Global dynamics of the parabolic Choquard equation with asymptotically linear nonlinearity

This article investigates the global dynamics of the semilinear parabolic Choquard equation $$ \begin{aligned} \partial _{t} u(x,t) - \Delta u(x,t) + u(x,t) &= \big( I_{\alpha }* |u(\cdot ,t)|^{p} \big)(x)\, |u(x,t)|^{p-2} u(x,t) \\ &\quad + f\big(u(x,t)\big), \qquad (x,t)\in \mathbb{R}^{N}\times (0,\infty ) . \end{aligned} $$ where $N \ge 3$ , $\alpha \in (N-4,N)$ , $p \in \bigl( 2, \frac{N+\alpha}{N-2} \bigr)$ , and f is an asymptotically linear nonlinearity. The above equation represents the natural parabolic counterpart of the elliptic Choquard equation studied recently by Wang–Lai–Guo (2025), who constructed multiple radial nodal stationary solutions using shooting–gluing methods and topological degree theory. In this work we establish the global well-posedness of the problem in $H^{1}(\mathbb{R}^{N})$ by combining optimal Hardy–Littlewood–Sobolev estimates with energy methods and dissipation identities. In particular, we show that the associated Choquard energy functional is strictly decreasing along every non-stationary trajectory. This property allows us to obtain a precise characterization of the ω-limit set and to prove convergence of the parabolic flow toward stationary Choquard states. A main contribution of the present paper is that it provides, to the best of our knowledge, the first rigorous analysis of the parabolic Choquard equation involving an asymptotically linear nonlinearity. Furthermore, we show that, for suitable initial data, the solution converges to the k-th radial nodal stationary solution obtained in the corresponding elliptic theory. Finally, numerical simulations illustrate the long-time dynamics of the flow and confirm the dynamical selection of nodal stationary structures.