About A Class Of Semilinear Wave Equations With Time Decay Dissipation

In this thesis, we are concerned with the following semi-linear wave equation with time-dependent dissipation:where t ?[0, ?),??(0,1), ??(0,1), and ?> 0 is a constant.In this thesis, we will devote to prove the global existence of the solution to problem (0.3). In the process of proving the global existence of solution to problem(0.3) with time-dependent dissipation,we need to use the Matsumura-type esti-mate of solutions of the following linear wave equation (0.4) with time-dependent dissipation:where t ?[0, ?), s?[0,t],?? (0,1), ?? (0,1), and ?> 0 is a constant.At first, we give the Matsumura-type estimate of the solution to problem (0.4)with time-dependent dissipation, and then give the global existence of the solution of problem (0.3) with time-dependent dissipation. Finally, we obtain the following two main conclusions:Theorem 2.0.3:Let 1 +?-2?> 0, and t?s?0. Then the solution u of problem (0.4) and its derivatives satisfy the following Matsumura-type estimates:||(?)xju||2(?){1+(1+t)1+?-2?-(1+s)1+?-2?}-1/2(||u1|H1+||u2||L2),||(?)t(?)xju||2(?)(1+t)?-2?{1+(1+t)1+?-2?-(1+s)1+?-2?}-j/2-1(||u1||H1+||u2||L2),where j =0,1.Theorem 2.0.4:Assume 1+?-2?>0,n?1 and p>4/n(1+?-2?)+1.Suppose that there is a constant 0 <?<<1 such that||(u1,u2)||A1,1??.Then problem (0.3) has a unique solution u ? C([0, ?), H1) ? C1([0,?), L2).Moreover, there exists a constant C > 0 independent of t, such that u satisfies the following estimates:||v(t,·)||L2?C||(u1,u2)||A1,1,||?u(t,·)||L2?C||(u1,u2)||A1,1(1+t)-1/2(1+?-2?),||ut(t,·)||L2?C||(u1,u2)||A1,1(1 +t)-1.In Chapter 1, we will give some introdution to Theorem 2.0.3 and Theorem 2.0.4. The detailed proof of Theorem 2.0.3 and Theorem 2.0.4 are proved in Chapter 2. In chapter 3, we will summarize this thesis and give some outlook.