American Institute of Mathematical Sciences

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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

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

J. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford Univ. Press, New York, 1985.  Google Scholar

[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.  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]

T. Kato, Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Univ. Tokyo, 25 (1970), 241-258.  Google Scholar

[12]

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

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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

W. Rudin, "Real and Complex Analysis," $3^{rd}$ edition, McGraw-Hill Inc., New York, 1987.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

J. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford Univ. Press, New York, 1985.  Google Scholar

[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.  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]

T. Kato, Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Univ. Tokyo, 25 (1970), 241-258.  Google Scholar

[12]

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

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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

W. Rudin, "Real and Complex Analysis," $3^{rd}$ edition, McGraw-Hill Inc., New York, 1987.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

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