I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions:
\begin{array}{c} \varphi _{t}+\Delta \varphi =0\text{ in }\Omega \times (0,T) \\ \varphi =0\text{ on } \partial \text{}\Omega \times (0,T)\text{} \\ \varphi (T)=\varphi _{T}\text{ }% \end{array} The observability inequality in some books is given by: $$ \left\Vert \varphi (0)\right\Vert ^{2}\leq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx $$ and some times $$\left\Vert \varphi (T)\right\Vert ^{2}\leq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx$$ and $$\int_{\omega }\int_{T/4}^{3T/4}\varphi (x,t)dtdx\leq c\int_{\omega }\int_{0}^{T}\varphi (x,t)dtdx$$ To passe from the firt to the second I guess that they made the substitution $s=T-t$, however, for the thir it comes from the golobal Carleman estimates but I can not see the equivalence between them. Thank you.
1) First, all your inequalities have the wrong sign and missing the square of the absolute value (you need the $L^2$-norm in the integrals with $\omega$).
2) the first inequality is not equivalent to the second one. The first one (with reversed sign) is made for null-controllability whereas the second one is made for exact controllability, two non equivalent notions.
3) To pass from the third inequality to the first one, you use the dissipation of the equation.