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

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

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.
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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