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∂t2u−γ∂tΔxu−Δxu+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≡0ϕ≡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|5)|f(u)|⩽C(1+|u|5).