A Final Value Problem with a Non-local and a Source Term: Regularization by Truncation
研究了一个带非线性源项和非局部项的抛物型方程终值问题，证明其不适定性，通过截断傅里叶展开得到正则化近似解，并在不同Gevrey光滑性假设下给出参数选择策略和误差估计。
Abstract 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 Grönwalls’ inequality for iterated integrals, which perhaps, is proposed and analysed for the first time.