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A general approach to stability in free boundary problems. (English) Zbl 0973.35200
The authors consider the free boundary problem $u_t=L u+f, \quad x\in\Omega_t\subset\mathbb{R}^N,\;t>0, \quad u=g_1,\quad \frac{\partial u}{\partial n}=g_2 \quad \text{on }\partial \Omega_t,\tag{1}$ where $$L$$ is a uniformly (time independent) elliptic operator and $$f,g_1,g_2$$ are given functions. The coefficients of $$L$$ and the data are supposed to have sufficient regularity. The authors’ aim is to study the stability of the equilibrium solutions to the problem above, assuming that the corresponding steady state problem is solvable. The main results are: local existence and uniqueness for the evolution problem, the stability theorem, the structure of the stable and unstable manifolds in case of instability (the latter result extending the classical saddle point theorem).
The problem is reformulated in a fixed domain by means of a suitable transformation and linearization in the neighborhood of the equilibrium solution is introduced. The new form taken by (1) is a boundary-value problem for a fully nonlinear parabolic equation with fully nonlinear boundary conditions. The stability analysis is based on the properties of the spectrum of the operator $$L$$, which is the realization of $$L$$ with homogeneous boundary conditions in $$C(\overline \Omega)$$. Some applications are illustrated.

##### MSC:
 35R35 Free boundary problems for PDEs 35B35 Stability in context of PDEs 35K55 Nonlinear parabolic equations
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##### References:
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