Feb 3, 1976 - As is well known, $\phi_{j}\in C^{4m+[n/2]+1}(\overline{\Omega})$ ...... $+\Vert\nu P_{\gamma}f(\cdot, 0)+P_{\gamma}D_{t}f(\cdot, 0)$ ...
J. Math. Soc. Japan Vol. 30, No. 3, 1978
Bounded, periodic and almost periodic
classical solutions of some nonlinear wave equations with a dissipative term By Mitsuhiro NAKAO (Received Feb. 3, 1976) (Revised June 20, 1977)
Introduction. This paper deals with the questions of the existence and asymptotics of the bounded, periodic and almost periodic classical solutions for the equations (E)
$\frac{\partial^{2}}{\partial t^{2}}u+\sum_{||\alpha|.\beta^{|\leqq m}}(-1)^{|\alpha|}D^{\alpha}(a_{\alpha,\beta}(x)D^{\beta}u(x, t))+\nu\frac{\partial}{\partial t}u$
$+F(x, t, u)=0$
on
(or
$\Omega\times R$
$\Omega+R^{+}$
)
together with the boundary condition (B)
$D^{\alpha}u|_{\partial\Omega}=0$
for
$|\alpha|\leqq m-1$
,
is a bounded domain in n-dimensional Euclidwhere is a positive constant, ean space and its boundary. We use the following notations $\Omega$
$\nu$
$R^{n}$
$\partial\Omega$
$\alpha=(\alpha_{1}, \cdots, \alpha_{n})$
,
$|\alpha|=\sum_{i=1}^{n}|\alpha_{i}|$
$x=(x_{1}, \cdots, x_{n})$
,
$D^{\alpha}=\prod_{i=1}^{n}(\frac{\partial}{\partial x_{i}})^{\alpha_{i}}$
,
etc.
The functions to be considered are all real valued and throughout the paper we make the following assumptions: $H_{1}$
.
$A=\sum_{||\alpha|,\beta|\leqq m}(-1)^{1a1}D^{\alpha}(a_{\alpha},\mu_{X})D^{\beta}u)$
is formally selfadjoint and coercive on constants such that $c_{0},$
,
$i$
.
$e.$
, there
exist some positive
$c_{1}$
c’ where we put
$\mathring{H}_{m}$
$\Vert u\Vert_{H_{m}}^{2}0\geqq\langle Au, u\rangle\geqq c_{0}^{2}\Vert u\Vert_{\dot{H}_{m}}^{2}$
for
$u\in\mathring{H}_{m}$
M. NAKAO
376 \langle Au,
$ v\rangle$
$=\sum_{||\alpha|.\beta^{|\leqq m}}\int_{\rho}a_{\alpha\beta}(x)D^{\beta}u(x)D^{\alpha}v(x)dx$
,
$v\in H_{m}^{\circ}$
$u,$
and $\Vert u\Vert_{\mathring{H}_{m}}=(\sum_{|\alpha|\leq m}\int_{\Omega}|D^{\alpha}u|^{2}d_{X})^{1/2}$
$H_{2}$
.
$F(x, t, u)$
is of the form $F(x, t, u)=g(x, u)+f(x, t)$
and
$g,$
$f$
satisfy the conditions: $g\in C^{\max([n/2]+1,3)}(\overline{\Omega}\times R)$
and where
$|\frac{\partial^{j}}{\partial u^{j}}g(x, u)|\leqq K_{0}\sum_{i=12}.|u|^{\max(r_{i}+1-j.0)}$
$(j=0,1,2,3)$
are constants such that
$r_{i}(i=1,2)$
if
$ 0
