## A nonlinear parabolic equation with varying domain.(English)Zbl 0569.35048

This paper considers the problem $$u_ t=\Delta u+f(\lambda,u)$$ in $$\Omega_{\epsilon}$$ $$\partial u/\partial n=0$$ on $$\partial \Omega_{\epsilon}$$ where $$0<\epsilon \leq 1$$, $$\Omega_{\epsilon}$$ is a domain in $$R^ n$$ and $$\lambda$$ a small real parameter. The objective is to discuss the equilibrium solutions and their stability as a function of ($$\lambda$$,$$\epsilon)$$ in a neighborhood of (0,0), when the domain $$\Omega_{\epsilon}$$ increases in $$\epsilon$$, $$| \Omega_{\epsilon}|$$ is continuous in $$\epsilon$$, $$| \Omega_{\epsilon}\setminus \Omega_ 0| \to 0$$ as $$\epsilon$$ $$\to 0$$ and $$\Omega_ 0$$ is the union of connected domains $$\Omega^ L_ 0$$ and $$\Omega^ R_ 0$$ with disjoint closures. The approach is a Lyapunov-Schmidt method devised as follows. For $$\Omega_ 0$$, 0 is a double eigenvalue of the Laplacian with Neumann boundary conditions. Using some more assumptions on $$\Omega_{\epsilon}$$, the discussion of the equilibrium solutions is reduced to a pair of bifurcation equations depending upon the parameter $$\epsilon$$. The technical difficulties occur in showing the smoothness properties of the bifurcation equation.
Reviewer: J.Mawhin

### MSC:

 35K55 Nonlinear parabolic equations 35B32 Bifurcations in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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### References:

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