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

Abstract

We present a new method of investigating the so-called quasilinear strongly-damped wave equationsdomains. 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 ).

Large Time Stability Control for a Class of Quasilinear Parabolic and Hyperbolic Equations

“…Thus, we can discuss and conclude some results with lemmas in the next text [4][5][6]. It turns out that this has an influence on the control of weak solution with structural damping.…”

“…Li prove the existence of weak periodic decay strong solution on the periodicity condition, X.K. Su and JL Zhang [3][4][5][6] proved the controllability of a smooth solution in the method of Cauchy problems in the case of smooth and small data.…”

Weak Decay Stability Control on a Class of Quasilinear Wave Equations

“…Y.M. Qin, Ebihara, Xin Liu [1][2][3][4][5][6][7][8] [3][4][5][6][7][8][9][10]. TG Wang, Ming Zhang, MJ Wang simplified the above arguments and give the proof of control with exponential decay.…”

“…Make use of combining p L -theorem of Soblev space and semi group theorem of operators, Nakao [5] and T. Kato [6] devised certain decay rate of energy of global solutions with large data under a specific condition which is certainly satisfied if the mean curvature of the boundary ∂Ω is non-positive. For 1 n  and 1 f  , nonlinear elliptic equation with periodicity conditions was studied [7][8][9][10],…”

A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor

“…This equation is a generalization of an autonomous model which has been investigated in many articles by several authors (see, for example, [6,7,3,13,14,21,24,25]), with particular emphasis on its asymptotic behaviour. Assuming (critical) growth restrictions on the non-linear term f , the global existence of solutions in the autonomous case has been established in [6], based on the theory of ε-regular solutions.…”