On a non-local Kirchhoff type equation with random terminal observation.

In this work, we are concerned with the terminal value problem for the time fractional equation (in the sense of Conformable fractional derivative) with a nonlocal term of the Kirchhoff type$ \partial_t^\alpha u = K\Big(\|\nabla u\|_{L^2(\mathcal{D})}\Big)\Delta u + f(x,t), \quad (x,t) \in (0,T)\times \mathcal{D} $subject to the final data which is blurred by random Gaussian white noise. The principal goal of this article is to recover the solution $ u $. This problem is severely ill-posed in the sense of Hadamard, because of the violation of the continuous dependence of the solution on the data (the solution's behavior does not change continuously with the final condition). By applying non-parametric estimates of the value data from observation data and the truncation method for the Fourier series, we obtain a regularized solution. Under some priori assumptions, we derive an error estimate between a mild solution and its regularized solution. [ABSTRACT FROM AUTHOR]