Certain inverse problems for parabolic equations

In the paper, we study the inverse problem of finding the solution $u$ and the coefficient $q$ from the following data:
\begin{gather*} Mu=u_t-L(x,t,D_x)u+g(x,t,u,\nabla u)+q(x,t)u(x,t)=f(x,t), \\ (x,t)\in Q=G\times(0,T), \\ u|_{S}=\varphi(x,t),\quad \frac{\partial u}{\partial n}\biggr|_{S}=\psi(x,t),\quad u|_{t=0}=u_0(x),\quad S=\Gamma\times(0,T), \end{gather*}
where $G\subset\mathbb R^n$ is a bounded domain with boundary $\Gamma$ and $L$ is a second-order elliptic operator. We prove that the problem is solvable locally in time or in the case where the norms of its data are sufficiently small.