Well-posedness for nonlocal damping exponents at or below minus one

Establish a well-posedness theory for the abstract viscoelastic wave equation with nonlocal damping term \(\|u_t\|^{\rho}u_t\) in the regime \(\rho\leq -1\), where the damping operator is no longer continuous and monotone at the origin.

Background

The paper proves global well-posedness and uniqueness for the abstract viscoelastic wave equation only when $\rho>-1$. In that range, the nonlocal damping operator $l(v)=\|v\|^{\rho}v$ is continuous, monotone, and semicontinuous, which permits the use of the Faedo–Galerkin method and monotone-operator arguments.

For $\rho\leq -1$, the damping operator loses the continuity and monotonicity properties required by the paper’s approach at the origin. Although the dissipation rate may remain nonnegative for part of this range, the authors explicitly leave the development of a corresponding well-posedness theory unresolved.