On a backward problem for the Kirchhoff's model of parabolic type

Abstract We study for the first time the backward problem for nonlocal nonlinear boundary value problem of Kirchhoff’s model of parabolic type. First, we show that the problem is severely ill-posed in the sense of Hadamard. We propose two methods: the Fourier truncation method for stabilizing the problem with homogeneous source and the quasi-reversibility method for regularizing the problem with nonlinear source. Under some priori assumptions on the exact solution, we establish some stability estimates in the H 0 1 norm.

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