Let $|\cdot|$ a norm in $L^2$ and $\|\cdot\|$ a norm in $H_0^1$. Then
$\begin{align} &|u_h^n|^2 + 2 \theta \Delta t h^{-1} |u_h^n|\|u_h^{n+\theta}\| |u_h^{n+ \theta} - u_h^n| +2 \theta \Delta t |f^n| \|u_h^{n+\theta}\| + 2 \theta \Delta t |g^n|_{L^\infty} \|u_h^{n+\theta}\| \\ & \leq |u_h^n|^2 + \nu \theta \Delta t \|u_h^{n+\theta}\|^2 + \frac{1}{2}|u_h^{n+\theta}-u_h^n|^2 + C_0(\Delta t h^{-1})^2 |u_h^n|^2 \|u_h^n\|^2 + \frac{C_o \Delta t}{\nu}(|f^n|^2 + |g|^2_{L^\infty}). \end{align} $ where $\theta, \nu, h, \Delta t >0.$
I'm struggling with second side of this inequality, and i don't know how article reached it.
I found this expression to second side, $$|u_h^n|^2 + \frac{1}{2}|u_h^{n+\theta} - u_h^n|^2 +2(\theta h^{-1} \Delta t)^2|u_h^n| \|u_h^n\|^2 + \nu \| u_h^{n+\theta}\|^2 + \frac{2(\theta \Delta t)^2}{\nu}(|g^n|^2_{L^\infty} + |f^n|^2), $$ in this case $C_0 = 2\theta ^2.$
This is in article http://www.math.ualberta.ca/ijnam/Volume-7-2010/No-4-10/2010-04-12.pdf where i found this.
How could i apply Young inequality to achieve which this article got?
The left-hand side is $$L=|u_h^n|^2+I_1+I_2+I_3,$$ where, by Young's inequality, \begin{align*} I_1&=2 \theta \Delta t h^{-1} |u_h^n|\|u_h^{n+\theta}\| |u_h^{n+ \theta} - u_h^n|\leq C_{\varepsilon}\left(2 \theta \Delta t h^{-1} |u_h^n|\|u_h^{n+\theta}\|\right)^2+\varepsilon|u_h^{n+ \theta} - u_h^n|^2,\\\\ I_2&=2 \theta \Delta t |f^n| \|u_h^{n+\theta}\| \leq 2 \theta \Delta t \left(C_\delta|f^n|^2+{\delta} \|u_h^{n+\theta}\|^2\right),\\\\ I_3&=2 \theta \Delta t |g^n|_{L^\infty} \|u_h^{n+\theta}\|\leq2 \theta \Delta t \left(C_\delta|g^n|_{L^\infty}^2+{\delta} \|u_h^{n+\theta}\|^2\right). \end{align*}
Therefore,
\begin{align*}L\leq&\;|u_h^n|^2+ C_{\varepsilon}\left(2 \theta \Delta t h^{-1} |u_h^n|\|u_h^{n+\theta}\|\right)^2+\varepsilon|u_h^{n+ \theta} - u_h^n|^2+2 \theta \Delta t \left(C_\delta|f^n|^2+{\delta} \|u_h^{n+\theta}\|^2\right)\\ &+2 \theta \Delta t \left(C_\delta|g^n|_{L^\infty}^2+{\delta} \|u_h^{n+\theta}\|^2\right)\\ =&\;|u_h^n|^2 +4{\delta} \theta \Delta t \|u_h^{n+\theta}\|^2 +\varepsilon|u_h^{n+ \theta} - u_h^n|^2 +C_{\varepsilon}4 \theta^2\left( \Delta t h^{-1}\right)^2 |u_h^n|^2\|u_h^{n+\theta}\|^2\\ &+2 \theta C_\delta \Delta t\left(|f^n|^2+|g^n|_{L^\infty}^2\right). \end{align*} Taking $\delta=\frac{\nu}{4}$ and $\varepsilon=\frac{1}{2}$, we obtain the desired result with $C_0=\max\{2\theta,2\theta^2\}$.