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The location and stability of interface solutions of an inhomogeneous parabolic problem. (English) Zbl 0983.35015

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 \[ \varepsilon^2 w_\tau= \varepsilon^2 w_{xx}+ f^2(x) (g^2(x)- w^2)w;\;w(x,0)= \Phi(x),\;x\in (0,1)\tag{1} \] 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.

MSC:

35B25 Singular perturbations in context of PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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