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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:247

Article

Kalantarov, Varga¹  Zelik, Sergey² 

[1] Koc Univ, Dept Math, TR-34450 Istanbul, Turkey

[2] Univ Surrey, Dept Math, Guildford GU2 7XH, Surrey, England

关键词: Quasi-linear strongly-damped wave equation;    Energy solutions;    Uniqueness;    Regularity;    Global attractor;   

DOI  :  10.1016/j.jde.2009.04.010

来源: Elsevier

【 摘 要 】

We present a new method of investigating the so-called quasi-linear strongly-damped wave equations partial derivative(2)(t)u - gamma partial derivative(t)Delta(x)u - Delta(x)u + f(u) = del(x).phi'(del(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 phi 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 phi equivalent to 0 which corresponds to the so-called semi-linear strongly-damped wave equation, our result allows to remove the long-standing growth restriction vertical bar f(u)vertical bar <= C(1 + vertical bar u vertical bar(5)). (C) 2009 Published by Elsevier Inc.

【 授权许可】

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