A problem with nonlocal conditions for a one-dimensional parabolic equation

In present paper, we consider a problem with nonlocal conditions for parabolic equation and show that there exists a unique weak solution in Sobolev space. The main tool to prove the existence of a unique weak solution to the problem is a priori estimates derived by authors. We also note a connection between Steklov nonlocal conditions and first kind integral conditions. This connection enables interpret the problem under consideration as a problem with perturbed Steklov nonlocal conditions. Obtained results may be useful for certain class of problems including inverse problems.