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.

[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.

[3]

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

[4]

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

[5]

N. Dunford and J. Schwartz, Linear Operators, Part II,, John Wiley and Sons, (1957).

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

R. Lattes and J. L. Lions, The Method of Quasireversibility, Applications to Partial Differential Equations,, Amer. Elsevier, (1969).

[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.

[13]

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

[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., (1972), 161.

[15]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).

[16]

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

[17]

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

[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.

[19]

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

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.

[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.

[3]

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

[4]

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

[5]

N. Dunford and J. Schwartz, Linear Operators, Part II,, John Wiley and Sons, (1957).

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

R. Lattes and J. L. Lions, The Method of Quasireversibility, Applications to Partial Differential Equations,, Amer. Elsevier, (1969).

[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.

[13]

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

[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., (1972), 161.

[15]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).

[16]

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

[17]

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

[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.

[19]

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

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