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

2008/07/31 by Kalantarov, Varga, Zelik, Sergey · 2 citations
#35Q20 #37L30 #73F15 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.0807.5078

Abstract

We present a new method of investigating the so-called quasi-linear strongly damped wave equations ∂_t²u-γ∂_tΔ_x u-Δ_x u+f(u)= ∇_x⋅ ϕ'(∇_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 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|⁵).

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