# American Institute of Mathematical Sciences

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. Google Scholar [2] K. A. Ames, "Comparison Results for Related Properly and Improperly Posed Problems, with Applications to Mechanics,", Ph.D. Thesis, (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. 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. Google Scholar [5] N. Dunford and J. Schwartz, "Linear Operators, Part II,", John Wiley and Sons, (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. 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. Google Scholar [8] J. A. Goldstein, "Semigroups of Linear Operators and Applications,", Oxford Univ. Press, (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. Google Scholar [10] Y. Huang and Q. Zheng, Regularization for a class of ill-posed Cauchy problems,, Proc. Amer. Math. Soc., 133-10 (2005), 133. Google Scholar [11] T. Kato, Linear evolution equations of "hyperbolic" type,, J. Fac. Sci. Univ. Tokyo, 25 (1970), 241. Google Scholar [12] R. Lattes and J. L. Lions, "The Method of Quasireversibility, Applications to Partial Differential Equations,", Amer. Elsevier, (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. Google Scholar [14] I. V. Mel'nikova and A. I. Filinkov, "Abstract Cauchy Problems: Three Approaches,", Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., (2001). Google Scholar [15] K. Miller, Stabilized quasi-reversibility and other nearly-best-possible methods for non-well-posed problems,, in, (1972), 161. Google Scholar [16] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer-Verlag, (1983). Google Scholar [17] W. Rudin, "Real and Complex Analysis,", $3^{rd}$ edition, (1987). Google Scholar [18] R. E. Showalter, The final value problem for evolution equations,, J. Math. Anal. Appl., 47 (1974), 563. 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), 1. 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), 1. Google Scholar

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##### References:
 [1] S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach space,, Comm. Pure Appl. Math., 16 (1963), 121. Google Scholar [2] K. A. Ames, "Comparison Results for Related Properly and Improperly Posed Problems, with Applications to Mechanics,", Ph.D. Thesis, (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. 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. Google Scholar [5] N. Dunford and J. Schwartz, "Linear Operators, Part II,", John Wiley and Sons, (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. 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. Google Scholar [8] J. A. Goldstein, "Semigroups of Linear Operators and Applications,", Oxford Univ. Press, (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. Google Scholar [10] Y. Huang and Q. Zheng, Regularization for a class of ill-posed Cauchy problems,, Proc. Amer. Math. Soc., 133-10 (2005), 133. Google Scholar [11] T. Kato, Linear evolution equations of "hyperbolic" type,, J. Fac. Sci. Univ. Tokyo, 25 (1970), 241. Google Scholar [12] R. Lattes and J. L. Lions, "The Method of Quasireversibility, Applications to Partial Differential Equations,", Amer. Elsevier, (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. Google Scholar [14] I. V. Mel'nikova and A. I. Filinkov, "Abstract Cauchy Problems: Three Approaches,", Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., (2001). Google Scholar [15] K. Miller, Stabilized quasi-reversibility and other nearly-best-possible methods for non-well-posed problems,, in, (1972), 161. Google Scholar [16] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer-Verlag, (1983). Google Scholar [17] W. Rudin, "Real and Complex Analysis,", $3^{rd}$ edition, (1987). Google Scholar [18] R. E. Showalter, The final value problem for evolution equations,, J. Math. Anal. Appl., 47 (1974), 563. 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), 1. 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), 1. Google Scholar
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