I need a proof for three results about the following system of differential equations. We can consider any regularity for the functions. The system is given by:
$u_i'=-\lambda (u_i-u_{i-1}) - \lambda (u_i - u_{i+1})$
The results that I need are:
- For a solution $u_1$, ... , $u_N$, proof that
$\frac{d}{dt}\sum_{i=1}^{N}u_i^2 \leq 0$
- For a solution $u_1$, ... , $u_N$, with $u_i(t) > 0 \quad \forall t \geq 0, \quad i=1,..., N$ proof that
$\frac{d}{dt}\sum_{i=1}^{N}u_i log \, u_i \leq 0$
This last sum represent the negavity of the entropy in the physic system.
- Proof that the last expression is also true for a solution $u_1$, ... , $u_N$ of the system given by $u_i' = -\lambda u_i + \lambda u_{i-1}$, with $u_i(t) > 0 \quad \forall t \geq 0, \quad i=1,..., N$