| 概要 | We consider N-component systems of diagonal diffusion
equations under dynamic boundary conditions on bounded
smooth domains. The features of our system are as follows.
(1) Each component diffuses freely (independently from
other components) inside the domain (there is no
interaction in bulk);
(2) There are dynamic interactions between the components
involved on the boundary.
We are interested in the stability of trivial solutions of
such a system, and the linearized problem is characterized
by the diagonal diffusion matrix D (which acts in the
interior of domain), together with the mass transfer and
boundary rate matrices J and W, respectively, which act on
the boundary. We emphasize that the matrices D and W have
to be positive definite by nature. As regard to the
linearized eigenvalue problem, we obtained the following
results.
(a) When J is symmetric, we obtained a variational
characterization of all eigenvalues:
(a-1) All eigenvalues of our problem are real.
(a-2) If J is negative definite, then all eigenvalues of
our problem are negative.
(a-3) If J has a positive eigenvalue, then our problem
has corresponding positive eigenvalues.
(b) When the diffusion matrix D and the boundary rate
matrix are constant multiples of the N by N identity
matrix, then we identified a smooth region U in the
complex plane C for which we have:
(b-1) If all eigenvalues of J belong to the interior of
U, then the eigenvalues of our problem have
negative real part.
(b-2) If J has an eigenvalue belonging to the exterior
of U, then our problem has a corresponding
eigenvalue with positive real part.
(b-3) If J has an eigenvalue on the boundary of U, then
our problem has a corresponding pure-imaginary
eigenvalue.
One of the key ingredients to obtain these result is to
formulate our problem in terms of Steklov-eigenvalue
problems. This is a joint work with Ciprian Gal of Florida
International University. |