Global well-posedness and continuity results for fourth-order parabolic equations with time-integral nonlinearity

This paper studies a fourth-order parabolic equation with memory effects in a bounded domain. We introduce new weighted Lebesgue spaces and use improved $ L^p $-$ L^q $ estimates for analytic semigroups to prove that mild solutions exist globally, are unique, and have the required regularity under both global and local Lipschitz conditions on the nonlinear source term. Moreover, we rigorously show that as the memory parameter tends to zero, the solutions of the equation with memory converge to those of the corresponding memory-free equation. Finally, we derive precise gradient estimates and apply our results to classical models such as the Cahn–Hilliard and Kuramoto–Sivashinsky equations with memory. Our methods and ideas can also be extended to a wide variety of other problems involving parabolic equations.