Dimension of attractors and invariant sets of damped wave equations in unbounded domains

zbMATH Open1292.35058arXiv1107.2589MaRDI QIDQ378462

Publication date: 11 November 2013

Published in: Topological Methods in Nonlinear Analysis (Search for Journal in Brave)

Abstract: Under fairly general assumptions, we prove that every compact invariant set

m a t h c a l I

of the semiflow generated by the semilinear damped wave equation u_{tt}+alpha u_t+�eta(x)u-Deltau = f(x,u), (t,x)in[0,+infty[ imesOmega, u = 0, (t,x)in[0,+infty[ imespartialOmega in

H_{0}^{1} (O m e g a) i m e s L^{2} (O m e g a) h a s f i n i t e H a u s d o r f f a n d f r a c t a l d i m e n s i o n . H e r e

Omega

i s a r e g u l a r, p o s s i b l y u n b o u n d e d, d o m a i n i n

R^3

a n d

f(x,u)

i s a n o n l i n e a r i t y o f c r i t i c a l g r o w t h . T h e n o n l i n e a r i t y

f(x,u)

n e e d s n o t t o s a t i s f y a n y d i s s i p a t i v e n e s s a s s u m p t i o n a n d t h e i n v a r i a n t s u b s e t

mathcal I

n e e d s n o t t o b e a n a t t r a c t o r . I f

f(x,u)

i s d i s s i p a t i v e a n d

mathcal I

i s t h e g l o b a l a t t r a c t o r, w e g i v e a n e x p l i c i t b o u n d o n t h e H a u s d o r f f a n d f r a c t a l d i m e n s i o n o f

mathcal I$ in terms of the structure parameters of the equation.

Full work available at URL: https://arxiv.org/abs/1107.2589

zbMATH Keywords

Hausdorff dimension fractal dimension critical growth

Mathematics Subject Classification ID

Attractors (35B41) Initial-boundary value problems for second-order hyperbolic equations (35L20) Second-order semilinear hyperbolic equations (35L71)