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

TL;DR

This work analyzes a three-dimensional damped wave equation with displacement-dependent damping and a super-cubic nonlinearity. The authors derive refined space-time a priori estimates, establish well-posedness of weak solutions via Faedo-Galerkin, and prove an energy equality that yields dissipativity. They then prove the associated semigroup is norm-to-weak continuous, show asymptotic compactness, and obtain the existence of a global attractor in the energy space $\mathcal{H}=H^1_0(\Omega)\times L^2(\Omega)$. Under an additional strong-damping condition, they prove the global attractor is also bounded in the higher-regularity phase space $\mathcal{V}=(H^2\cap H^1_0)\times H^1_0$, via a cutting-off decomposition and uniform strong-solution estimates. Collectively, the results illuminate the long-time behavior and regularity of solutions to nonlinear waves with displacement-dependent damping, with implications for the qualitative understanding of such dissipative systems in physics and engineering.

Abstract

This work investigates the semilinear wave equation featuring the displacement dependent term $σ(u)\partial_t u $ and nonlinearity $f(u)$. By developing refined space-time a priori estimates under extended ranges of the nonlinearity exponents with $σ(u)$ and $f(u)$, the well-posedness of the weak solution is established. Furthermore, the existence of a global attractor in the naturally phase space $H^1_0(Ω)\times L^2(Ω)$ is obtained. Moreover, the regularity of the global attractor is established, implying that it is a bounded subset of $(H^2(Ω)\cap H^1_0(Ω))\times H^1_0(Ω)$.

Theorems & Definitions (27)

• Definition 2.1: Weak solution
• Theorem 2.2: The a priori estimate of weak solution
• proof
• Remark 2.3
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Definition 3.3: zhong
• Proposition 3.4