This paper is concerned with recovering the solution of a final value problem associated with a parabolic equation involving a non linear source and a non-local term, which to the best of our knowledge has not been studied earlier. It is shown that the considered problem is ill-posed, and thus, some regularization method has to be employed in order to obtain stable approximations. In this regard, we obtain regularized approximations by solving some non linear integral equations which is derived by considering a truncated version of the Fourier expansion of the sought solution. Under different Gevrey smoothness assumptions on the exact solution, we provide parameter choice strategies and obtain the error estimates. A key tool in deriving such estimates is a version of Gr{\"o}nwalls' inequality for iterated integrals, which perhaps, is proposed and analysed for the first time.

翻译：本文研究了一类与带有非线性源项和非局部项的抛物方程相关的终值问题的解恢复，据我们所知，该问题此前尚未被研究。结果表明，该问题是不适定的，因此需采用某种正则化方法以获得稳定的近似解。为此，我们通过求解一些非线性积分方程得到了正则化近似解，这些方程是通过考虑所求解的傅里叶展开的截断版本推导得出的。在精确解的不同Gevrey光滑性假设下，我们提供了参数选择策略并获得了误差估计。推导此类估计的一个关键工具是迭代积分形式的Grönwall不等式的一个版本，这可能是首次被提出和分析。