A multivariable calculus exercise in the weak formulation of the Navier Stokes Equations The following is an excerpt from Robinson's An introduction to the classical theory of the
Navier–Stokes equations.

Here (1.1):

Using integration by parts I can get the $(\nabla u,\nabla\varphi)$ term from the second term of (1.1). But I don't get $\int_0^T(u,\varphi_t)$ and $-(u_0,\varphi)$. Also, the signs in (1.8) are confusing. Would anyone help me to derive (1.8) from (1.1)?

It is because $\phi(0,x)$ is not necessarily zero, so 
$$\int_0^T\int_\Omega u_t\phi=-\int_0^T\int_\Omega u\phi_t+\int_\Omega u_0\phi(0,x).$$
