Well-posedness and global attractor for wave equation with nonlinear damping and super-cubic nonlinearity

TL;DR

This work addresses the long-time behavior of a semilinear wave equation with nonlinear damping $g(u_t)$ and a super-cubic nonlinearity $f(u)$ in a 3D bounded domain. It develops space-time a priori estimates following Todorova's approach to overcome the lack of Strichartz estimates in the nonlinear damping setting, establishes existence and uniqueness of weak and strong solutions within an expanded exponent region (Region III), and proves the existence of a global attractor in $\mathcal{H}=H^1_0(\Omega)\times L^2(\Omega)$. Furthermore, it proves that the global attractor is regular, i.e., bounded in $\mathcal{V}=(H^2(\Omega)\cap H^1_0(\Omega))\times H^1_0(\Omega)$, using a uniform strong-solution bound, partial regularity of trajectories, and a finite-cutoff decomposition around equilibria. These results advance understanding of the asymptotic dynamics for nonlinear damped wave equations and establish robust regularity properties of attractors under broader growth conditions.

Abstract

This study investigates a semilinear wave equation characterized by nonlinear damping $g(u_t) $ and nonlinearity $f(u)$. First, the well-posedness of weak solutions across broader exponent ranges for $g$ and $f$ is established, by utilizing a priori space-time estimates. Moreover, the existence of a global attractor in the phase space $H^1_0(Ω)\times L^2(Ω)$ is obtained. Furthermore, it is proved that this global attractor is regular, implying that it is a bounded subset of $(H^2(Ω)\cap H^1_0(Ω))\times H^1_0(Ω)$.

Figures (1)

• Figure 1: Well-posedness region of the weak solution

Theorems & Definitions (33)

• Definition 2.1: Weak solution
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4: Gronwall's lemma
• Lemma 3.1: The a priori estimate of weak solutions
• proof
• Remark 3.2
• Theorem 3.3: Uniqueness of the weak solution