2010/02/16 by Arnaldo Simal do Nascimento, Nascimento, Arnaldo Simal do
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations #math.AP
paper · pdf · doi:10.48550/arxiv.1002.3071
This paper has been withdrawn by the author due to a scaling error which invalidate the ilustration provided for the convex domain..
openalex publication_date 2010/02/16 · arxiv created 2010/03/15 · arxiv updated 2010/03/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We address the question of existence of nonconstant stable stationary solution to the heat equation on a class of convex domains subject to nonlinear boundary flux involving a positive parameter. Such solutions which were known to exist in dumbbell-type domains and not to exist in N-dimensional balls are shown to exist in some convex domains obtained by smoothing out, in a convenient way, the edges and corners of a cube. Therefore convexity of the domain is not a necessary condition for nonexistence of this type of solutions. If the parameter is small enough we prove that the only equilibria are the constant ones by using the Implicit Function Theorem in a special setting. Symmetry inheritance by local minimizers from symmetry properties of the domain is also addressed.