May  2022, 21(5): 1715-1734. doi: 10.3934/cpaa.2022043

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

Department of Economic Mathematics, Banking University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

Received  July 2021 Revised  January 2022 Published  May 2022 Early access  February 2022

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 Seidman [T.I. Seidman, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., 33 (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 $ H^q(\mathbb{R}^n) $.

Citation: Dinh Nguyen Duy Hai. Hölder-Logarithmic type approximation for nonlinear backward parabolic equations connected with a pseudo-differential operator. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1715-1734. doi: 10.3934/cpaa.2022043
References:
[1]

J. V. Beck, B. Blackwell and C. R. S. Clair, Inverse Heat Conduction, Ill-Posed Problems, New York, Wiley, 1985. doi: 10.1007/978-1-4612-0873-0.

[2]

H. ChengC. L. FuG. H. Zheng and J. Gao, A regularization for a Riesz-Feller space-fractional backward diffusion problem, Inverse Probl. Sci. Eng., 22 (2014), 860-872.  doi: 10.1080/17415977.2013.840298.

[3]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Dordrecht, Kluwer, 1996. doi: 10.1007/978-1-4612-0873-0.

[4]

R. E. Ewing, The approximation of certain parabolic equations backward in time by Sobolev equations, SIAM J. Math. Anal., 6 (1975), 283-294.  doi: 10.1137/0506029.

[5]

B. Guo, X. Pu and F. Huang, Fractional partial differential equations and their numerical solutions, Publishing Co. Pvt. Ltd., Hackensack, NJ, 2015. doi: 10.1007/978-1-4612-0873-0.

[6]

J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Dover, New York, 1953. doi: 10.1007/978-1-4612-0873-0.

[7]

D. N. D. HaiN. H. TuanL. D. Long and L. G. Q. Thong, Inverse problem for nonlinear backward space-fractional diffusion equation, J. Inverse Ill-Posed Probl., 25 (2016), 423-443.  doi: 10.1515/jiip-2015-0065.

[8]

D. N. Hao and N. V. Duc, Stability results for the heat equation backward in time, J. Math. Anal. Appl., 353 (2009), 627-641.  doi: 10.1016/j.jmaa.2008.12.018.

[9]

M. KarimiF. Moradlou and M. Hajipour, Regularization technique for an inverse space-fractional backward heat conduction problem, J. Sci. Comput., 83 (2020), 440-455.  doi: 10.1007/s10915-020-01211-2.

[10]

T. T. Khieu and V. H. Hung, Recovering the historical distribution for nonlinear space-fractional diffusion equation with temporally dependent thermal conductivity in higher dimensional space, J. Comput. Appl. Math., 345 (2019), 114-126.  doi: 10.1016/j.cam.2018.06.018.

[11]

F. MainardiY. Luchko and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Cacl. Appl. Anal., 4 (2001), 153-192. 

[12]

T. I. Seidman, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., 33 (1996), 162-170.  doi: 10.1137/0733010.

[13]

R. E. Showalter, Cauchy problem for hyper-parabolic partial differential equations, in: Trends in the Theory and Practice of Non-Linear Analysis, Elsevier, 1983. doi: 10.1007/978-1-4612-0873-0.

[14]

U. Tautenhahn and T. Schröter, On optimal regularization methods for the backward heat equation, Z. Anal. Anwend., 15 (1996), 475-493.  doi: 10.4171/ZAA/711.

[15]

U. Tautenhahn, Optimality for ill-posed problems under general source conditions, Num. Funct. Anal. Optim., 19 (1998), 377-398.  doi: 10.1080/01630569808816834.

[16]

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Winston, Washington, 1977. doi: 10.1007/978-1-4612-0873-0.

[17]

L. M. TrietP. H. QuanD. D. Trong and N. H. Tuan, A backward parabolic equation with a time-dependent coefficient: Regularization and error estimates, J. Comput. Appl. Math., 237 (2013), 432-441.  doi: 10.1016/j.cam.2012.06.012.

[18]

D. D. TrongD. N. D. Hai and N. D. Minh, Stepwise regularization method for a nonlinear Riesz-Feller space-fractional backward diffusion problem, J. Inverse Ill-Posed Probl., 27 (2019), 759-775.  doi: 10.1515/jiip-2018-0033.

[19]

N. H. TuanD. N. D. HaiL. D. LongN. V. Thinh and M. Kirane, On a Riesz - Feller space fractional backward diffusion problem with a nonlinear source, J. Comput. Appl. Math., 312 (2017), 103-126.  doi: 10.1016/j.cam.2016.01.003.

[20]

N. H. TuanL. D. Thang and D. Lesnic, A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source, J. Math. Anal. Appl., 434 (2016), 1376-1393.  doi: 10.1016/j.jmaa.2015.09.085.

[21]

F. YangX. X. LiD. G. Li and L. Wang, The simplified Tikhonov regularization method for solving a Riesz-Feller space-fractional backward diffusion problem, Math. Comput. Sci., 11 (2017), 91-110.  doi: 10.1007/s11786-017-0292-6.

[22]

J. ZhaoS. Liu and T. Liu, An inverse problem for space-fractional backward diffusion problem, Math. Methods Appl. Sci., 37 (2014), 1147-1158.  doi: 10.1002/mma.2876.

[23]

G. H. Zheng and T. Wei, Two regularization methods for solving a Riesz-Feller space-fractional backward diffusion problem, Inverse Probl., 26 (2010), 115017, 22 pp. doi: 10.1088/0266-5611/26/11/115017.

[24]

G. H. Zheng and Q. G. Zhang, Recovering the initial distribution for space-fractional diffusion equation by a logarithmic regularization method, Appl. Math. Lett., 61 (2016), 143-148.  doi: 10.1016/j.aml.2016.06.002.

show all references

References:
[1]

J. V. Beck, B. Blackwell and C. R. S. Clair, Inverse Heat Conduction, Ill-Posed Problems, New York, Wiley, 1985. doi: 10.1007/978-1-4612-0873-0.

[2]

H. ChengC. L. FuG. H. Zheng and J. Gao, A regularization for a Riesz-Feller space-fractional backward diffusion problem, Inverse Probl. Sci. Eng., 22 (2014), 860-872.  doi: 10.1080/17415977.2013.840298.

[3]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Dordrecht, Kluwer, 1996. doi: 10.1007/978-1-4612-0873-0.

[4]

R. E. Ewing, The approximation of certain parabolic equations backward in time by Sobolev equations, SIAM J. Math. Anal., 6 (1975), 283-294.  doi: 10.1137/0506029.

[5]

B. Guo, X. Pu and F. Huang, Fractional partial differential equations and their numerical solutions, Publishing Co. Pvt. Ltd., Hackensack, NJ, 2015. doi: 10.1007/978-1-4612-0873-0.

[6]

J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Dover, New York, 1953. doi: 10.1007/978-1-4612-0873-0.

[7]

D. N. D. HaiN. H. TuanL. D. Long and L. G. Q. Thong, Inverse problem for nonlinear backward space-fractional diffusion equation, J. Inverse Ill-Posed Probl., 25 (2016), 423-443.  doi: 10.1515/jiip-2015-0065.

[8]

D. N. Hao and N. V. Duc, Stability results for the heat equation backward in time, J. Math. Anal. Appl., 353 (2009), 627-641.  doi: 10.1016/j.jmaa.2008.12.018.

[9]

M. KarimiF. Moradlou and M. Hajipour, Regularization technique for an inverse space-fractional backward heat conduction problem, J. Sci. Comput., 83 (2020), 440-455.  doi: 10.1007/s10915-020-01211-2.

[10]

T. T. Khieu and V. H. Hung, Recovering the historical distribution for nonlinear space-fractional diffusion equation with temporally dependent thermal conductivity in higher dimensional space, J. Comput. Appl. Math., 345 (2019), 114-126.  doi: 10.1016/j.cam.2018.06.018.

[11]

F. MainardiY. Luchko and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Cacl. Appl. Anal., 4 (2001), 153-192. 

[12]

T. I. Seidman, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., 33 (1996), 162-170.  doi: 10.1137/0733010.

[13]

R. E. Showalter, Cauchy problem for hyper-parabolic partial differential equations, in: Trends in the Theory and Practice of Non-Linear Analysis, Elsevier, 1983. doi: 10.1007/978-1-4612-0873-0.

[14]

U. Tautenhahn and T. Schröter, On optimal regularization methods for the backward heat equation, Z. Anal. Anwend., 15 (1996), 475-493.  doi: 10.4171/ZAA/711.

[15]

U. Tautenhahn, Optimality for ill-posed problems under general source conditions, Num. Funct. Anal. Optim., 19 (1998), 377-398.  doi: 10.1080/01630569808816834.

[16]

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Winston, Washington, 1977. doi: 10.1007/978-1-4612-0873-0.

[17]

L. M. TrietP. H. QuanD. D. Trong and N. H. Tuan, A backward parabolic equation with a time-dependent coefficient: Regularization and error estimates, J. Comput. Appl. Math., 237 (2013), 432-441.  doi: 10.1016/j.cam.2012.06.012.

[18]

D. D. TrongD. N. D. Hai and N. D. Minh, Stepwise regularization method for a nonlinear Riesz-Feller space-fractional backward diffusion problem, J. Inverse Ill-Posed Probl., 27 (2019), 759-775.  doi: 10.1515/jiip-2018-0033.

[19]

N. H. TuanD. N. D. HaiL. D. LongN. V. Thinh and M. Kirane, On a Riesz - Feller space fractional backward diffusion problem with a nonlinear source, J. Comput. Appl. Math., 312 (2017), 103-126.  doi: 10.1016/j.cam.2016.01.003.

[20]

N. H. TuanL. D. Thang and D. Lesnic, A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source, J. Math. Anal. Appl., 434 (2016), 1376-1393.  doi: 10.1016/j.jmaa.2015.09.085.

[21]

F. YangX. X. LiD. G. Li and L. Wang, The simplified Tikhonov regularization method for solving a Riesz-Feller space-fractional backward diffusion problem, Math. Comput. Sci., 11 (2017), 91-110.  doi: 10.1007/s11786-017-0292-6.

[22]

J. ZhaoS. Liu and T. Liu, An inverse problem for space-fractional backward diffusion problem, Math. Methods Appl. Sci., 37 (2014), 1147-1158.  doi: 10.1002/mma.2876.

[23]

G. H. Zheng and T. Wei, Two regularization methods for solving a Riesz-Feller space-fractional backward diffusion problem, Inverse Probl., 26 (2010), 115017, 22 pp. doi: 10.1088/0266-5611/26/11/115017.

[24]

G. H. Zheng and Q. G. Zhang, Recovering the initial distribution for space-fractional diffusion equation by a logarithmic regularization method, Appl. Math. Lett., 61 (2016), 143-148.  doi: 10.1016/j.aml.2016.06.002.

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