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 .

Proof:

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

and

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

.

Thus,

.

By choosing

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).

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## About baotangquoc

Lecturer
School of Applied Mathematics and Informatics
Hanoi University of Science and Technology
No 1, Dai Co Viet Street, Hanoi