On the source function recovering in quazilinear parabolic problems with pointwise overdetermination conditions

In the article we examine the question of well-posedness in the Sobolev spaces of the inverse source problem in the case of a quasilinear parabolic system of the second order. These problems arise when describing heat and mass transfer, diffusion, filtration, and in many other fields. The main part of the operator is linear. The unknown functions depending on time occur in the nonlinear right-hand side. In particular, this class of problems includes the coefficient inverse problems on determination of the lower order coefficients in a parabolic equation or a system. The overdetermination conditions are the values of the solution at some collection of points lying inside the spacial domain. The Dirichlet and oblique derivative problems are taken as boundary conditions. The problems are studied in a bounded domain with a smooth boundary. However, the results can be generalized to the case of unbounded domains as well for which the corresponding solvability theorems hold. The conditions ensuring local in time well-posedness of the problem in the Sobolev classes are exposed. The conditions on the data are minimal. The results are sharp. The problem is reduced to an operator equation whose solvability is proven with the use of a priori bounds and the fixed point theorem. The solution possesses all generalized derivatives occurring in the system which belong to the space $L_p$ with $p>n+2$ and some additional necessary smoothness in some neighborhood about the overdetermination points.