Abstract:

This paper presents a discussion on the long-time dynamical behavior of solutions for the classical reaction-diffusion equation with time-dependent memory kernel when nonlinear term adheres to subcritical growth and the external force term $g(x,t) $ belongs to the space $ L^{2}_{{\mathrm{loc}}}(\mathbb{R};L^{2}(\varOmega)) $ in the time-dependent space $ L^2(\varOmega)\times L_{\mu_{t}}^2(\mathbb{R}_{+}; H_{0}^1(\varOmega)) $. Within the new theorical framework, the well-posedness and the regularity of the solution, as well as the existence of the time-dependent pullback attractors are established. This is achieved by applying the delicate integral estimation method and decomposition techniques.

Key words: classical reaction-diffusion equation, time-dependent memory kernel, well-posedness, time-dependent pullback attractors, existence