Existence, regularity and asymptotic analysis for conformable parabolic equations with history-dependent diffusion

This paper investigates a diffusion equation incorporating memory effects and a conformable fractional derivative. We begin by deriving an explicit representation of the mild solution and analyzing its regularity properties. Next, we explore the continuous dependence of the solution on key parameters, highlighting its sensitivity to changes. As the order of the conformable derivative approaches one, we demonstrate that the mild solution converges to that of the classical diffusion equation. Furthermore, we establish a finite-time blow-up criterion under specific nonlinear conditions. Employing fixed-point theorems, we prove the existence and uniqueness of mild solutions and identify the conditions necessary to ensure these results. Our findings offer new insights into how conformable fractional derivatives and memory terms influence solution behavior and stability.

کلیدواژه ها:

conformable derivative ، Memory terms ، Mild solution ، Regularity ، Blow-up criterion