Well-posedness for a deep-sea mining pipeline equation with nonlocal stiffness

We study a fourth-order evolution equation describing the transverse vibrations of a deep-sea mining pipeline with a nonlocal stiffness coefficient. The bending rigidity depends on the total deformation of the pipeline through the integral $$ \int_0^L |v_s(\sigma,t)|^2d\sigma. $$ The equation also contains axial loading, gyroscopic effects, submerged weight and linear velocity damping. We prove local-in-time existence of weak solutions by the Faedo-Galerkin method, using energy estimates, compactness arguments and the positivity of the nonlocal stiffness coefficient. We also derive a continuation criterion and establish a small-energy continuation result, which yields existence on arbitrary prescribed finite time intervals under a suitable smallness condition on the initial energy. Under additional regularity and compatibility assumptions, we prove the existence of local strong solutions. Finally, we establish uniqueness in the strong class and prove continuous dependence on the initial data in a regular energy class. For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/85/abstr.html