Global dynamic bifurcation of local semiflows and nonlinear evolution equations.

In this work new global dynamic bifurcation theorems are established for local semiflows on complete metric spaces. These theorems are applied to the nonlinear evolution equation u t + A u = f λ (u) in a Banach space X , where A is a sectorial operator with compact resolvent, and f λ (0) = 0 for all λ ∈ R. In particular, if f λ takes the form f λ (u) = λ u + f (u) , it is shown that the global dynamic bifurcation branch of a bifurcation point (0 , λ 0) is necessarily unbounded without assuming the "crossing odd-multiplicity" condition. It has long been recognized that dynamical methods can be helpful in solving static problems. As an example, the bifurcation of elliptic equations on bounded domains associated with homogenous Dirichlet boundary condition is addressed. By considering the corresponding nonclassical parabolic flows and applying the global dynamic bifurcation theorems given here for local semiflows, some new results with global features are derived. [ABSTRACT FROM AUTHOR]