## 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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