Topological instabilities in families of semilinear parabolic problems subject to nonlinear perturbations

Mickaël D. Chekroun

doi:10.3934/dcdsb.2018075

Article Contents

2018, Volume 23, Issue 9: 3723-3753. Doi: 10.3934/dcdsb.2018075

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Topological instabilities in families of semilinear parabolic problems subject to nonlinear perturbations

Mickaël D. Chekroun

Department of Atmospheric & Oceanic Sciences, University of California, Los Angeles, Los Angeles, CA 90095-1565, USA

Received: October 2017

Early access: January 2018

Published: September 2018

The author is grateful to Thierry Cazenave for the stimulating discussions concerning the reference [20] at the start of this project. The author thanks also Lionel Roques, Jean Roux and Eric Simonnet for their interests in this work, and Honghu Liu for his help in preparing Figure 1. This work was partially supported by the grant N00014-16-1-2073 from the Multidisciplinary University Research Initiative (MURI) of the Office of Naval Research, and by the National Science Foundation grants OCE-1658357 and DMS-1616981.

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Abstract

Semilinear parabolic problems are considered for which we prove their topological sensitivity to arbitrarily small perturbations of the nonlinear term. This instability result is a consequence of the sensitivity of the multiplicity of solutions of the corresponding nonlinear elliptic problems. As shown here, it is indeed always possible (in dimension $d = 1$ or $d = 2$ ) to find an arbitrary small perturbation that e.g. generates locally an S on the global bifurcation diagram, substituting thus a single solution by several ones. Such an increase in the local multiplicity of the solutions to the elliptic problem results then into a topological instability for the corresponding parabolic problem.

The rigorous proof of this instability result requires though to revisit the classical concept of topological equivalence to encompass important cases for applications such as semi-linear parabolic problems for which the semigroup may exhibit non-global dissipative properties, allowing for the coexistence of blow-up regions and local attractors in the phase space; cases that arise e.g. in combustion theory. A revised framework of topological robustness is thus introduced in that respect within which the main topological instability result is then proved for continuous, locally Lipschitz but not necessarily $C^1$ nonlinear terms, that prevent in particular the use of linearization techniques.

Keywords:

Semilinear elliptic and parabolic problems,
nonlinear eigenvalue problems,
Leray-Schauder degree,
$S$-shaped bifurcation,
structural stability,
topological instability,
perturbed bifurcation theory.

Mathematics Subject Classification: Primary: 35J61, 35B30, 35B32, 35B20, 35K58, 35A16; Secondary: 37K50, 37C20, 37H20, 37J20, 47H11.

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Figure 1. Schematic of some typical situations dealt with Theorem 3.1. The left panel corresponds to case (i), the right panel corresponds to case (ⅱ), and the middle panel corresponds to case (ⅲ). In each case, either a multiple-point or a new fold-point can be created (locally) by arbitrary small perturbations of the nonlinearity $g$ in (25), as described in Theorem 3.1. The appearance of such singular points implies a topological instability——in the sense of Definition 2.5——of the one-parameter family of semigroups associated with the corresponding family of parabolic problems.

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Figure 2. Bifurcation diagrams for the perturbed problem (red curve) and the unperturbed one (blue curve). The fold-points are indicated by the green dots.

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