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Geometrical evolution of developed interfaces. (English) Zbl 0840.35010

The authors study some striking spatial property of the solution of the initial value problem of the following semilinear parabolic equation, \[ {\partial u\over \partial t}- h^2 \Delta u+ \phi(u)= 0\quad\text{in}\quad \mathbb{R}^N\times (0, \infty),\quad u(x, 0)= u_0(x)\quad\text{in} \quad \mathbb{R}^N, \] where the parameter \(h\) is small. \(\phi\) is some nonlinear term, the most typical case of which is \(\phi(u)= u^3- u\). They prove in the higher-dimensional case \((N\geq 2)\) that some layer arises and the solution has an interesting shape. Because of the smallness of \(h\), the solution \(u(x, t)\) approaches 1 in the region \(\{u_0(x)< 0\}\) and \(- 1\) in \(\{u_0(x)> 0\}\) in a finite time. From this, \(u\) has a very steep slope around the zero set of \(u_0\), which is called an interface. The authors give more elaborate analysis on this procedure and deduce qualitative information. Layer phenomena have been frequently observed in many physical situations and described by PDE’s with a small parameter. The methods developed in this paper will be applied to many similar problems.
Reviewer: S.Jimbo (Sapporo)

MSC:

35B25 Singular perturbations in context of PDEs
35K57 Reaction-diffusion equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K15 Initial value problems for second-order parabolic equations
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