2015, 2015(special): 479-488. doi: 10.3934/proc.2015.0479

Estimates for solutions of nonautonomous semilinear ill-posed problems

1. 

Division of Science and Engineering, Penn State Abington, 1600 Woodland Road, Abington, PA 19001

Received  September 2014 Revised  January 2015 Published  November 2015

The nonautonomous, semilinear problem $\frac{du}{dt}=A(t)u(t)+h(t,u(t))$, $0 \leq s \leq t < T$, $u(s)=\chi$ in Hilbert space with a Lipschitz condition on $h$, is generally ill-posed under prescribed conditions on the operators $A(t)$. Hence, regularization techniques are sought out in order to estimate known solutions of the problem. We study two quasi-reversibility methods of approximation which have successfully established regularization in the linear case, and provide an estimate on a solution $u(t)$ of the problem under these approximations in the nonlinear case. The results apply to partial differential equations of arbitrary even order including the nonlinear backward heat equation with a time-dependent diffusion coefficient.
Citation: Matthew A. Fury. Estimates for solutions of nonautonomous semilinear ill-posed problems. Conference Publications, 2015, 2015 (special) : 479-488. doi: 10.3934/proc.2015.0479
References:
[1]

K. A. Ames and R. J. Hughes, Structural stability for ill-posed problems in Banach space, Semigroup Forum, 70 (2005), 127-145.  Google Scholar

[2]

N. Boussetila and F. Rebbani, A modified quasi-reversibility method for a class of ill-posed Cauchy problems, Georgian Math J., 14 (2007), 627-642.  Google Scholar

[3]

B. Campbell Hetrick and R. J. Hughes, Continuous dependence on modeling for nonlinear ill-posed problems, J. Math. Anal. Appl., 349 (2009), 420-435.  Google Scholar

[4]

G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well-posed problems, Electron. J. Diff. Eqns., 1994 (1994), 1-9.  Google Scholar

[5]

N. Dunford and J. Schwartz, Linear Operators, Part II, John Wiley and Sons, Inc., New York, 1957.  Google Scholar

[6]

M. A. Fury, Regularization for ill-posed inhomogeneous evolution problems in a Hilbert space, Discrete and Continuous Dynamical Systems, 2013 (2013), Issue special, 259-272. Google Scholar

[7]

M. A. Fury, Modified quasi-reversibility method for nonautonomous semilinear problems, Electron. J. Diff. Eqns., Conf. 20 (2013), 65-78. Google Scholar

[8]

M. Fury and R. J. Hughes, Continuous dependence of solutions for ill-posed evolution problems, Electron. J. Diff. Eqns., Conf. 19 (2010), 99-121.  Google Scholar

[9]

Y. Huang, Modified quasi-reversibility method for final value problems in Banach spaces, J. Math. Anal. Appl. 340 (2008) 757-769.  Google Scholar

[10]

Y. Huang and Q. Zheng, Regularization for a class of ill-posed Cauchy problems, Proc. Amer. Math. Soc., 133-10 (2005), 3005-3012.  Google Scholar

[11]

R. Lattes and J. L. Lions, The Method of Quasireversibility, Applications to Partial Differential Equations, Amer. Elsevier, New York, 1969.  Google Scholar

[12]

N. T. Long and A. P. N. Dinh, Approximation of a parabolic non-linear evolution equation backwards in time, Inverse Problems, 10 (1994), 905-914.  Google Scholar

[13]

I. V. Mel'nikova, General theory of the ill-posed Cauchy problem, J. Inverse and Ill-posed Problems, 3 (1995), 149-171.  Google Scholar

[14]

K. Miller, Stabilized quasi-reversibility and other nearly-best-possible methods for non-well-posed problems, in Symposium on Non-Well-Posed Problems and Logarithmic Convexity (Heriot-Watt Univ., Edinburgh, 1972), 161-176, Springer Lecture Notes in Mathematics, Volume 316, Springer, Berlin, 1973.  Google Scholar

[15]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.  Google Scholar

[16]

R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl., 47 (1974), 563-572.  Google Scholar

[17]

D. D. Trong and N. H. Tuan, Regularization and error estimates for nonhomogeneous backward heat problems, Electron. J. Diff. Eqns., 2006 (2006), 1-10.  Google Scholar

[18]

D. D. Trong and N. H. Tuan, Stabilized quasi-reversibility method for a class of nonlinear ill-posed problems, Electron. J. Diff. Eqns., 2008 (2008), 1-12.  Google Scholar

[19]

N. H. Tuan and D. D. Trong, On a backward parabolic problem with local Lipschitz source, J. Math. Anal. Appl. 414 (2014), 678-692.  Google Scholar

show all references

References:
[1]

K. A. Ames and R. J. Hughes, Structural stability for ill-posed problems in Banach space, Semigroup Forum, 70 (2005), 127-145.  Google Scholar

[2]

N. Boussetila and F. Rebbani, A modified quasi-reversibility method for a class of ill-posed Cauchy problems, Georgian Math J., 14 (2007), 627-642.  Google Scholar

[3]

B. Campbell Hetrick and R. J. Hughes, Continuous dependence on modeling for nonlinear ill-posed problems, J. Math. Anal. Appl., 349 (2009), 420-435.  Google Scholar

[4]

G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well-posed problems, Electron. J. Diff. Eqns., 1994 (1994), 1-9.  Google Scholar

[5]

N. Dunford and J. Schwartz, Linear Operators, Part II, John Wiley and Sons, Inc., New York, 1957.  Google Scholar

[6]

M. A. Fury, Regularization for ill-posed inhomogeneous evolution problems in a Hilbert space, Discrete and Continuous Dynamical Systems, 2013 (2013), Issue special, 259-272. Google Scholar

[7]

M. A. Fury, Modified quasi-reversibility method for nonautonomous semilinear problems, Electron. J. Diff. Eqns., Conf. 20 (2013), 65-78. Google Scholar

[8]

M. Fury and R. J. Hughes, Continuous dependence of solutions for ill-posed evolution problems, Electron. J. Diff. Eqns., Conf. 19 (2010), 99-121.  Google Scholar

[9]

Y. Huang, Modified quasi-reversibility method for final value problems in Banach spaces, J. Math. Anal. Appl. 340 (2008) 757-769.  Google Scholar

[10]

Y. Huang and Q. Zheng, Regularization for a class of ill-posed Cauchy problems, Proc. Amer. Math. Soc., 133-10 (2005), 3005-3012.  Google Scholar

[11]

R. Lattes and J. L. Lions, The Method of Quasireversibility, Applications to Partial Differential Equations, Amer. Elsevier, New York, 1969.  Google Scholar

[12]

N. T. Long and A. P. N. Dinh, Approximation of a parabolic non-linear evolution equation backwards in time, Inverse Problems, 10 (1994), 905-914.  Google Scholar

[13]

I. V. Mel'nikova, General theory of the ill-posed Cauchy problem, J. Inverse and Ill-posed Problems, 3 (1995), 149-171.  Google Scholar

[14]

K. Miller, Stabilized quasi-reversibility and other nearly-best-possible methods for non-well-posed problems, in Symposium on Non-Well-Posed Problems and Logarithmic Convexity (Heriot-Watt Univ., Edinburgh, 1972), 161-176, Springer Lecture Notes in Mathematics, Volume 316, Springer, Berlin, 1973.  Google Scholar

[15]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.  Google Scholar

[16]

R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl., 47 (1974), 563-572.  Google Scholar

[17]

D. D. Trong and N. H. Tuan, Regularization and error estimates for nonhomogeneous backward heat problems, Electron. J. Diff. Eqns., 2006 (2006), 1-10.  Google Scholar

[18]

D. D. Trong and N. H. Tuan, Stabilized quasi-reversibility method for a class of nonlinear ill-posed problems, Electron. J. Diff. Eqns., 2008 (2008), 1-12.  Google Scholar

[19]

N. H. Tuan and D. D. Trong, On a backward parabolic problem with local Lipschitz source, J. Math. Anal. Appl. 414 (2014), 678-692.  Google Scholar

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