Well-posedness and continuity results for a pseudo-parabolic equation with memory term

In this paper, we are interested in studying a pseudo-parabolic equation with a memory term. This type of equation has lots of appearance in many fields, such as: physics, mechanical engineering, and heat conduction theory. For the homogeneous case, we prove well-posedness and convergence as the memory parameter $ b \to 0 $. In the global Lipschitz case, we establish the global existence and uniqueness of mild solution by using the Banach fixed point theorem and show convergence as $ b \to 0 $. Furthermore, when the nonlinear source is locally Lipschitz, we extend the solutions to the interval $ [0, \infty) $ and establish the existence of a global strong solution under specific conditions. In this context, the primary techniques for obtaining mild solutions involve first proving the global existence of solutions to the truncated problem and then analyzing the convergence of these truncated solutions using Aubin - Lions' compactness theorem to achieve a global solution to our problem. To the best of our knowledge, this paper is one of the first result to consider the global strong solution to pseudo-parabolic equations with a memory term.