Weakly viscous nonlocal wave equations: forward theory, Lebesgue Runge approximation, and inverse problems

Abstract

We study forward and inverse problems for weakly viscous nonlocal wave equations with elastic term $(-Δ)^s u$ and dissipative term $(-Δ)^r\partial_tu$, where $0\leq r<s$. For time-dependent potentials in suitable $L^\infty(0,T;L^p(Ω))$ classes, we establish finite energy well-posedness, strong continuity in time, and an energy identity by parabolic regularization and time-regularization arguments. Under explicit nonlocality and nonresonance assumptions on the fractional orders, we prove Runge approximation in $L^2(0,T;\widetilde H^r(Ω))$ and hence in $L^2(Ω\times(0,T))$. As an application, partial exterior Dirichlet-to-Neumann measurements uniquely determine the potential in all three exponent regimes of the Runge theorem, including integer elastic orders with noninteger dissipative orders. The recovery argument combines equality of solutions with the same exterior input, unique continuation or the entanglement principle, and a single Lebesgue-space Runge approximation. It applies to the full admissible potential class, without spatial boundedness, sign, or smallness assumptions.