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November  2013, 33(11&12): 5107-5141. doi: 10.3934/dcds.2013.33.5107

Partial reconstruction of the source term in a linear parabolic initial problem with Dirichlet boundary conditions

Davide Guidetti 1,

1. 

Dipartimento di Matematica, Piazza di Porta S. Donato, 5, 40126 Bologna

Received  October 2011 Revised  August 2012 Published  May 2013

We consider the problem of the reconstruction of the source term in a parabolic Cauchy-Dirichlet system in a cylindrical domain. The supplementary information, necessary to determine the unknown part of the source term together with the solution, is given by the knowledge of an integral of the solution with respect to some of the space variables.
Keywords: Reconstruction of the source term, cylindrical domains, parabolic mixed problems..
Mathematics Subject Classification: Primary: 35K20, 35K30; Secondary: 35J2.
Citation: Davide Guidetti. Partial reconstruction of the source term in a linear parabolic initial problem with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5107-5141. doi: 10.3934/dcds.2013.33.5107
References:
[1]

P. Acquistapace and B. Terreni, Hölder classes with boundary conditions as interpolation spaces, Math. Z., 195 (1987), 451-471. doi: 10.1007/BF01166699.  Google Scholar

[2]

H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186 (1997), 5-56. doi: 10.1002/mana.3211860102.  Google Scholar

[3]

H. Amann, Vector-Valued Distributions and Fourier Multipliers, manuscript (2003), University of Zürich. Google Scholar

[4]

Y. E. Anikonov and A. Lorenzi, Explicit representation for the solution to a parabolic differential identification problem in a Banach space, J. Inv. Ill-Posed Problems, 15 (2007), 669-681. doi: 10.1515/jiip.2007.037.  Google Scholar

[5]

Y. Y. Belov, "Inverse problems for Partial Differential Equations," Inverse Ill-posed Probl. Ser., VSP, 2002. doi: 10.1515/9783110944631.  Google Scholar

[6]

M. Di Cristo, D. Guidetti and A. Lorenzi, Abstract parabolic equations with applications to problems in cylindrical space domains, Adv. Diff. Eq., 15 (2010), 1-42.  Google Scholar

[7]

P. Grisvard, Commutativité de deux foncteurs d'interpolation et applications, J. Math. Pures et appl., 45 (1966), 143-206.  Google Scholar

[8]

P. Grisvard, Spazi di tracce e applicazioni, Rend. Mat., 6 (1972), 657-729.  Google Scholar

[9]

D. Guidetti, On interpolation with boundary conditions, Math. Z., 207 (1991), 439-460. doi: 10.1007/BF02571401.  Google Scholar

[10]

D. Guidetti, The parabolic mixed Cauchy-Dirichlet problem in spaces of functions which are Hölder continuous with respect to space variables, Rend. Mat. Acc. Lincei, 7 (1996), 161-168.  Google Scholar

[11]

D. Guidetti, Partial reconstruction of the source term in a linear parabolic initial problem with first order boundary conditions, to appear in Applicable Analysis (2013). Google Scholar

[12]

D. Guidetti and S. Piskarev, On real interpolation, finite differences, and estimates depending on a parameter for discretizations of elliptic boundary value problems, Abstr. Appl. Anal., 18 (2003), 1005-1035. doi: 10.1155/S1085337503306359.  Google Scholar

[13]

A. Hasanov, Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: Weak solutions approach, J. Math. Anal. Appl., 330 (2007), 766-779. doi: 10.1016/j.jmaa.2006.08.018.  Google Scholar

[14]

N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Hölder Spaces," Graduate Studies in Mathematics vol. 12, American Mathematical Society, 1996.  Google Scholar

[15]

A. Lorenzi and A. I. Prilepko, Fredholm-type results for integrodifferential identification parabolic problems, Differential Integral Equations, 6 (1993), 535-552.  Google Scholar

[16]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems" Birkhäuser, 1995. doi: 10.1007/978-3-0348-9234-6.  Google Scholar

[17]

A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, "Methods for Solving Inverse Problems in Mathematical Physics," Marcel Dekker, 1999.  Google Scholar

[18]

W. Rundell, Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data, Applicable Anal., 10 (1980), 231-242. doi: 10.1080/00036818008839304.  Google Scholar

[19]

L. Schwartz, "Mixed Problems in Partial Differential Equations and Representations of Semigroups," Tata Institute of Fundamental Research, 1964. Google Scholar

[20]

B. Stewart, Generation of analytic semigroups by strongly elliptic operators, Trans. Am. Math. Soc., 199 (1974), 141-162. doi: 10.1090/S0002-9947-1974-0358067-4.  Google Scholar

[21]

H. Tanabe, "Equations of Evolution," Pitman, 1979.  Google Scholar

[22]

W. von Wahl, Gebrochene Potenzen eines elliptischen Operators und parabolische Differentialgleichungen in Rumen hölderstetiger Funktionen, Nachr. Akad. Wiss. Göttingen II, Math. Phys. Klasse Jahrgang, 11 (1972), 231-258.  Google Scholar

show all references

References:
[1]

P. Acquistapace and B. Terreni, Hölder classes with boundary conditions as interpolation spaces, Math. Z., 195 (1987), 451-471. doi: 10.1007/BF01166699.  Google Scholar

[2]

H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186 (1997), 5-56. doi: 10.1002/mana.3211860102.  Google Scholar

[3]

H. Amann, Vector-Valued Distributions and Fourier Multipliers, manuscript (2003), University of Zürich. Google Scholar

[4]

Y. E. Anikonov and A. Lorenzi, Explicit representation for the solution to a parabolic differential identification problem in a Banach space, J. Inv. Ill-Posed Problems, 15 (2007), 669-681. doi: 10.1515/jiip.2007.037.  Google Scholar

[5]

Y. Y. Belov, "Inverse problems for Partial Differential Equations," Inverse Ill-posed Probl. Ser., VSP, 2002. doi: 10.1515/9783110944631.  Google Scholar

[6]

M. Di Cristo, D. Guidetti and A. Lorenzi, Abstract parabolic equations with applications to problems in cylindrical space domains, Adv. Diff. Eq., 15 (2010), 1-42.  Google Scholar

[7]

P. Grisvard, Commutativité de deux foncteurs d'interpolation et applications, J. Math. Pures et appl., 45 (1966), 143-206.  Google Scholar

[8]

P. Grisvard, Spazi di tracce e applicazioni, Rend. Mat., 6 (1972), 657-729.  Google Scholar

[9]

D. Guidetti, On interpolation with boundary conditions, Math. Z., 207 (1991), 439-460. doi: 10.1007/BF02571401.  Google Scholar

[10]

D. Guidetti, The parabolic mixed Cauchy-Dirichlet problem in spaces of functions which are Hölder continuous with respect to space variables, Rend. Mat. Acc. Lincei, 7 (1996), 161-168.  Google Scholar

[11]

D. Guidetti, Partial reconstruction of the source term in a linear parabolic initial problem with first order boundary conditions, to appear in Applicable Analysis (2013). Google Scholar

[12]

D. Guidetti and S. Piskarev, On real interpolation, finite differences, and estimates depending on a parameter for discretizations of elliptic boundary value problems, Abstr. Appl. Anal., 18 (2003), 1005-1035. doi: 10.1155/S1085337503306359.  Google Scholar

[13]

A. Hasanov, Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: Weak solutions approach, J. Math. Anal. Appl., 330 (2007), 766-779. doi: 10.1016/j.jmaa.2006.08.018.  Google Scholar

[14]

N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Hölder Spaces," Graduate Studies in Mathematics vol. 12, American Mathematical Society, 1996.  Google Scholar

[15]

A. Lorenzi and A. I. Prilepko, Fredholm-type results for integrodifferential identification parabolic problems, Differential Integral Equations, 6 (1993), 535-552.  Google Scholar

[16]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems" Birkhäuser, 1995. doi: 10.1007/978-3-0348-9234-6.  Google Scholar

[17]

A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, "Methods for Solving Inverse Problems in Mathematical Physics," Marcel Dekker, 1999.  Google Scholar

[18]

W. Rundell, Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data, Applicable Anal., 10 (1980), 231-242. doi: 10.1080/00036818008839304.  Google Scholar

[19]

L. Schwartz, "Mixed Problems in Partial Differential Equations and Representations of Semigroups," Tata Institute of Fundamental Research, 1964. Google Scholar

[20]

B. Stewart, Generation of analytic semigroups by strongly elliptic operators, Trans. Am. Math. Soc., 199 (1974), 141-162. doi: 10.1090/S0002-9947-1974-0358067-4.  Google Scholar

[21]

H. Tanabe, "Equations of Evolution," Pitman, 1979.  Google Scholar

[22]

W. von Wahl, Gebrochene Potenzen eines elliptischen Operators und parabolische Differentialgleichungen in Rumen hölderstetiger Funktionen, Nachr. Akad. Wiss. Göttingen II, Math. Phys. Klasse Jahrgang, 11 (1972), 231-258.  Google Scholar

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