MURPHYS-HSFS-2014 - 7th International Workshop on Multi-Rate Processes & Hysteresis, 2nd International Workshop on Hysteresis and Slow-Fast Systems, April 7-11, 2014 - Abstract
Kopteva, Natalia
he semilinear reaction-diffusion equation $-varepsilon^2triangle u+b(x,u)=0$ with Dirichlet boundary conditions is considered in a convex polygonal domain. The singular perturbation parameter $varepsilon$ is arbitrarily small, and the “reduced equation” $b(x,u_0(x))=0$ may have multiple solutions. An asymptotic expansion for $u$ is constructed that involves boundary and corner layer functions. By perturbing this asymptotic expansion, we obtain certain sub- and super-solutions and thus show the existence of a solution $u$ that is close to the constructed asymptotic expansion. Note that the polygonal boundary causes some complications in the analysis in that some “corner layer functions” must be used in the construction of an asymptotic expansion. These corner layer functions are related to the nonlinear autonomous elliptic equation $-triangle z+f(z)=0$ posed in an infinite sector subject to a constant boundary condition. For this problem, we establish stability in the sense that the principal eigenvalue of its linearization about a certain solution $z$ of interest is bounded away from zero. This publication has emanated from research conducted with the financial support of Science Foundation Ireland. References. R.B. Kellogg and N. Kopteva, A singularly perturbed semilinear reaction-diffusion problem in a polygonal domain, J. Differential Equations, 248 (2010), 184-208.