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.
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