Exponential growth of solution for a couple of semi-linear pseudo-parabolic equations with memory and source terms

This work is concerned with coupled semi-linear pseudo-parabolic equations with memory terms in both equations, associated with the homogeneous Dirichlet boundary condition. We show that the solution grows exponentially under specific conditions regarding the relaxation functions and initial energy. In order to prove the result, we use the energy method based on the construction of a suitable Lyapunov function.The most important behavior of the evolution system is the exponential growth phenomena because of its wide range of applications in modern science, such as chemistry, biology, ecology, and other areas of engineering and physical sciences.