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November 2018, 23(9): 3723-3753. doi: 10.3934/dcdsb.2018075

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

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

Received  October 2017 Published  January 2018

Fund Project: 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

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.

Citation: Mickaël D. Chekroun. Topological instabilities in families of semilinear parabolic problems subject to nonlinear perturbations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3723-3753. doi: 10.3934/dcdsb.2018075
References:
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show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, Addison-Wesley Publishing Company, Inc., 1978. doi: 10.1090/chel/364.

[2]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review, 18 (1976), 620-709. doi: 10.1137/1018114.

[3]

A. A. Andronov and L. S. Pontryagin, Systémes grossiers, Dokl. Akad. Nauk. SSSR, 14 (1937), 247-250.

[4]

V. I. Arnol'd, Singularity Theory, vol. 53, Cambridge University Press, 1981.

[5]

V. I. Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations, Grundlehren der Mathematischen Wissenschaften, vol. 250, Springer-Verlag, New York, 1983, Translated from the Russian by Joseph Szücs, Translation edited by Mark Levi. doi: 10.1007/978-1-4612-1037-5.

[6]

J. Bebernes and D. Eberly, Mathematical Problems From Combustion Theory, Springer-Verlag, 1989. doi: 10.1007/978-1-4612-4546-9.

[7]

N. Ben-Gal, Grow-Up Solutions and Heteroclinics to Infinity for Scalar Parabolic PDEs, Ph. D. Thesis, Division of Applied Mathematics, Brown University, 2010.

[8]

M. BergerP. Church and J. Timourian, Folds and cusps in Banach spaces, with applications to nonlinear partial differential equations. Ⅰ, Indiana Univ. Math. J., 34 (1985), 1-19. doi: 10.1512/iumj.1985.34.34001.

[9]

______, Folds and cusps in Banach spaces, with applications to nonlinear partial differential equations. Ⅱ, Trans. Amer. Math. Soc., 307 (1988), 225-244. doi: 10.1090/S0002-9947-1988-0936814-8.

[10]

H. Brezis and H. Berestycki, On a free boundary problem arising in plasma physics, Nonlinear Analysis, 4 (1980), 415-436. doi: 10.1016/0362-546X(80)90083-8.

[11]

H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Analysis: Theory, Methods and Applications, 10 (1986), 55-64. doi: 10.1016/0362-546X(86)90011-8.

[12]

H. BrezisT. CazenaveY. Martel and A. Ramiandrisoa, Blow up for $u_t - Δ u = g(u)$ revisited, Adv. Differential Equations, 1 (1996), 73-90.

[13]

H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Compl. Madrid, 10 (1997), 443-469.

[14]

F. Brezzi and H. Fujii, Numerical imperfections and perturbations in the approximation of nonlinear problems, The Mathematics of Finite Elements and Applications, Ⅳ (Uxbridge, 1981), 431-452, Academic Press, London-New York, 1982.

[15]

F. BrezziJ. Rappaz and P. A. Raviart, Finite dimensional approximation of nonlinear problems, Numerische Mathematik, 36 (1980/81), 1-25. doi: 10.1007/BF01395985.

[16]

K. J. BrownM. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, Nonlinear Anal.: Theory, Methods, and Applications, 5 (1981), 475-486. doi: 10.1016/0362-546X(81)90096-1.

[17]

P. Brunovský and P. Poláčik, The Morse-Smale structure of a generic reaction-diffusion equation in higher space dimension, J. Differential Equations, 135 (1997), 129-181. doi: 10.1006/jdeq.1996.3234.

[18]

A. Castro and R. Shivaji, Uniqueness of positive solutions for a class of elliptic boundary value problems, Proc. Royal Soc. Edinburgh Sect. A: Mathematics, 98 (1984), 267-269. doi: 10.1017/S0308210500013445.

[19]

T. Cazenave, An Introduction to Semilinear Elliptic Equations, Editora do Instituto de Matemática, Universidade Federal do Rio de Janeiro, 2006.

[20]

T. CazenaveM. Escobedo and A. Pozio, Some stability properties for minimal solutions of $-Δ u = λ g(u)$, Portugaliae Mathematica, 59 (2002), 373-391.

[21]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, vol. 13, The Clarendon Press Oxford University Press, New York, 1998, Translated from the 1990 French original by Yvan Martel and revised by the authors.

[22]

S. C. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover Publ., N. Y., 1957.

[23]

M. D. ChekrounM. GhilJ. Roux and F. Varadi, Averaging of time-periodic systems without a small parameter, Disc. Cont. Dyn. Syst. A, 14 (2006), 753-782. doi: 10.3934/dcds.2006.14.753.

[24]

M. D. Chekroun and J. Roux, Homeomorphism groups of normed vector space: The conjugacy problem and the Koopman operator, Disc. Cont. Dyn. Syst. A, 33 (2013), 3957-3980. doi: 10.3934/dcds.2013.33.3957.

[25]

M. D. ChekrounE. Park and R. Temam, The Stampacchia maximum principle for stochastic partial differential equations and applications, J. Differential Equations, 260 (2016), 2926-2972. doi: 10.1016/j.jde.2015.10.022.

[26]

M. D. ChekrounA. Kroener and H. Liu, Galerkin approximations of nonlinear optimal control problems in Hilbert spaces, Electron. J. Differential Equations, 2017 (2017), 1-40.

[27]

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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.
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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