Inhomogeneous fast reaction, slow diffusion and weighted curve shortening (Q2730686)

The authors study the following reaction-diffusion equation, which generalizes the well-known Allen-Cahn equation: NEWLINE\[NEWLINE\begin{matrix} \r & {\l}\qquad & {\l}\\ u_t - \varepsilon\nabla\cdot (k(x)\nabla u) & =\frac 1\varepsilon\;f(x)^2(g(x)^2-u^2)u, & x\in\Omega,\;t>0,\\ \frac{\partial u}{\partial n} & = 0, & x\in\partial\Omega,\;t>0,\\ u(x,0) & =\phi(x), & x\in\Omega,\end{matrix}NEWLINE\]NEWLINE where \(0 <\varepsilon\ll 1\), and \(\Omega\subset\mathbb{R}^n\) is a bounded domain. Furthermore, \(f,g,k > 0\) in \(\overline{\Omega}\). Typically, the solution \(u = u(x,t,\varepsilon)\) will -- after an initial period of time -- be close to \(g(x)\) in an open domain \(\Omega_1(t)\), close to \(-g(x)\) in an open domain \(\Omega_2(t)\) \((\Omega_1(t)\cap\Omega_2(t) = \emptyset)\), and take values between \(-g(x)\) and \(g(x)\) in a transition region \(\Gamma_\varepsilon(t)\) of width \(O(\varepsilon)\). Using formal asymptotics, the authors derive the following equation for an equilibrium surface \(\Gamma_0\), where \(\Gamma_\varepsilon\to \Gamma_0\) as \(\varepsilon\to 0\): NEWLINE\[NEWLINEK_0=\frac{\partial}{\partial n}\left(\ln(\sqrt{k}f g^3)\right).NEWLINE\]NEWLINE Here, \(K_0\) is the mean curvature of \(\Gamma_0\), and \(n\) is the outward normal to \(\Gamma_0\). In addition, the authors investigate the local stability of the equilibrium surfaces and present results of numerical calculations in 2 space dimensions.