TY - JOUR
T1 - A singularly perturbed semilinear reaction-diffusion problem in a polygonal domain
AU - Kellogg, R. Bruce
AU - Kopteva, Natalia
PY - 2010/1/1
Y1 - 2010/1/1
N2 - The semilinear reaction-diffusion equation - ε2 Δ u + b (x, u) = 0 with Dirichlet boundary conditions is considered in a convex polygonal domain. The singular perturbation parameter ε is arbitrarily small, and the "reduced equation" b (x, u0 (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. The polygonal boundary forces the study of the nonlinear autonomous elliptic equation - Δ z + f (z) = 0 posed in an infinite sector, and then well-posedness of the corresponding linearized problem.
AB - The semilinear reaction-diffusion equation - ε2 Δ u + b (x, u) = 0 with Dirichlet boundary conditions is considered in a convex polygonal domain. The singular perturbation parameter ε is arbitrarily small, and the "reduced equation" b (x, u0 (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. The polygonal boundary forces the study of the nonlinear autonomous elliptic equation - Δ z + f (z) = 0 posed in an infinite sector, and then well-posedness of the corresponding linearized problem.
UR - http://www.scopus.com/inward/record.url?scp=70349787368&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2009.08.020
DO - 10.1016/j.jde.2009.08.020
M3 - Article
AN - SCOPUS:70349787368
SN - 0022-0396
VL - 248
SP - 184
EP - 208
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 1
ER -