Finite-dimensional attractors for the quasi-linear strongly-damped wave equation

Abstract

We present a new method of investigating the so-called quasi-linear strongly-damped wave equations partial derivative(2)(t)u - gamma partial derivative(t)Delta(x)u - Delta(x)u + f(u) = del(x).phi'(del(x)u) + g in bounded 3D domains. This method allows us to establish the existence and uniqueness of energy solutions in the case where the growth exponent of the non-linearity phi is less than 6 and f may have arbitrary polynomial growth rate. Moreover, the existence of a finite-dimensional global and exponential attractors for the solution semigroup associated with that equation and their additional regularity are also established. In a particular case phi equivalent to 0 which corresponds to the so-called semi-linear strongly-damped wave equation, our result allows to remove the long-standing growth restriction vertical bar f(u)vertical bar <= C(1 + vertical bar u vertical bar(5)).