关于具有时间依赖记忆核的经典反应扩散方程, 当非线性项满足次临界增长, 外力项$ g(x, t)\in $$ L^{2}_{{\mathrm{loc}}}(\mathbb{R};L^{2}(\varOmega)) $时, 在时间依赖空间$ L^{2}(\varOmega)\times L_{\mu_{t}}^2(\mathbb{R}_{+}; H_{0}^1(\varOmega)) $中讨论了解的长时间动力学行为. 在新的理论框架下, 利用积分估计方法以及分解技术证明了解的适定性和正则性, 进而证明了时间依赖拉回吸引子的存在性.

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.