Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations

1. Front Matter

   Pages i-xvii

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2. Preliminaries

   • Messoud Efendiev

   Pages 1-70

3. Trajectory Dynamical Systems and Their Attractors

   • Messoud Efendiev

   Pages 71-162

4. Symmetry and Attractors: The Case N ≤ 3

   • Messoud Efendiev

   Pages 163-175

5. Symmetry and Attractors: The Case N ≤ 4

   • Messoud Efendiev

   Pages 177-186

6. Symmetry and Attractors

   • Messoud Efendiev

   Pages 187-198

7. Symmetry and Attractors: Arbitrary Dimension

   • Messoud Efendiev

   Pages 199-224

8. The Case of p-Laplacian Operator

   • Messoud Efendiev

   Pages 225-252

9. Back Matter

   Pages 253-258

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This book deals with a systematic study of a dynamical system approach to investigate the symmetrization and stabilization properties of nonnegative solutions of nonlinear elliptic problems in asymptotically symmetric unbounded domains. The usage of infinite dimensional dynamical systems methods for elliptic problems in unbounded domains as well as finite dimensional reduction of their dynamics requires new ideas and tools. To this end, both a trajectory dynamical systems approach and new Liouville type results for the solutions of some class of elliptic equations are used. The work also uses symmetry and monotonicity results for nonnegative solutions in order to characterize an asymptotic profile of solutions and compares a pure elliptic partial differential equations approach and a dynamical systems approach. The new results obtained will be particularly useful for mathematical biologists.

• Sobolev spaces
• Nemytskii operator
• trajectory attractor
• Laplace operator
• symmetry and attractors
• stabilization properties
• reduction of dynamics
• partial differential equations

“The author does a very good job with the difficult and somewhat unexpected task of pairing together elliptic PDE’s with dynamical systems methods specific to finite dimensions.” (Florin Catrina, zbMATH 1445.35005, 2020)

• Institute of Computational Biology, Helmholtz Center Munich, Neuherberg, Germany

  Messoud Efendiev

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