Front Interaction and Nonhomogeneous Equilibria for Tristable Reaction-Diffusion Equations
Jacob Rubinstein, Peter Sternberg, Joseph Bishop Keller · SIAM Journal on Applied Mathematics · 1993
The authors investigate the interfacial dynamics associated with three-phase systems and the possible occurrence of steady multiphase patterns in one and higher space dimensions. The model for this study is a Ginzburg–Landau model for phase transitions given by the partial differential equation\[ u_1 = \varepsilon \Delta u - \frac{1}{\varepsilon } V_u^\varepsilon ( u ) \] defined for x in a domain $\Omega \subset R$ and $t > 0$. The potential $V^\varepsilon $ is assumed to possess three wells having depths within $\mathcal{O}( \varepsilon )$ of each other. The evolution of the solution can be described by tracking the motion of fronts connecting adjacent wells of $V^\varepsilon $. The primary interest is to understand how the self-induced motion of a given front is affected by the presence of a second nearby front. This interaction is characterized precisely using the formal method of matched asymptotic expansions. In some settings, this leads to the prediction of steady waves linking the outer two phases across a thin region of the intermediate phase. A rigorous proof of the existence of stable equilibria having this profile is given using the techniques of Gamma-convergence.