THE EXISTENCE OF SMOOTH ATTRACTORS OF DAMPED AND DRIVEN NONLINEAR WAVE EQUATIONS

by Björn Birnir and Kenneth Nelson

We consider a nonlinear wave equation of the form $u_{tt}-u_{xx} +\alpha u_t +g(u) = \gamma f(x,t)$ defined on a bounded domain $\Omega \subset {\rm {\bf R}}^3$, with initial conditions in ${\cal H}^1(\Omega)\times {\cal L}^2(\Omega)$. The nonlinear term g is assumed to have polynomial growth of degree less than 5, and f is a smooth driving function. If solutions to this equation satisfy periodic boundary conditions on $\Omega$, then there is a compact attractor of bounded sets of ${\cal H}^1(\Omega)\times {\cal L}^2(\Omega)$ under the dynamical system defined by this equation. This attractor lies in ${\cal H}^{m+1}(\Omega)\times{\cal H}^m(\Omega)$, where m depends entirely upon the smoothness of g. Additionally, we show that if g is a smooth function, then so is the attractor. These results generalize the work of other authors by establishing a series of estimates on the solution which shows that for any t>0, and an integrating factor $\zeta(t) = e^{\lambda t}$, then $\zeta u \in {\cal L}^r([0,t];{\cal L}^q(\Omega))$, as long as r and q satisfy the relationship $6 \leq r = \frac{2q}{q-6} < \infty$.