Estimates for solutions of nonautonomous semilinear ill-posed problems

Matthew A. Fury

doi:10.3934/proc.2015.0479

Article Contents

2015, Volume 2015: 479-488. Doi: 10.3934/proc.2015.0479 open access

This issue Previous Article Remark on a semirelativistic equation in the energy space Next Article Blow-up for nonlinear inequalities with gradient terms and singularities on unbounded sets

Estimates for solutions of nonautonomous semilinear ill-posed problems

Matthew A. Fury^1,

1.
Division of Science and Engineering, Penn State Abington, 1600 Woodland Road, Abington, PA 19001

Received: August 31, 2014

Revised: December 31, 2014

Published: October 2015

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 47D03, 46C99; Secondary: 35K55.

Citation:

Related Papers

Cited by

References

[1]	K. A. Ames and R. J. Hughes, Structural stability for ill-posed problems in Banach space, Semigroup Forum, 70 (2005), 127-145.
[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.
[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.
[4]	G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well-posed problems, Electron. J. Diff. Eqns., 1994 (1994), 1-9.
[5]	N. Dunford and J. Schwartz, Linear Operators, Part II, John Wiley and Sons, Inc., New York, 1957.
[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.
[7]	M. A. Fury, Modified quasi-reversibility method for nonautonomous semilinear problems, Electron. J. Diff. Eqns., Conf. 20 (2013), 65-78.
[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.
[9]	Y. Huang, Modified quasi-reversibility method for final value problems in Banach spaces, J. Math. Anal. Appl. 340 (2008) 757-769.
[10]	Y. Huang and Q. Zheng, Regularization for a class of ill-posed Cauchy problems, Proc. Amer. Math. Soc., 133-10 (2005), 3005-3012.
[11]	R. Lattes and J. L. Lions, The Method of Quasireversibility, Applications to Partial Differential Equations, Amer. Elsevier, New York, 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), 905-914.
[13]	I. V. Mel'nikova, General theory of the ill-posed Cauchy problem, J. Inverse and Ill-posed Problems, 3 (1995), 149-171.
[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.
[15]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
[16]	R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl., 47 (1974), 563-572.
[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.
[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.
[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.