l,0},E(0) = 2l|ui||2+5l|Vuo||2-^||wo||pir1,fljo(n) = {w e /^(n): v\Ti = 0}In chapter 3, in order to obtain the existence of global solutions to problem (5)-(8), we use the Faedo Galerkin method and to get the uniform decay rates of the energy , we use the perturbed energy method. The main result is the following:Theorem 2 Suppose that(At) /o G C1^), |/0(s)| < /3|s|, |V/o| < L, s G RN, where 0 and L are positive constants. (A2) g G C°{R),g'{s) > 0,g(s)s > 0,s G R;CMP < |jff(?)| < CjM1/?, when|s| < 1; C3\a\ < \9(s)\ < Ci\s\, when|s| > 1; where Ci(i = l,2,3,4)are positive constants, p > 1 is a constant. (^3) h G C°(R), |/i(s)I < M(l + |s|), s G R, where M is a positive constant. (At) h: R+ -* R+is nonnegative and bounded C2 function such that l-Jjf ft(s)ds = ? > 0, /i(0) = 0 and there exist positive constants ï¿¡1,62, ï¿¡3 such that for some to > 0, — ï¿¡ih(t) < h'(t) < —&h(t), h"(t) < ï¿¡3h(t), for alH > t0.{A5) (uo,ui) G (H^0{n) n H2{n))2, verifying the compatibility condition ^ = 0 on Ti. then for any T > 0, the problem (5)-(8) has at least one global generalized solution u = u(x, t) satisfyingu e W^^T-H^n)), utt G X°°(0,T;L2(n))Moreover, assuming that p = 1 in (A2) and considering ||/i||z,i(o,oo) and /? (give by {A{)) sufficiently small ,the energy determined by the solution u decays exponentialy, that isE(t) = Un(\Mt)\2 + \Vu(t)\2)dxt0 for some positive constants c and 7. In addition, if /1 is globally Lipschitz ,the solution is unique.