The location and stability of interface solutions of an inhomogeneous parabolic problem (Q2706086)

This paper is concerned with interface solutions of a reaction-diffusion problem with a small diffusion coefficient. After a change of timescale, the problem is NEWLINE\[NEWLINE\varepsilon^2 w_\tau= \varepsilon^2 w_{xx}+ f^2(x) (g^2(x)- w^2)w;\;w(x,0)= \Phi(x),\;x\in (0,1)\tag{1}NEWLINE\]NEWLINE with the Neumann boundary condition; \(f\) and \(g\) are smooth strictly positive functions. The authors are mainly interested in the behavior of a single interface solution \(w\) of (1), which arises by an appropriate choice of the initial condition \(\Phi\). In the first part of the paper, they use asymptotic analysis methods to get a precise estimate of \(S(\tau,\varepsilon)- S_\infty\), where \(S(\tau,\varepsilon)\) is the zero of \(w\), and \(S_\infty\) a zero of the derivative of \(H(x)= \ln(f(x) g^3(x))\), such that \(A= H_{xx}(S_\infty)\neq 0\), and \(S(\tau,\varepsilon)- S_\infty\to 0\), as \(\varepsilon\to 0\). The sign of \(A\) determines the stability of this steady state interface. The second part of the paper is devoted to a numerical approach of (1), by means of finite difference methods.