Geometric Theory of Semilinear Parabolic Equations
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840 Dan Henry
Geometric Theory of Semi linear Parabolic Equations
Springer-Verlag Berlin Heidelberg New York 1981
Author
Daniel Henry Universidade de Sao Paulo Instituto de Matematica e Estatistica Caixa Postal 20570 Agencia Iguatemi 05508 Sao Paulo (SP) Brazil
AMS Subject Classifications (1980): 35A30, 35 Bxx, 35 Kxx, 35010
ISBN 3-540-10557-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10557-3 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
TABLE OF CONTENTS Page INTRODUCTION . . . . . . .
1
CHAPTER 1:
3
1.1
1.2 1.3 1.4 1.5 CHAPTER 2: 2.1 2.2 2.3 2.4 2.5 2.6
2.7
CHAPTER 3: 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 CHAPTER 4: 4.1 4.2 4.3 CHAPTER 5:
PRELIMINARIES. What is geometric theory? Basic facts and notation. .. Sectorial operators and analytic semigroups. Fractional powers of operators . . . . . . . Invariant subspaces and exponential bounds .
6.1 6.2 6.3 6.4
16 24
30
EXAMPLES OF NONLINEAR PARABOLIC EQUATIONS IN PHYSICAL, 41 BIOLOGICAL AND ENGINEERING PROBLEMS . . . . . Nonlinear heat equation. . . . . . . . . . . . Flow of electrons and holes in a semiconductor Hodgekin-Hux1ey equations for the nerve axon Chemical reactions in a catalyst pellet. Population genetics . Nuclear reactor dynamics . . . . . . . Navier-Stokes and related equations . . EXISTENCE, UNIQUENESS AND CONTINUOUS DEPENDENCE. Examples and counterexamples . . . . . . . . . . . The linear Cauchy problem. . . . . . . . . . . . . Local existence and uniqueness . Continuous and differentiable dependence of solutions. Smoothing action of the differential equation. Example: Ut = 6u + f(t,x,u,grad u) . Example: u t = 6u - Au 3(on IRn) . Example: the Navier-Stokes equation . DYNAMICAL SYSTEMS AND LIAPUNOV STABILITY
41 42 42 43 43 44 45 47 47 49
52 62 70 75 77
79 82
Dynamical systems and Liapunov functions Converse theorem on asymptotic stability Invariance principle . . . . . .
82 86
NEIGHBORHOOD OF AN EQUILIBRIUM POINT
98
5.1 Stability and instability by the linear approximation. 5.2 The saddle-point property . . . . . . . 5.3 The Chafee-Infante problem: u t = u + Af(u), and gradient flows . . . . . . . . . . . . . . . . . 5.4 Traveling waves of parabolic equations . . . . . . Appendix. Essential spectrum of some ordinary differential operators. . . . . . . . . . . . . . . . . . . . CHAPTER 6:
3 6
INVARIANT MANIFOLDS NEAR AN EQUILIBRIUM POINT. Existence and stability of an invariant manifold Critical cases of stability. . . . . . . . . . . . . Bifurcation and tr
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