Regularization for ill-posed inhomogeneous evolution problems in a Hilbert space

Matthew A. Fury

doi:10.3934/proc.2013.2013.259

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2013, 2013(special): 259-272. doi: 10.3934/proc.2013.2013.259

Regularization for ill-posed inhomogeneous evolution problems in a Hilbert space

Matthew A. Fury ^1,

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

Received July 2012 Published November 2013

Abstract
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Under a Creative Commons license

We prove regularization for ill-posed evolution problems that are both inhomogeneous and nonautonomous in a Hilbert Space $H$. We consider the ill-posed problem $du/dt = A(t,D)u(t)+h(t)$, $u(s)=\chi$, $0\leq s \leq t< T$ where $A(t,D)=\sum_{j=1}^ka_j(t)D^j$ with $a_j\in C([0,T]:\mathbb{R}^+)$ for each $1\leq j\leq k$ and $D$ a positive, self-adjoint operator in $H$. Assuming there exists a solution $u$ of the problem with certain stabilizing conditions, we approximate $u$ by the solution $v_{\beta}$ of the approximate well-posed problem $dv/dt = f_{\beta}(t,D)v(t)+h(t)$, $v(s)=\chi$, $0\leq s \leq t< T$ where $0<\beta <1$. Our method implies the existence of a family of regularizing operators for the given ill-posed problem with applications to a wide class of ill-posed partial differential equations including the inhomogeneous backward heat equation in $L^2(\mathbb{R}^n)$ with a time-dependent diffusion coefficient.

Keywords: Regularizing families of operators, ill-posed problems, evolution equations, backward heat equation..

Mathematics Subject Classification: Primary: 47D06, 46C99; Secondary: 35K0.

Citation: Matthew A. Fury. Regularization for ill-posed inhomogeneous evolution problems in a Hilbert space. Conference Publications, 2013, 2013 (special) : 259-272. doi: 10.3934/proc.2013.2013.259

References:

[1]	S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach space, Comm. Pure Appl. Math., 16 (1963), 121-151.
[2]	K. A. Ames, "Comparison Results for Related Properly and Improperly Posed Problems, with Applications to Mechanics," Ph.D. Thesis, Cornell University, Ithaca, NY, 1980.
[3]	K. A. Ames and R. J. Hughes, Structural stability for ill-posed problems in Banach space, Semigroup Forum, 70 (2005), 127-145.
[4]	B. Campbell Hetrick and R. J. Hughes, Continuous dependence results for inhomogeneous ill-posed problems in Banach space, J. Math. Anal. Appl., 331 (2007), 342-357.
[5]	N. Dunford and J. Schwartz, "Linear Operators, Part II," John Wiley and Sons, Inc., New York, 1957.
[6]	M. Fury and R. J. Hughes, Continuous dependence of solutions for ill-posed evolution problems, Electron. J. Diff. Eqns., Conf. 19 (2010), 99-121.
[7]	M. A. Fury and R. J. Hughes, Regularization for a class of ill-posed evolution problems in Banach space, Semigroup Forum, 85 (2012), 191-212, (DOI) 10.1007/s00233-011-9353-3.
[8]	J. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford Univ. Press, New York, 1985.
[9]	Y. Huang and Q. Zheng, Regularization for ill-posed Cauchy problems associated with generators of analytic semigroups, J. Differential Equations, 203 (2004), 38-54.
[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]	T. Kato, Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Univ. Tokyo, 25 (1970), 241-258.
[12]	R. Lattes and J. L. Lions, "The Method of Quasireversibility, Applications to Partial Differential Equations," Amer. Elsevier, New York, 1969.
[13]	I. V. Mel'nikova, General theory of the ill-posed Cauchy problem, J. Inverse and Ill-posed Problems, 3 (1995), 149-171.
[14]	I. V. Mel'nikova and A. I. Filinkov, "Abstract Cauchy Problems: Three Approaches," Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., vol. 120, Chapman & Hall, Boca Raton, FL, 2001.
[15]	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.
[16]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New York, 1983.
[17]	W. Rudin, "Real and Complex Analysis," $3^{rd}$ edition, McGraw-Hill Inc., New York, 1987.
[18]	R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl., 47 (1974), 563-572.
[19]	D. D. Trong and N. H. Tuan, Regularization and error estimates for nonhomogeneous backward heat problems, Electron. J. Diff. Eqns., 2006 (2006) No. 4, 1-10.
[20]	D. D. Trong and N. H. Tuan, A nonhomogeneous backward heat problem: regularization and error estimates, Electron. J. Diff. Eqns., 2008 (2008) No. 33, 1-14.

show all references

References:

[1]	S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach space, Comm. Pure Appl. Math., 16 (1963), 121-151.
[2]	K. A. Ames, "Comparison Results for Related Properly and Improperly Posed Problems, with Applications to Mechanics," Ph.D. Thesis, Cornell University, Ithaca, NY, 1980.
[3]	K. A. Ames and R. J. Hughes, Structural stability for ill-posed problems in Banach space, Semigroup Forum, 70 (2005), 127-145.
[4]	B. Campbell Hetrick and R. J. Hughes, Continuous dependence results for inhomogeneous ill-posed problems in Banach space, J. Math. Anal. Appl., 331 (2007), 342-357.
[5]	N. Dunford and J. Schwartz, "Linear Operators, Part II," John Wiley and Sons, Inc., New York, 1957.
[6]	M. Fury and R. J. Hughes, Continuous dependence of solutions for ill-posed evolution problems, Electron. J. Diff. Eqns., Conf. 19 (2010), 99-121.
[7]	M. A. Fury and R. J. Hughes, Regularization for a class of ill-posed evolution problems in Banach space, Semigroup Forum, 85 (2012), 191-212, (DOI) 10.1007/s00233-011-9353-3.
[8]	J. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford Univ. Press, New York, 1985.
[9]	Y. Huang and Q. Zheng, Regularization for ill-posed Cauchy problems associated with generators of analytic semigroups, J. Differential Equations, 203 (2004), 38-54.
[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]	T. Kato, Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Univ. Tokyo, 25 (1970), 241-258.
[12]	R. Lattes and J. L. Lions, "The Method of Quasireversibility, Applications to Partial Differential Equations," Amer. Elsevier, New York, 1969.
[13]	I. V. Mel'nikova, General theory of the ill-posed Cauchy problem, J. Inverse and Ill-posed Problems, 3 (1995), 149-171.
[14]	I. V. Mel'nikova and A. I. Filinkov, "Abstract Cauchy Problems: Three Approaches," Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., vol. 120, Chapman & Hall, Boca Raton, FL, 2001.
[15]	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.
[16]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New York, 1983.
[17]	W. Rudin, "Real and Complex Analysis," $3^{rd}$ edition, McGraw-Hill Inc., New York, 1987.
[18]	R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl., 47 (1974), 563-572.
[19]	D. D. Trong and N. H. Tuan, Regularization and error estimates for nonhomogeneous backward heat problems, Electron. J. Diff. Eqns., 2006 (2006) No. 4, 1-10.
[20]	D. D. Trong and N. H. Tuan, A nonhomogeneous backward heat problem: regularization and error estimates, Electron. J. Diff. Eqns., 2008 (2008) No. 33, 1-14.

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