Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 35K20, 35K30; Secondary: 35J25.

 Citation:

•  [1] P. Acquistapace and B. Terreni, Hölder classes with boundary conditions as interpolation spaces, Math. Z., 195 (1987), 451-471.doi: 10.1007/BF01166699. [2] H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186 (1997), 5-56.doi: 10.1002/mana.3211860102. [3] H. Amann, Vector-Valued Distributions and Fourier Multipliers, manuscript (2003), University of Zürich. [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. [5] Y. Y. Belov, "Inverse problems for Partial Differential Equations," Inverse Ill-posed Probl. Ser., VSP, 2002.doi: 10.1515/9783110944631. [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. [7] P. Grisvard, Commutativité de deux foncteurs d'interpolation et applications, J. Math. Pures et appl., 45 (1966), 143-206. [8] P. Grisvard, Spazi di tracce e applicazioni, Rend. Mat., 6 (1972), 657-729. [9] D. Guidetti, On interpolation with boundary conditions, Math. Z., 207 (1991), 439-460.doi: 10.1007/BF02571401. [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. [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). [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. [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. [14] N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Hölder Spaces," Graduate Studies in Mathematics vol. 12, American Mathematical Society, 1996. [15] A. Lorenzi and A. I. Prilepko, Fredholm-type results for integrodifferential identification parabolic problems, Differential Integral Equations, 6 (1993), 535-552. [16] A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems" Birkhäuser, 1995.doi: 10.1007/978-3-0348-9234-6. [17] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, "Methods for Solving Inverse Problems in Mathematical Physics," Marcel Dekker, 1999. [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. [19] L. Schwartz, "Mixed Problems in Partial Differential Equations and Representations of Semigroups," Tata Institute of Fundamental Research, 1964. [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. [21] H. Tanabe, "Equations of Evolution," Pitman, 1979. [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.