### Abstract

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 −Dz + f (z) = 0

posed in an infinite sector, and then well-posedness of the corresponding linearized problem.

Original language | English |
---|---|

Pages (from-to) | 184-208 |

Number of pages | 25 |

Journal | Journal of Differential Equations |

Volume | 248 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2010 |

### Fingerprint

### Keywords

- singularly perturbed
- semilinear
- reaction-diffusion problem
- polygonal domain

### Cite this

*Journal of Differential Equations*,

*248*(1), 184-208. https://doi.org/10.1016/j.jde.2009.08.020

}

*Journal of Differential Equations*, vol. 248, no. 1, pp. 184-208. https://doi.org/10.1016/j.jde.2009.08.020

**A singularly perturbed semilinear reaction-diffusion problem in a polygonal domain.** / Kellogg, R. Bruce; Kopteva, Natalia.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A singularly perturbed semilinear reaction-diffusion problem in a polygonal domain

AU - Kellogg, R. Bruce

AU - Kopteva, Natalia

PY - 2010

Y1 - 2010

N2 - The semilinear reaction-di®usion equation ¡"24u+b(x; u) = 0 with Dirichlet bound-ary 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 −Dz + f (z) = 0 posed in an infinite sector, and then well-posedness of the corresponding linearized problem.

AB - The semilinear reaction-di®usion equation ¡"24u+b(x; u) = 0 with Dirichlet bound-ary 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 −Dz + f (z) = 0 posed in an infinite sector, and then well-posedness of the corresponding linearized problem.

KW - singularly perturbed

KW - semilinear

KW - reaction-diffusion problem

KW - polygonal domain

UR - http://dx.doi.org/10.1016/j.jde.2009.08.020

U2 - 10.1016/j.jde.2009.08.020

DO - 10.1016/j.jde.2009.08.020

M3 - Article

VL - 248

SP - 184

EP - 208

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 1

ER -