A singular perturbation result for a class of periodic-parabolic BVPs

Abstract

In this article, we obtain a very sharp version of some singular perturbation results going back to Dancer and Hess [Behaviour of a semilinear periodic-parabolic problem when a parameter is small, Lecture Notes in Mathematics, Vol. 1450, Springer-Verlag, Berlin, 1990, pp. 12–19] and Daners and López-Gómez [The singular perturbation problem for the periodic-parabolic logistic equation with indefinite weight functions, J. Dynam. Differential Equations 6 (1994), 659–670] valid for a general class of semilinear periodic-parabolic problems of logistic type under general boundary conditions of mixed type. The results of Dancer and Hess [Behaviour of a semilinear periodic-parabolic problem when a parameter is small, Lecture Notes in Mathematics, Vol. 1450, Springer-Verlag, Berlin, 1990, pp. 12–19] and [The singular perturbation problem for the periodic-parabolic logistic equation with indefinite weight functions, J. Dynam. Differential Equations 6 (1994), 659–670] were found, respectively, for Neumann and Dirichlet boundary conditions with L=−Δ{\mathfrak{L}}=-\Delta . In this article, L{\mathfrak{L}} stands for a general second-order elliptic operator.

Keywords

• positive solutions
• periodic-parabolic problems
• singular perturbations
• 35b09
• 35b10
• 35b25