A minimizing Movement approach to a class of scalar reaction–diffusion equations

Abstract

The purpose of this paper is to introduce a Minimizing Movement approach to scalar reaction-diffusion equations of the form \partial_t u \ = \ \Lambda\cdot \mathrm{div}[u(\nabla F'(u) + \nabla V)] \ - \ \Sigma\cdot (F'(u) + V) u, \quad \text{ in } (0, +\infty)\times\Omega, with parameters $\Lambda, \Sigma > 0$ and no-flux boundary condition u(\nabla F'(u) + \nabla V)\cdot {\sf n} \ = \ 0, \quad \text{ on } (0, +\infty)\times\partial\Omega, which is built on their gradient-flow-like structure in the space $\mathcal{M}(\bar{\Omega})$ of finite nonnegative Radon measures on $\bar{\Omega}\subset\xR^d$, endowed with the recently introduced Hellinger-Kantorovich distance $\HK_{\Lambda, \Sigma}$. It is proved that, under natural general assumptions on $F: [0, +\infty)\to\xR$ and $V:\bar{\Omega}\to\xR$, the Minimizing Movement scheme \mu_\tau^0:=u_0\mathscr{L}^d \in\mathcal{M}(\bar{\Omega}), \quad \mu_\tau^n \text{ is a minimizer for } \mathcal{E}(\cdot)+\frac{1}{2\tau}\HK_{\Lambda, \Sigma}(\cdot, \mu_\tau^{n-1})^2, \ n\in\xN, for \mathcal{E}: \mathcal{M}(\bar{\Omega}) \to (-\infty, +\infty], \ \mathcal{E}(\mu):= \begin{cases} \int_\Omega{[F(u(x))+V(x)u(x)]\xdif x} &\text{ if } \mu=u\mathscr{L}^d, \\ +\infty &\text{ else}, \end{cases} yields weak solutions to the above equation as the discrete time step size $\tau\downarrow 0$. Moreover, a superdifferentiability property of the Hellinger-Kantorovich distance $\HK_{\Lambda, \Sigma}$, which will play an important role in this context, is established in the general setting of a separable Hilbert space.

Fine Properties of Geodesics and Geodesic $$\lambda $$-Convexity for the Hellinger–Kantorovich Distance

“…In [ 15 , 23 ], the JKO scheme (minimizing movement scheme) for a gradient system is considered, i.e., for we iteratively define and consider the limit (along subsequences) to obtain generalized minimizing movements (GMM) (cf. [ 2 ]).…”

“…[ 2 ]). Under suitable conditions, including the assumption with and E superlinear, it is shown in [ 15 , Thm. 3.4] that all GMM have the form , and the density c is a weak solution of the reaction-diffusion equation In [ 24 ], the equation is studied, whose solutions are steady states for HK gradient flows for .…”