The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces (Q998149)

The authors introduce for weakly dissipative problems (in particular for weakly damped non-autonomous hyperbolic equation) a new class of functions, which are more general than translation compact so far used in the study of long-time behaviour of non-autonomous equations of mathematical physics. Subsequently, they study the uniform attractors for weakly damped non-autonomous hyperbolic equations with this new class (satisfying so-called \(C^*\)-condition) of time-dependent external forces \(g(t,x)\) and prove the existence of the uniform attractors for the equation \[ {\partial^2 u\over\partial t^2}+\alpha{\partial u\over\partial t}- \Delta_x u+ f(u)= g(t, x),\quad u|_{\partial\Omega}= 0, \] \[ u(\tau,x)= u_\tau(x),\;\partial_t u(\tau, x)= p_\tau(x),\quad \alpha> 0, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) and \(f\), \(g\), \(u_\tau\), \(p_\tau\) satisfying some natural conditions.