TITLE:
Long-Time Behavior of Solutions to the Classical Diffusion Equation with a Time-Dependent Memory Kernel
AUTHORS:
Xuan Wang, Xinyu Ma
KEYWORDS:
Classical Reaction-Diffusion Equation, Time-Dependent Memory Kernel, Time-Dependent Global Attractors, Supercritical Nonlinearity
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.1,
January
15,
2026
ABSTRACT: This paper investigates a class of classical reaction-diffusion equations with a time-dependent memory kernel, where the nonlinear term satisfies a supercritical growth condition. Under a new theoretical framework, we avoid the use of Sobolev embeddings in the treatment of the supercritical nonlinearity. Since the Sobolev control fails in the supercritical case, we establish the well-posedness of solutions by means of integral-type energy estimates combined with the intrinsic structure of the system. Furthermore, by employing a refined solution decomposition technique, we prove the existence and regularity of a time-dependent global attractor. Our results extend the existing conclusions for reaction–diffusion equations with subcritical or critical nonlinearities.