Inverse problems for a quasilinear parabolic system with integral overdetermination conditions

The question of well-posedness in Sobolev spaces of inverse problems of recovering the source in a quasilinear parabolic system of the second order is examined. The main part of the operator is linear. The integral overdetermination conditions are considered. It is shown that in the case of at most linear growth of the nonlinear summand in its arguments, there exists a unique solution to this problem globally in time and the problem itself is well-posed in the Sobolev classes. The conditions on the data are sharp.