Finite-dimensional Attractors for the Quasi-linear Strongly-damped Wave Equation

We present a new method of investigating the so-called quasi-linear strongly damped wave equations ∂ t u− γ∂t∆xu−∆xu+ f(u) = ∇x · φ ′(∇xu) + 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 φ 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 φ ≡ 0 which corresponds to the so-called semi-linear strongly damped wave equation, our result allows to remove the long-standing growth restriction |f(u)| ≤ C(1 + |u|).