I want to solve the following exercise: We consider the example of the two molecules $U_1$ reaction reversible to one molecule $U_2$:
\begin{equation} \begin{pmatrix} 2 & 0\\\ 0 & 1 \end{pmatrix}. \begin{pmatrix} U_1 \\\ U_2 \end{pmatrix} \longrightarrow \begin{pmatrix} 0 & 1\\\ 2 & 0 \end{pmatrix}. \begin{pmatrix} U_1 \\\ U_2 \end{pmatrix} \end{equation} If the substance $U_i$ (i=1,2) diffuse in $\Omega$ with constant diffusivities $d_i$ and the reaction rates are modelled by the mass action law and independent of temperature, we obtain the system
\begin{equation} \partial _t u_1 - d_1 \Delta u_1=-2(u_1 ^2-u_2),\\\ \partial _t u_2 - d_2 \Delta u_2=u_1 ^2-u_2 \end{equation}
together with homogeneous Neumann conditions $n(x)\nabla _{x} u_i(t,x)=0$ on $\partial\Omega$ and $n(x)$ denoting the unit outer normal of a bounded domain $\Omega$. Calculate the entropy of given system and the corresponding entropy decay.
I am not really familiar with this topic. Hence, I am not sure whether my approach is correct or not. I did the following: I defined the entropy as \begin{equation} \epsilon (t)=\int_{\Omega} (-u_1 ln(u_1)-u_2 ln(2)) dx \end{equation} In order to analyze the entropy decay, I calculated the time derivative of $\epsilon$. \begin{equation} \frac{d\epsilon}{dt}=-\int(\partial _t u_1 ln(u_1)+\partial _t u_2 ln(u_2)) \end{equation} Then, I substituted the given system equations \begin{equation} \frac{d\epsilon}{dt}=\int(d_1\Delta u_1 ln(u_1)+d_2\Delta u_2 ln(u_2))+2(u_1 ^2-u_2)ln(u_1)+(u_1 ^2-u_2)ln(u_2))dx. \end{equation} Next I used the Neumann boundary conditions and integration by parts \begin{equation} \frac{d\epsilon}{dt}=\int(-d_1\nabla (u_1 \nabla ln(u_1))-d_2 \nabla (u_2 \nabla ln(u_2))+2(u_1 ^2-u_2)ln(u_1)+(u_1 ^2-u_2)ln(u_2))dx. \end{equation} After that I used the divergence theorem for the first two terms \begin{equation} \frac{d\epsilon}{dt}=-\int_{\Omega}(2(u_1 ^2-u_2)ln(u_1)+(u_1 ^2-u_2)ln(u_2))dx-\int_{\partial\Omega} (d_1 u_1\nabla (ln(u_1)).n+d_2 u_2 \nabla ln(u_2).n )ds \end{equation} Finally, I got the following expression \begin{equation} \frac{d\epsilon}{dt}=-\int_{\Omega}[2(u_1 ^2-u_2)ln(u_1)+(u_1 ^2-u_2)ln(u_2)]dx \end{equation}
Is my solution correct or are there any mistakes? Thanks!
