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

非线性波动方程边值问题（内域）的时间全局经典解

• 批准号: 10640191
• 负责人: KUBO Akisato
• 金额: $ 1.09万
• 依托单位: FUJITA HEALTH UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640191/
• 关键词: Semillinear wave equation; Time global classical solution; Decay rate; Boundary value problem; Optimality; Euler-Posson-Darboux; Galerkin's method; Spontaneous break down; 時間大域解; 古典解; 減衰解; 混合問題; オイラー・ポアソン・ダルブー; 半線型波動方程式; ディリクレ条件; 中性スカラー場の自発

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

调查人员对该项目进行了研究，我们得到了以下结果。对具有光滑边界的有界区域Ω D1 n D1，（t，x）∈[0，∞）×Ω，我们有D2 tt D1 m D1，m= 2，3，.，α∈R，μ>0（具有自相互作用的中性标量场对称性的自发破缺）. (1)U D2tt D2-Δ+2β（t+T）D1-1 D1u D2t D2=αu D2m D2，β，T>0，（Euler-Poisson-Darboux型方程）. (2)U_u_d ~ 2_t_t_d ~ 2-Δu+（μ+β（β+1）（1+ t）_d ~ 1 -2_t_d ~ 1 -1_t_d ~ 1）u=α_u_d ~ 1_m_d ~ 1，μ=λ_d ~ 21_d ~ 2_t_d ~ 2 +γ_d ~ 12_d ~ 1，β∈R，λ_d ~ 21_d ~ 2_d ~ 2：-Δ的第一本征值. (3)U=0 on [0，∞）× <$Ω（Dirichlet condition）. (4)U=φ（x），U = φ（x），t=0时，U =φ（x），U = D2 t =φ（x）。（初始条件）.（5）1.边值问题（1）-（4）.设μ=λ+γ D_（12）D_（1），λ是-Δ的一个特征值.在m，γ，n和μ的一定条件下，我们成功地得到了一个时间整体经典解，该解对于λ对应的特征函数满足e ∈ D1γt ∈ D1 U →φ（x）. v（t，x）=u（t，x）-e ∈ D1-γt ∈ D1φ（x）是通过求解v中的一个约化问题得到的 ...更多信息 在时间上倒退。其次，在此基础上，利用Galerkin方法成功地构造了无穷多个解，并得到了解的一些结构。“二”边值问题（3）-（4）. (3)其一般形式为（1）。取比“I”中小得多的μ，通过改进“I”后半部分的方法，求出了时间整体经典解，并更精确地计算了衰减率。“三”。（2）-（4）和（2）-（4）-（5）。通过改进“I“和“II”中的方法，得到解u（t，x）=t D1-β D1 f（t，x）+v（t，x），其中（t，s）是概周期函数，E[v]=0（t D1-β D1）.众所周知，（2）-（4）-（5）的任何解w（t，s）。衰减速度大于或等于t D1-β D1。由于u被看作是混合问题（2）-（4）-（5）的解，由u的衰减性质可以得出：（2）-（4）-（5）的解的最大和最小衰减率恰好等于t D 1-β D 1.“四”考虑具有非线性耗散项的波动方程Utt-Δu+u_（？）D_（13）D_（？）D_（1？）D_（2？）(6)By应用在“III”中所用的方法，我们证明了混合问题（6）（M. Nakao）是最佳的。少

项目成果

KUBO A.: "On the existence of a global solution of the boundary value problem for □u-μu+auィイD1mィエD1=0 in the interior domain."Mathematical Methods in the Applied Sciences. 21. 781-795 (1998)

KUBO A.：“关于内部域中 □u-μu+auiD1mieD1=0 边值问题的全局解的存在性。”应用科学中的数学方法 21. 781-795 (1998)。

Hoshino, H.: "Non negative global solutions to a class of strongly coupled reaction-diffusion systems."Advanced in Differential Equations. (To appear).

Hoshino, H.：“一类强耦合反应扩散系统的非负全局解。”微分方程高级。

星野弘喜: "Nonnegative global solutions to a class of strongly coupled reaction - diffusion systems"To appear in Advanced in Differential Equations.

Hiroki Hoshino：“一类强耦合反应-扩散系统的非负全局解”出现在《高级微分方程》中。

KUBO A.: "On the spherically symmetric solution to the mixed problem for a weakly hyperbolic equation of second order."Publ. RIMS.Kyoto Univ.. (To appear). (2000)

KUBO A.：“关于二阶弱双曲方程混合问题的球对称解”。

KUBO A.: "Asymptotic behavior and lower bounds for semilinear wave equations with a dissipative term in the interior domain."Proceeding of The fourth workshop on differential equations in Korea. 105-109 (1999)

KUBO A.：“内部域中具有耗散项的半线性波动方程的渐近行为和下界。”韩国第四届微分方程研讨会论文集。