| From 1900s, M. Nakao and R. Ikcehata have investigated on semilinear wave equations in exterior domains. By use of multiplier method, M. Nakao solved the problem on boundary of domain, but index a had a strong restriction. R. Dcehata derived energy decay of a radically symmetric solution in the exterior domain, but the results was not successful in N (N>2) dimensional Euclidean spaces. This paper means to expand the work of M. Nakao and R. Dcehata, and studies on fourth order wave equations further.In chapter 1, the background and current advancement of wave equations in exterior domain are introduced, and some results concerned are given.Chapter 2 is concerned with exterior problem of a damped wave equation. Firstly, total energy decay for linear damped wave equation is derived. This can be applied to the proof the global existence forsemilinear wave equation with nonlinear term |u|~α u satisfying1<α≤ 4/[N - 4]~+. For this purpose we shall deal with a radicaDysymmetric solution in N-D (3 ≤ N ≤ 7) exterior domain.In chapter 3, we consider fourth order wave equations in exteriordomains with nonlinearity f(u) like | u |~α u, α > 0. Firstly, we derivedenergy decay in general exterior domain without any geometrical condition imposed on the boundary. Furthermore, if the regularity of the initial data is improved, we can derive a better decay results.In chapter 4, using multiplier method and weighted function method, we derived local energy decay of the wave equations with localized dissipation while the condition of compact support of original data was removed. |
