Irregular convergence of mild solutions of semilinear equations

TL;DR

This paper demonstrates that irregular convergence of operator semigroups ensures similar convergence of mild solutions in semilinear equations, with applications to biological models such as shadow systems and neurotransmitter dynamics.

Contribution

It establishes a novel link between irregular semigroup convergence and mild solution convergence in semilinear equations, extending previous results.

Findings

Irregular semigroup convergence implies mild solution convergence.

Applications to biological models like shadow systems and neurotransmitter dynamics.

Provides new insights into the behavior of solutions under irregular limits.

Abstract

We prove that even irregular convergence of semigroups of operators implies similar convergence of mild solutions of the related semi-linear equations with Lipschitz continuous nonlinearity. This result is then applied to three models originating from mathematical biology: shadow systems, diffusions on thin layers, and dynamics of neurotransmitters