A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in is given in terms of an infinite-dimensional dynamical system. The dynamical system is on the space of boundary data for the PDE. This is a novel approach to elliptic problems that enables the use of dynamical systems tools in studying the corresponding PDE. The dynamical system is ill-posed, meaning solutions do not exist forwards or backwards in time for generic initial data. We offer a framework in which this ill-posed system can be analyzed. This can be viewed as generalizing the theory of spatial dynamics, which applies to the case of an infinite cylindrical domain.
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Christopher Jones, Yuri Latushkin, Alim Sukhtayev, Margaret Beck, Graham Cox, A dynamical approach to semilinear elliptic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 2, pp. 421–450DOI 10.1016/J.ANIHPC.2020.08.001