On the time global clasical solution to the boundary value problem (in the interior domain) for nonlinear wave equations

调查员已经研究了该项目，我们已经获得了以下结果。对于一个边界域Ω〔Ryie D1 n yie D1〕和（t，x）μ [0，∞）×Ω我们认为Uyie D2 tt yie D2-μ u+μu=α uyie D1 m yie D1，m=2，3，...，带有self-interaction的中性标量字段（Spontaneous break down of symmetry of neutral scalar field）...（1） U型D2 tt型D2-Δ+2β（t+T）D1-1 u型D2 t型D2=αu型D2， β，T>0，（Euler-Poisson-Darboux type of equation）...（2） UイイD2-Δu+（μ+β（β+1）（1+t）イイD1 -2イエD1+2γβ（1+ t）イD1-1イエD1）u=αuイD1mイエD1，μ=イD21イエD2+イD1， β∈R，イD21イエD2：first eigen value of -Δ ...（3） U=0 on [0，±）× ±（Dirichlet condition） ...（4） U=φ（x）， UイイD2tイエD2=φ（x） at t=0。（初始条件）...（五）1.边界值问题（1）-（4）。Let？=？+？D12？D1 and？be an eigen value of -？。Under some condition on m，逗号， n and逗号， we succeeded in obtaining a time global classical solution satisfying e D1逗号D1 U ￣（x） for an eigen function corresponding to逗号。v（t， x）=u（t， x）-e D1-γt D1 Σ（x）在v中被解决了一个减少的问题 ... More 在时间里倒退。在这一过程中，“单一超声波操作员”扮演了一个重要的角色。下一个，基于这一方法，我们成功地构建了无数的解决方案，并利用Galerkin方法获得了一些结构。“II”。边界值问题（3）-（4）。（3）以一般形式（1）。在“I”、wee seek time global classical solution and calculate the decay rate of it more precisely by improving the method used in the latter part of "I"。“III”。（2）-（4）和（2）-（4）-（5）。We obtain the solution u （t， x）= t-D1--D1f （t， x）+v（t， x） by improving the method in“I”and“II”where（t， s） is an almost periodic function and E[v]=0（t-D1-D1）。我们都知道（2）-（4）-（5）的任何解决方案w（t， s）。decays faster than or equal to t D1-D1。自从你被规范为混合问题的解决方案（2）-（4）-（5），从你的衰变特性到（2）-（4）-（5）解决方案的最大和最小衰变率（2）-（4）-（5）实际上等于T D1-β D1。“IV”。We consider the following wave equation with nonlinear dissipation. Utt-Δu+u y D13 y D2 t y D2=g（t， x） ...（6）By applying the method used in“III”， we show that the decay estimate of the solution to the mixed problem to （6） （M. Nakao） is optimal。Less（低）

The investigators has researched the project and we have obtained the following results. For a bounded domain Ω⊂RィイD1nィエD1 with smooth boundary∂Ωand (t,x)∈[0,∞)×Ωwe considerUィイD2ttィエD2-Δu+μu=αuィイD1mィエD1、m=2,3,...,α∈R, μ>0 (Spontaneous break down of symmetry of neutral scalar field with self-interaction) ...(1) UィイD2ttィエD2-Δ+2β(t+T)ィイD1-1ィエD1uィイD2tィエD2=αuィイD2mィエD2, β、T>0,(Euler-Poisson-Darboux type of equation) ...(2) UィイD2ttィエD2-Δu+(μ+β(β+1)(1+t)ィイD1-2ィエD1+2γβ(1+t)ィイD1-1ィエD1)u=αuィイD1mィエD1,μ=λィイD21ィエD2+γィイD12ィエD1, β∈R, λィイD21ィエD2:first eigen value of -Δ ...(3) U=0 on [0,∞)×∂Ω (Dirichlet condition) ...(4) U=φ(x), UィイD2tィエD2=φ(x) at t=0. (Initial conditions) ...(5)1. Boundary value problem (1)-(4). Let μ=λ+γィイD12ィエD1 and λbe an eigen value of -Δ. Under some condition on m,γ, n and μ, we succeeded in obtaining a time global classical solution satisfying eィイD1γtィエD1U→φ(x) for an eigen function corresponding to λ. v(t, x)=u(t, x)-eィイD1-γtィエD1φ(x) is obtained by solving a reduced problem in v … More backward in time. In this process 'Singular hyperbolic operator' plays an important role.Next, based on this method, we succeeded in constructing infinitely many solutions and obtaining some structure of them by Galerkin method."II". Boundary value problem(3)-(4). (3) is in the general form of (1). Taking μmuch smaller than in " I", wee seek time global classical solution and calculate the decay rate of it more precisely by improving the method used in the latter part of "I"."III". (2)-(4) and (2)-(4)-(5). We obtain the solution u (t, x)=tィイD1-βィエD1f (t, x)+v(t, x) by improving the method in "I "and "II" where(t, s) is an almost periodic function and E[v]=0(tィイD1-βィエD1). It is well known that any solution w(t, s) of (2)-(4)-(5). decays faster than or equal to tィイD1-βィエD1. Since u is regarded as the solution of the mixed problem (2)-(4)-(5), from the decay property of u it is followed that the maximal and minmal decay rates of the solutions of (2)-(4)-(5) are exactly equal to tィイD1-βィエD1."IV". We consider the following wave equation with nonlinear dissipation.Utt-Δu+uィイD13ィエD1ィイD2tィエD2=g(t, x) ...(6)By applying the method used in "III", we show that the decay estimate of the solution to the mixed problem to (6) (M. Nakao) is optimal. Less