Weak-space asymptotic behavior without kinetic-energy damping

Characterize the asymptotic behavior in the weak energy space H_0^1(Ω) × L^2(Ω) of the nonlocal degenerate wave equation u_tt − Δu + ||∇u(t)||^2u_t + f(u) = 0 with Dirichlet boundary conditions and without a kinetic-energy term in the damping coefficient, for weak initial data.

Background

The paper identifies models whose damping coefficient depends only on the potential-energy quantity ||∇u(t)||2, omitting the kinetic-energy term ||u_t(t)||2, as especially difficult. Standard multiplier arguments do not directly recover the missing kinetic control or the decay inequalities needed for asymptotic analysis.

For the displayed degenerate model studied by Cavalcanti et al., decay was established for more regular initial data in (H2(Ω) ∩ H_01(Ω)) × H_01(Ω). The unresolved question concerns asymptotic behavior in the weaker natural energy space H_01(Ω) × L2(Ω) under weak initial data.

References

By contrast, results concerning the asymptotic behavior of the nonlocal degenerate model model-cavalcanti in the weak topological space $H1_0(\Omega) \times L2(\Omega)$, under weak initial data, remain an open question.

model-cavalcanti:

utt−Δu+∥∇u(t)∥2ut+f(u)=0,u_{tt} - \Delta u + \|\nabla u(t)\|^2 u_t + f(u) = 0,

— Global Well-Posedness and Asymptotic Behavior of Energy-Damped Subquintic Wave Equations  (2609.29778 - Lasiecka et al., 24 Sep 2026) in Section 1, subsection “General comments,” subsection “Quintic and subquintic forcings”

However, the extension of the approach developed in to nonlinear nonlocal dissipative models of the form considered here remains largely open in the critical quintic regime.

— Global Well-Posedness and Asymptotic Behavior of Energy-Damped Subquintic Wave Equations  (2609.29778 - Lasiecka et al., 24 Sep 2026) in Section 1, subsection “How do we handle the problem?”