Hölder-logarithmic type approximation for nonlinear backward parabolic equations connected with a pseudo-differential operator. (English) Zbl 1487.35224
Summary: In this paper, we deal with the backward problem for nonlinear parabolic equations involving a pseudo-differential operator in the \(n\)-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 T. I. Seidman [SIAM J. Numer. Anal. 33, No. 1, 162–170 (1996; Zbl 0851.65066)]. 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 \(H^q (\mathbb{R}^n)\).
MSC:
| 35K58 | Semilinear parabolic equations |
| 35S16 | Initial-boundary value problems for PDEs with pseudodifferential operators |
| 35R25 | Ill-posed problems for PDEs |
| 47J06 | Nonlinear ill-posed problems |
| 60H50 | Regularization by noise |
Keywords:
nonlinear parabolic equations; pseudo-differential operator; ill-posed problem; regularization; error estimatesCitations:
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