Boundary Value Problems Posed in Unbounded Domains

Jason Morris University of Pittsburgh, USA

A system of ordinary differential equations posed on the half line [0, ∞) with conditions at both “ends” can model a quantity u(t) ∈ R^{n} that changes with time t and that is expected to eventually settle toward a value u_{∞}. If this convergence can take arbitrarily long depending on the initial value u(0), the problem should not be reposed on a bounded interval [0, T]. Similarly, a parabolic-type Cauchy problem such as u_{t} − Δ_{x}u + G(t, u) = 0; u(0, ·) = g(·) might be posed in a semi-infinite cylinder [0, ∞)×Ω. Again, this is done in order to insist upon a specific asymptotic behavior for solutions u(t, x) as t → ∞.

When proving that such problems are well posed, certain difficulties arise from the unboundedness of the domain. These difficulties preclude the use of previously existing techniques, including methods based on the Leray-Schauder topological degree. It is indicated how these difficulties are overcome for certain classes of nonlinear differential equations, both ordinary and partial. The main tool is a topological degree for Fredholm operators, developed by Fitzpatrick, Pejsachowicz, and Rabier in 1992, and later improved by Pejsachowicz and Rabier in 1998.

This is work from the speaker’s doctoral dissertation, completed earlier this year under the supervision of Patrick Rabier.