2000 Fiscal Year Final Research Report Summary
Stabilization problem for nonlinear wave eq
Project/Area Number 
10440053

Research Category 
GrantinAid for Scientific Research (B).

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  KYUSHU UNIVERSITY 
Principal Investigator 
NAKAO Mitsuhiro KYUSHU UNIVERSITY, Mathematics, Professor, 大学院・数理学研究院, 教授 (10037278)

CoInvestigator(Kenkyūbuntansha) 
SHIBATA Yoshihiro Waseda Univ., Mathematics, Professor, 理工学部, 教授 (50114088)
KATO Hisaku KYUSHU UNIVERSITY, Mathematics, Professor, 大学院・数理学研究院, 教授 (00038457)
KAWASHIWA Shuichi KYUSHU UNIVERSITY, Mathematics, Professor, 大学院・数理学研究院, 教授 (70144631)

Project Period (FY) 
1998 – 2000

Keywords  Nonlinear wave / Global existence / Energy decay / Smoothing effect / Exterior domain 
Research Abstract 
The head investigator. Nakao. has considered the stabilization problem for the nonlinear wave equations in interior and exterior domains and also the behaviours of solutions for nonlinear heat equations. For exterior problems. we first proved the local energy decay for linear wave equations. and on the basis of this we have derived L^p estimates. Further. by use of these estimates we have discussed on the global existence of semilinear wave equations. We note that in our argument no geometrical conditions on the shape of obstacles. For interior problems. we have proved global existence of smooth solutions for the quasilinear wave equations with a very weak dissipative term. In this procedure we have showed a unique continuation property for the wave equation with a variable coefficient. Nakao's inequality was used for the decay estimate. which is an originality of this paper. Concernibg nonlinear heat equations we have treated meancurvature type and mlaplacian type quasilinear equations under various nonlinear perturbations. We have derived sharp estimates of solutions including asymptotics as t → ∞ and smoothing effects near t = O. Investigator Kawashima has mainly treated the equations concerning gas dynamics and shown many interesting results. Investigator Shibata has considered the viscoelastic wave equations and also exterior problems concerning fluid dynamicsand prove many new results by use of spectral analysis. Investigator Kato has discussed on the global solutions of a nonNewtonian flow equation.

Research Products
(13 results)