In this post, I simply give a proof of the well posedness of a “simple” coupled PDE-ODE system (which we mentioned in this post).
The idea is based on “localized method”.
Let is a bounded set with smooth boundary . We consider the following system
Theorem: Suppose that and . Then there exists such that system (1) has a unique weak solution .
Denote by . We construct a mapping as follows. Take any . We denote by is a unique weak solution of the following reaction diffusion equation with inhomogeneous Robin boundary condition
We know that , thus, . Hence, solve the ODE
we have get a unique weak solution . Now we define as
We will prove that is a contraction mapping. Denote by where . We also denote by solutions of (2) w.r.t respectively. For the differences, we write . Then from (2) and (3), we have
Multiply (4) by then integrate on , we get
Integrating on , we get
Now, multiplying (5) by then integrating over , doing some integrations by parts, we find that
Integrate on , we get, inparticular
Apply this to (6) we have, for all
we see that is a contraction mapping. Then has a unique fixed point . Insert to (2) to get a unique . It is easy to verify that is the solution of the system (1).
The uniqueness of solutions is standard since the system is linear, so we omit it here (or leave it to the readers).