Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs

作者：Gameiro Marcia^*; Lessard Jean Philippe

来源：Journal of Differential Equations, 2010, 249(9): 2237-2268.

DOI：10.1016/j.jde.2010.07.002

In this paper we extend the ideas of the so-called validated continuation technique to the context of rigorously proving the existence of equilibria for partial differential equations defined on higher-dimensional spatial domains. For that effect we present a new set of general analytic estimates. These estimates are valid for any dimension and are used, together with rigorous computations, to construct a finite number of radii polynomials. These polynomials provide a computationally efficient method to prove, via a contraction argument, the existence and local uniqueness of solutions for a rather large class of nonlinear problems. We apply this technique to prove existence and local uniqueness of equilibrium solutions for the Cahn-Hilliard and the Swift-Hohenberg equations defined on two- and three-dimensional spatial domains.

• 出版日期2010-11-1