Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities

By Sergey Zelik. 2004

70 citations recorded by the source corpus.

We study the asymptotic behavior of weak energy solutions of the following damped hyperbolic equation in a bounded domain $\Omega\subset\R^3$: $\varepsilon\partial_t^2u+\gamma\partial_t u-\Delta_x u+f(u)=g,\quad u|_{\partial\Omega}=0,$ where $\gamma$ is a positive constant and $\varepsilon>0$ is a small parameter. We do not make any growth restrictions on the nonlinearity $f$ and, consequently, we do not have the uniqueness of weak solutions for this problem. We prove that the trajectory dynamical system acting on the space of all properly defined weak energy solutions of this equation possesses a global attractor $\mathcal A_\varepsilon^{tr}$ and verify that this attractor consists of global strong regular solutions, if $\varepsilon>0$ is small enough. Moreover, we also establish that, generically, any weak energy solution converges exponentially to the attractor $\mathcal A_\varepsilon^{tr}$.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/de3fee80f6d08ac3f1ead08c902552f706ce2fa1

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In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

On a class of degenerate parabolic equations with dynamic boundary conditions (2011, background): “According to general theory [14, 38, 56], we are only required to check that the trajectory dynamical system is continuous and that it possesses a compact and M-bounded absorbing set.”
Chapter 3 Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains (2008, methodology): “In this example, we consider, following [240], an alternative way to construct a trajectory attractor for the damped wave equation (6.”