  We consider the highly nonlinear and ill-posed inverse problem of determining
some general expression $F(x,t,u,\nabla_xu)$ appearing in the diffusion
equation $\partial_tu-\Delta_x u+F(x,t,u,\nabla_xu)=0$ on $\Omega\times(0,T)$,
with $T>0$ and $\Omega$ a bounded open subset of $\mathbb R^n$, $n\geq2$, from
measurements of solutions on the lateral boundary $\partial\Omega\times(0,T)$.
We consider both linear and nonlinear expression of $F(x,t,u,\nabla_xu)$. In
the linear case, the equation can be seen as a convection-diffusion equation
and our inverse problem corresponds to the unique recovery, in some suitable
sense, of a time evolving velocity field associated with the moving quantity as
well as the density of the medium in some rough setting described by non-smooth
coefficients on a Lipschitz domain. In the nonlinear case, we prove the
recovery of more general quasi-linear expression appearing in a nonlinear
parabolic equation associated with more complex model. Here the goal is to
determine the underlying physical low of the system associated with our
equation. In this paper, we consider for what seems to be the first time the
unique recovery of a general vector valued first order coefficient, depending
on both time and space variable. Moreover, we provide results of full recovery
of some general class of quasi-linear terms admitting evolution inside the
system independently of the solution from measurements at the boundary. These
last results improve earlier works of Isakov in terms of generality and
precision. In addition, our results give a partial positive answer, in terms of
measurements restricted to the lateral boundary, to an open problem posed by
Isakov in his classic book (\textit{Inverse Problems for Partial Differential
Equations}) extended to the recovery of quasi-linear terms.
