Loss Of Drrivatives On Two Classes Of Wave Equations

In this thesis, the Cauchy problem for wave equations with variable coefficients in the time variable is considered. Under some appropriate conditions on the coefficients a(t), by establishing the estimates of the energy functional, the loss of derivatives to the solution for the Cauchy problem is obtained.The main results of this thesis are stated as follows.(1) The coefficient a(t) satisfies the condition|a(t)|≤(?) with r∈[0,1] for re(0,T], If r∈[0,1), then there is no loss of derivatives, that is, the energy inequality Es (u)(t)≤CsEs (μ)(0) holds for s≥0; If r=1, then there is a finite loss of derivatives, that is, the energy inequality Es-s0(μ)(t)<CsEs (μ)(0) hold for large s with a positive constant s0.(2) The coefficient a(t) satisfies with|ak (t+τ)-ak (t)|≤M|τ||ln|τ||ω(|τ|), M>0,|τ|≤1, the function ω(ξ) has the property:ξ->0+,ω(ξ)↓0. Then, for every δ>0, the solution of the Cauchy problem for wave equation satisfies En-δ(u)(t)≤CδEs(u)(O), Cδ>o,t∈(0,T].