In this paper, we consider the non-autonomous nonclassical diffusion equations on \(\mathbb{R}^{N}\) with hereditary memory

The main characteristics of the model is that the equation does not contain a term of the form \(-\Delta u\), which contributes to an instantaneous damping. We first investigate the existence and uniqueness of weak solutions to the initial-boundary-value problem for above-mentioned equation. Next, we study the long-time dynamical behavior of the solutions in the weak topological space \(H^{1}(\mathbb{R}^{N}) \times L^{2}_{\mu }(\mathbb{R}^{+},H^{1}( \mathbb{R}^{N}))\), where the nonlinearity is critical and the time-dependent forcing term is only translation bounded instead of translation compact. The results in this paper will extend and improve some results in (Conti et al. in Commun. Pure Appl. Anal. 19:2035–2050, 2020) in the non-autonomous and unbouded domain cases which have not been studied before.

在本文中，我们考虑了具有遗传记忆的 \（\ mathbb {R} ^ {N} \）上的非自治非经典扩散方程

该模型的主要特征是该方程不包含形式为\（-\ Delta u \）的项，这会导致瞬时阻尼。我们首先研究上述方程的初边值问题的弱解的存在性和唯一性。接下来，我们研究弱拓扑空间\（H ^ {1}（\ mathbb {R} ^ {N}）\ times L ^ {2} _ {\ mu}（\ mathbb {R} ^ {+}，H ^ {1}（\ mathbb {R} ^ {N}））\），其中非线性是至关重要的，并且与时间相关的强迫项仅受翻译限制，而不是翻译紧凑。本文的结果将扩展和改进在（Conti等人，Commun。Pure Appl。Anal。19：2035-2050，2020）中在以前从未研究过的非自治和未绑定域中的结果。