Coupled Mkdv with Memory: Existence and Exponential Decay

Abstract

We investigate a coupled system of modified Korteweg-de Vries (MKdV) equations with memory effects through convolution integrals with relaxation kernels. Under suitable assumptions on the memory kernel positivity, monotonicity, and exponential decay we establish two main results. First, using Faedo-Galerkin approximations together with a priori estimates and compactness arguments, we prove local existence of weak solutions. Second, we construct an appropriate Lyapunov functional to exploit the dissipative nature of the memory terms, from which we derive an exponential decay estimate for the total energy. Our analysis shows how memory influences both solvability and long-term dynamics in this coupled dispersive system.