Budget Amount *help 
¥1,700,000 (Direct Cost : ¥1,700,000)
Fiscal Year 1999 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1998 : ¥1,100,000 (Direct Cost : ¥1,100,000)

Research Abstract 
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 consider UィイD2ttィエD2Δu+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 selfinteraction) ...(1) UィイD2ttィエD2Δ+2β(t+T)ィイD11ィエD1uィイD2tィエD2=αuィイD2mィエD2, β、T>0,(EulerPoissonDarboux type of equation) ...(2) UィイD2ttィエD2Δu+(μ+β(β+1)(1+t)ィイD12ィエD1+2γβ(1+t)ィイD11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D1γtィエD1φ(x) is obtained by solving a reduced problem in v
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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D1β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D1β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D1β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D1βィエD1. "IV". We consider the following wave equation with nonlinear dissipation. UttΔu+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
