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

The investigators已经研究出了一个项目，我们已经获得了追随者的成果。bounded domain Ω⊂R与smooth boundary∂Ωand (t,x)∈[0,∞)×Ωweconsideruィイd2ttィエd2-狄拉克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）U - D2tt D2-Δ+2β(t+ t) U -1 D1u - D2t - D2=α U - D2m - D2，β， T>0,（Euler-Poisson-Darboux type of equation）…(2)uィイd 2 ttィエd 2 -狄拉克δu +(μ+β(β+ 1)(1 + t)ィイd 1 - 2ィエd 1 + 2γ-β(1 + t)ィイd 1 - 1ィエd 1) u =αuィイd 1米ィエd, 1μ= 21颗心ィイdィエd 2 +γィイd 12ィエd 1,β∈r,颗心ィイd21ィエ以及可转换成d2: first eigen value of -狄拉克δ……(3)u = on[,∞)×∂ω- (dirichlet condition)……(4)u =φ主要(x),i D2t i D2=φ(x) at t=0. （Initial conditions）…(5)1. Boundary value problem (1)-(4). Letμ=λ+γ - D12 and λbe an eigen value of -Δ. Under some condition on m,γ, n and μ，我们succeeded in obtaining a time global classical solution satisfying e - D1γt - D1U→φ(x) for an eigenfunction corresponding to λ。v(t, x)=u(t，x)-e D1-γt φ(x) is obtained by solving a reduced problem in v…in this process 'Singular hyperbolic operator' plays an important More backward in time. in this process 'Singular hyperbolic operator' plays an importantrole.Next, based on this methodwe succeeded in constructing infinitely many solutions and obtaining some structure of them byGalerkin method."II". Boundary value problem(3)-(4). (3) is in the general form of (1). Taking μmuchsmaller than in " I"，wee seek time global classical solution and calculate the decay rate of it more precisely byimproving the method used in the latter part of "I"."III". (2)-(4) and (2)-(4)-(5). We obtain the解决方案u (t, x)=t 1-β D1f (t, x)+v(t, x) by improving the method in "I "and "II" where(t，s)是一个特定的periodic function and E[v]=0(t - b1 -β D1).这是一个很好的已知任何解决方案w(t，s) of (2)-(4)-(5). decays faster than or equal to t - 1-β D1.因为它是regarded as the solutionof the mixed problem (2)-(4)-(5)，from the decay property of u it is followed that the maximal and minmal decay rates of the solutionsof (2)-(4)-(5) are exactly equal to ti D1-β D1."IV". We consider the following wave equation with the following wave equationnonlinear dissipation.Utt-Δu+u - D13 - D2t - D2=g(t，x)…(6)By applying the method used in "III"，我们show that the decay estimate of the solution to the mixed problem to (6) （M. Nakao）是optimal.less

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