Positive Steady States of Evolution Equations with Finite Dimensional Nonlinearities

Citation

Calsina A & Farkas JZ (2014) Positive Steady States of Evolution Equations with Finite Dimensional Nonlinearities. SIAM Journal on Mathematical Analysis, 46 (2), pp. 1406-1426. https://doi.org/10.1137/130931199

Abstract
We study the question of existence of positive steady states of nonlinear evolution equations. We recast the steady state equation in the form of eigenvalue problems for a parametrised family of unbounded linear operators, which are generators of strongly continuous semigroups; and a xed point problem. In case of irreducible governing semigroups we consider evolution equations with non-monotone nonlinearities of dimension two, and we establish a new xed point theorem for set-valued maps. In case of reducible governing semigroups we establish results for monotone nonlinearities of any nite dimension n. In addition, we establish a non-quasinilpotency result for a class of strictly positive operators, which are neither irreducible nor compact, in general. We illustrate our theoretical results with examples of partial dierential equations arising in structured population dynamics. In particular, we establish existence of positive steady states of a size-structured juvenile- adult and a structured consumer-resource population model, as well as for a selection-mutation model with distributed recruitment process.

Keywords
Nonlinear evolution equations; positive steady states; xed points of multivalued maps; semigroups of operators; spectral theory of positive operators; structured populations

Journal
SIAM Journal on Mathematical Analysis: Volume 46, Issue 2