Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states

TL;DR

This paper demonstrates the ability to control nonlinear degenerate parabolic equations to change their sign structure using multiplicative controls, with applications in climatology and genetics.

Contribution

It introduces a method for steering degenerate reaction-diffusion systems between states with the same number of sign changes using multiplicative control techniques.

Findings

Systems can be steered between states with same sign change structure

Method applies to both weakly and strongly degenerate cases

Uses a technique involving shifting points of sign change via pure diffusion problems

Abstract

In this paper we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy problems. In particular, we consider a one-dimensional semilinear degenerate reaction-diffusion equation in divergence form governed via the coefficient of the \-reaction term (bilinear or multiplicative control). The above one-dimensional equation is degenerate since the diffusion coefficient is positive on the interior of the spatial domain and vanishes at the boundary points. Furthermore, two different kinds of degenerate diffusion coefficient are distinguished and studied in this paper: the weakly degenerate case, that is, if the reciprocal of the diffusion coefficient is summable, and the strongly degenerate case, that is, if that reciprocal isn't summable. In our main result we show that the above systems can be steered from an initial continuous state that admits a finite number of points of sign change to a target state with the same number of changes of sign in the same order. Our method uses a recent technique introduced for uniformly parabolic equations employing the shifting of the points of sign change by making use of a finite sequence of initial-value pure diffusion pro\-blems. Our interest in degenerate reaction-diffusion equations is motivated by the study of some \-energy balance models in climatology (see, e.g., the Budyko-Sellers model) and some models in population genetics (see, e.g., the Fleming-Viot model).