2011/07/13 by Martino Prizzi, Prizzi, Martino
Computer Science · Engineering · Mathematics · #35B41 #35L70 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1107.2589
openalex publication_date 2011/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Under fairly general assumptions, we prove that every compact\ninvariant set mathcal I of the semiflow generated\nby the semilinear damped wave equation\n\n
beginalignat2νtt+
alpha ut+
beta(x)u-
Delta uamp; =f(x,u),\namp;
quadamp;(t,x)
in[0,+
infty[
times
Omega,\n
\nuamp;=0,amp;
quad amp;(t,x)
in[0,+
infty[
times
partial
Omega,\n
endalignat\n\nin H10(\Ω)\× L2(\Ω) has finite Hausdorff and\nfractal dimension. Here \Ω is a regular, possibly unbounded,\ndomain in mathbb R3 and f(x,u) is a nonlinearity of critical\ngrowth. The nonlinearity f(x,u) needs not to satisfy any\ndissipativeness assumption and the invariant subset mathcal I\nneeds not to be an attractor. If f(x,u) is dissipative and\n mathcal I is the global attractor, we give an explicit bound on\nthe Hausdorff and fractal dimension of mathcal I in terms of\nthe structure parameters of the equation.