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 $${\partial }_t^{2}u-\gamma {\partial }_t{\Delta }_{x}u-{\Delta }_{x}u+f(u)={\nabla }_x\cdot {\varphi }^{\prime }({\nabla }_{x}u)+g$$ 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 $\varphi$ 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 $\varphi \equiv 0$ which corresponds to the so-called semi-linear strongly damped wave equation, our result allows to remove the long-standing growth restriction $|f(u)|\le C(1+|u{|}^5)$.

36 pages