| Research Abstract |
The author have obtained several results concerning with some problems arising in our project.
(1) When the dissipation is effective near the boundary in an exterior domain, the author and Prof. Ikehata (Hiroshima Univ.) have obtained that the total energy and L^2 norm of solutions for the wave equations decay to 0 as time goes to infinity. For the Cauchy problem in the whole space there is a work of Kawashima-Nakao-One (J.Math.Soc.Japan, 1995). Combining the usual energy method with the L^p-L^q estimate, they discussed the energy decay. Contrary to this method, we investigated the energy decay or L^2 bound in L^2 framework. Further, our method can be applicable to derive the L^2 decay for the density for the compressible Navier-Stokes equations in R^3, which was done with T.Kobayashi (Kyushu Inst.Tech.) and R.Ikehata.
(2) We have proved that when the dissipation is effective around the boundary, the energy does not in general decay for some specified initial data. Intuitively speaking,
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if the dissipation is effective in the trapping region, then the wave would decay there. Otherwise, the wave would escape. For the proof, we employed the weighted energy method due to Prof.K.Mochizuki to derive the integrability of space-time integral of the local energy and utilized its estimate to discuss the energy nondecay . Furthermore, it is noteworthy mentioning that this estimate can be applicable to improve the decay rate of the local energy, which was derived by Prof.M.Nakao (J.Diff.Eq.1998).
(3) Employing the weighted energy method, the author investigated the existence of classical solutions for the nonlinear dissipative wave equations in an exterior star-shaped domain in R^3. Moreover, we proved also that the energy does not in general decay. Our crucial idea is to treat the nonlinar dissipative term not only as a dissipation but also a perturbation. In particular, when the domain is the whole space R^3, we derived that the local energy decays to 0 at a certain rate. Its rate can be determined by the decay rate of the dissipation coefficient. Utilizing the representing formula of solutions to the free wave equations in R^3, we have proved the local energy decay. Unfortunately, since we do not know the representing formula of solutions to the free wave equations in exterior domains, it is open whether the local energy decays or not in exterior problem. The author lectured these results in Semiar on Partial Differential Equations held at several Japanese Universities. Less
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