Entropy for a PDE system with non-homogenous boundary condition

Consider a system of PDEs

$\begin{cases}L_t - \Delta L = 0, x\in \Omega, t>0\\ \dfrac{\partial L}{\partial \nu} + \beta L = \alpha l, x\in \partial \Omega, t>0\\l_t = -\alpha l + \beta L, x\in\partial \Omega, t>0\end{cases}$

Suppose that $(L,l)$ is a positive solution to the system (with sufficient smooth property).

Find the two convex function $\phi$ and $\psi$ such that the functional

$E(L,l) = \int_{\Omega}\phi(L)dx + \int_{\partial \Omega}\psi(l)dS$

is decreasing in time.

It is enough to find $\phi$ and $\psi$ such that $\dfrac{d}{dt}E(L,l) \leq 0$. We give a computation

$\dfrac{d}{dt}E(L,l) = \dfrac{d}{dt}(\int_{\Omega}\phi(L)dx + \int_{\partial \Omega}\psi(l)dS)$

$=\int_{\Omega}\phi'(L)L_tdx + \int_{\partial\Omega}\psi'(l)l_tdS$

$=\int_{\Omega}\phi'(L)\Delta Ldx +\int_{\partial\Omega}\psi'(l)(-\alpha l + \beta L)dS$

$= -\int_{\Omega}\phi''(L)|\nabla L|^2dx + \int_{\partial \Omega}\phi'(L)\dfrac{\partial L}{\partial \nu}dS +\int_{\partial\Omega}\psi'(l)(-\alpha l + \beta L)dS$

$= -\int_{\Omega}\phi''(L)|\nabla L|^2dx +\int_{\partial \Omega}(\phi'(L) - \psi'(l))(\alpha l -\beta L)dS$

The last relation suggests that if $(\phi'(x) - \psi'(y))(\alpha y - \beta x) \leq 0$, then we get what we claim since $\phi'' \geq 0$.

Two approriate choices of mine:

$\phi(x) = \beta x^2, \psi(x) = \alpha x^2$

and

$\phi(x) = x\log(\beta x) - \beta x, \psi(x) = x\log(\alpha x) - \alpha x$

[Update]

Since we want to use Csiszar-Kullback’s inequality, $\phi$ and $\psi$ should be

$\phi(x) = x(\log(\beta x) - \beta)+ \beta, \psi(x) = x(\log(\alpha x) - \alpha) + \alpha$