Counterintuitive dependence of temporal asymptotics on initial decay in a nonlocal degenerate parabolic equation arising in game theory

Abstract

We consider the degenerate parabolic equation with nonlocal source given by \[ u_t=u\Delta u + u \int_{\mathbb{R}^n} |\nabla u|^2, \] which has been proposed as model for the evolution of the density distribution of frequencies with which different strategies are pursued in a population obeying the rules of replicator dynamics in a continuous infinite-dimensional setting. Firstly, for all positive initial data $u_0\in C^0(\mathbb{R}^n)$ satisfying $u_0\in L^p(\mathbb{R}^n)$ for some $p\in (0,1)$ as well as $\int_{\mathbb{R}^n} u_0=1$, the corresponding Cauchy problem in $\mathbb{R}^n$ is seen to possess a global positive classical solution with the property that $\int_{\mathbb{R}^n} u(\cdot,t)=1$ for all $t>0$. Thereafter, the main purpose of this work consists in revealing a dependence of the large time behavior of these solutions on the spatial decay of the initial data in a direction that seems unexpected when viewed against the background of known behavior in large classes of scalar parabolic problems. In fact, it is shown that all considered solutions asymptotically decay with respect to their spatial $H^1$ norm, so that \[ \mathcal{E}(t):=\int_0^t \int_{\mathbb{R}^n} |\nabla u(\cdot,t)|^2, \qquad t>0, \] always grows in a significantly sublinear manner in that \begin{equation} \frac{\mathcal{E}(t)}{t} \to 0 \qquad \mbox{as } t\to\infty; \qquad (*) \end{equation} the precise growth rate of $\mathcal{E}$, however, depends on the initial data in such a way that fast decay rates of $u_0$ enforce rapid growth of $\mathcal{E}$. To this end, examples of algebraical and certain exponential types of initial decay are detailed, inter alia generating logarithmic and arbitrary sublinear algebraic growth rates of $\mathcal{E}$, and moreover indicating that (*) is essentially optimal.