I'm working on an a priori estimate, using equality's like Young, Cauchy,... But I'm stuck with my testfunction. I've got the following problem:
$\frac{\partial u}{\partial t} - \Delta u + \int_\Omega u = f$ in $(0,T)$ x $\Omega$
$u = 0$ on $[0,T]$ x $\partial \Omega$
$u(0,x) = u_0(x)$ in $\Omega$
I have to prove that
$$\max_{t \in [0,T]} \|u(t)\|² + \int_0^T \|\nabla u(t)\|² \leq C.$$
I'm starting with the first equation from my problem, I multiply by an testfunction $\phi$ and integrate over $\Omega$. I use the notation $(f,g) = \int_\Omega f(x)g(x)dx$. So I get:
$$(u_t, \phi) - (\Delta u,\phi) + (\int_\Omega u,\phi) = (f,\phi)$$
By using Green and the set V of testfunction $H^{0,1}(\Omega)$ this is the same as:
$$(u_t,\phi) + (\nabla u, \nabla \phi) + (\int_\Omega u,\phi) = (f,\phi) \tag{*}$$
I have to search now a good choice for my testfunction, so that I can use that $(f²)' = 2ff'$.
So for the first term from (*), I could use $u, u_t, u_{tt}$. For the second term $u, u_t, \int_0^tu$. But I'm stuck with the third one. I don't know how to work with the integral over $\Omega$. Can somebody help me?
Silke