This is the exercise I'm trying to solve:

Let $\displaystyle \Omega \subset \mathbb{R}^3$ be open and bounded. Suppose that $\displaystyle \overrightarrow{u} = \overrightarrow{u}(x,t)$ and $\displaystyle \theta = \theta(x,t)$ are regular for $\displaystyle t \geq 0$, and satisfy

$\displaystyle

\frac{\partial \overrightarrow{u}}{\partial t} + ( \overrightarrow{u} . \nabla ) \overrightarrow{u} - \nu \Delta \overrightarrow{u} + \nabla p = \overrightarrow{a}(\theta), \hspace{15pt} \nabla . \overrightarrow{u} = 0 \\

$

$\displaystyle

\frac{\partial \theta}{\partial t} + ( \overrightarrow{u} . \nabla ) \theta - k \Delta \theta = b(\overrightarrow{u}) \hspace{25pt} \overrightarrow{u}_{|\partial \Omega} = \overrightarrow{0} \text{ and } \theta_{| \partial \Omega} = 0

$

with $\displaystyle | \overrightarrow{a} (\theta) | \leq \alpha | \theta |$ and $\displaystyle | b(\overrightarrow{u})| \leq \beta |\overrightarrow{u}|$

The problem is: give sufficient conditions over the constants $\displaystyle \alpha, \beta, \nu, \kappa > 0$ so that there exists $\displaystyle \delta > 0$ and the following estimation holds, $\displaystyle \forall t > 0$:

$\displaystyle \int_{\Omega} \Big( |\overrightarrow{u}|^2 + \theta^2 \Big)(x,t) dx \leq e^{-\delta t} \int_{\Omega}\Big( |\overrightarrow{u}|^2 + \theta^2 \Big)(x,0) dx$

I though about multiply the first equation by $\displaystyle u$, second by $\displaystyle \theta$, integrate over $\displaystyle \Omega$ and try get somewhere, but I'm still lost. Any suggestions?

Thanks in advance.