Stability of solutions of a class of nonlinear fractional Laplacian parabolic problems

Abstract In this paper we study the forward and the backward in time problems for a class of nonlinear partial differential equations of fractional Laplace operator with nonlinear source function. The equations are generalized from lots of well-known equations such as the Burger equation, the Cahn–Hilliard equation. Herein we investigate the stability of the solution of the forward problem depended on the fractional order of the operator and the instability of the backward problem, and then, we establish a regularization method. Some numerical experiments are performed to verify efficiency and accuracy of our method.

[1]  Pham Hoang Quan,et al.  A Nonlinear Case of the 1-D Backward Heat Problem: Regularization and Error Estimate , 2007 .

[2]  Radu Precup,et al.  Methods in Nonlinear Integral Equations , 2002 .

[3]  Seth F. Oppenheimer,et al.  Quasireversibility Methods for Non-Well-Posed Problems , 1994 .

[4]  Dang Duc Trong,et al.  A two-dimensional backward heat problem with statistical discrete data , 2016, 1606.05463.

[5]  H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations , 2010 .

[6]  Michael Metcalf,et al.  Modern Fortran Explained , 2011, Oxford Scholarship Online.

[7]  Huy Tuan Nguyen,et al.  REGULARIZATION AND ERROR ESTIMATES FOR NONHOMOGENEOUS BACKWARD HEAT PROBLEMS , 2006 .

[8]  R. Temam,et al.  Gevrey class regularity for the solutions of the Navier-Stokes equations , 1989 .

[9]  Jun-Sheng Duan.  Time- and space-fractional partial differential equations , 2005 .

[10]  Robert Lattès,et al.  Méthode de quasi-réversibilbilité et applications , 1967 .

[11]  Le Minh Triet,et al.  On a backward nonlinear parabolic equation with time and space dependent thermal conductivity: Regularization and error estimates , 2014 .

[12]  Ralph E. Showalter,et al.  The final value problem for evolution equations , 1974 .

[13]  D. Hào,et al.  Stability estimates for Burgers-type equations backward in time , 2015 .

[14]  Dang Duc Trong,et al.  On a backward parabolic problem with local Lipschitz source , 2014 .

[15]  Norbert Heuer,et al.  Numerical Approximation of a Time Dependent, Nonlinear, Space-Fractional Diffusion Equation , 2007, SIAM J. Numer. Anal..

[16]  P. H. Quan,et al.  A backward nonlinear heat equation: regularization with error estimates , 2005 .

[17]  D. D. Trong,et al.  A nonlinearly backward heat problem: uniqueness, regularization and error estimate , 2006 .

[18]  George Marsaglia,et al.  The 64-bit universal RNG , 2004 .

[19]  A. Carasso,et al.  Computing Small Solutions of Burgers' Equation Backwards in Time* , 1977 .

[20]  Bui Thanh Duy,et al.  Backward heat equations with locally lipschitz source , 2015 .