Hölder-Logarithmic type approximation for nonlinear backward parabolic equations connected with a pseudo-differential operator

Abstract

<p style='text-indent:20px;'>In this paper, we deal with the backward problem for nonlinear parabolic equations involving a pseudo-differential operator in the <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional space. We prove that the problem is ill-posed in the sense of Hadamard, i.e., the solution, if it exists, does not depend continuously on the data. To regularize the problem, we propose two modified versions of the so-called optimal filtering method of Seidman [T.I. Seidman, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., <b>33</b> (1996), 162–170]. According to different a priori assumptions on the regularity of the exact solution, we obtain some sharp optimal estimates of the Hölder-Logarithmic type in the Sobolev space <inline-formula><tex-math id="M2">\begin{document}$ H^q(\mathbb{R}^n) $\end{document}</tex-math></inline-formula>.</p>