Long-term behavior for wave equation with nonlinear damping and super-cubic nonlinearity

TL;DR

The paper studies a semilinear damped wave equation on the 3-torus with nonlinear damping $g(u_t)$ and super-cubic nonlinearity $f(u)$ under a time-independent forcing. It combines Strichartz estimates in a lower-regularity space, energy methods, and a quasi-stability framework to prove well-posedness with extra regularity, and to establish both a global attractor and an exponential attractor for the associated semigroup, with the exponential attractor having finite fractal dimension and exponential attraction rates. Key contributions include extending the admissible exponent ranges for well-posedness and long-time behavior, deriving a detailed quasi-stability inequality, and proving Hölder continuity of trajectories via asymptotic regularity. These results advance the understanding of long-time dynamics for nonlinear damped wave equations and provide quantitative attraction rates essential for qualitative analysis of the system.

Abstract

In this paper, we consider the semilinear wave equation involving the nonlinear damping term $g(u_t) $ and nonlinearity $f(u)$. The well-posedness of the weak solution satisfying some additional regularity is achieved under the wider ranges of the exponents $g$ and $f$. Moreover, the existence of global attractor and exponential attractor are proved.

Theorems & Definitions (22)

• Definition 2.1
• Lemma 2.2: Strichartz estimate
• Definition 2.3: Regular solution
• Theorem 2.4: Well-posedness of regular solution in region IV
• proof
• Theorem 3.1
• Theorem 4.1
• Definition 4.2: chueshov15, Definition 3.4.1
• Theorem 4.3: chueshov15, Theorem 3.4.7
• Theorem 4.4: Quasi-stablity