Department of Mathematics
College of Science
Northern Border University
Arar 91431, Saudi Arabia
email: ridha.selmi@nbu.edu.sa, ridhaselmiridhaselmi@gmail.com
Ridha Selmi
Abstract:
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.
Submitted July 24, 2026. Published October 26, 2026.
Math Subject Classifications: 5A01, 35A02, 35D30, 35G31, 35B30, 35B35, 35B40, 74F10, 74K10.
Key Words: Deep-sea mining pipeline; Euler-Bernoulli beam;
nonlocal stiffness; Kirchhoff-type coefficient; weak solution;
strong solutions; small-energy continuation; continuous dependence.
DOI: 10.58997/ejde.2026.85
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Ridha Selmi Department of Mathematics College of Science Northern Border University Arar 91431, Saudi Arabia email: ridha.selmi@nbu.edu.sa, ridhaselmiridhaselmi@gmail.com |
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