Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations

Description

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. Front Matter ....Pages i-xvii Preliminaries (Messoud Efendiev)....Pages 1-70 Trajectory Dynamical Systems and Their Attractors (Messoud Efendiev)....Pages 71-162 Symmetry and Attractors: The Case N ≤ 3 (Messoud Efendiev)....Pages 163-175 Symmetry and Attractors: The Case N ≤ 4 (Messoud Efendiev)....Pages 177-186 Symmetry and Attractors (Messoud Efendiev)....Pages 187-198 Symmetry and Attractors: Arbitrary Dimension (Messoud Efendiev)....Pages 199-224 The Case of p-Laplacian Operator (Messoud Efendiev)....Pages 225-252 Back Matter ....Pages 253-258