# American Institute of Mathematical Sciences

September  2014, 3(3): 499-524. doi: 10.3934/eect.2014.3.499

## A strongly ill-posed integrodifferential singular parabolic problem in the unit cube of $\mathbb{R}^n$

 1 Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Saldini 50, 20133, Milano 2 Dipartimento di Matematica, Università degli Studi di Parma, Viale Parco Area delle Scienze 53/A, I-43124 Parma

Received  April 2013 Revised  May 2014 Published  August 2014

Via Carleman's estimates we prove uniqueness and continuous dependence results for a severely ill-posed linear integro-differential singular parabolic problems without initial conditions.
Citation: Alfredo Lorenzi, Luca Lorenzi. A strongly ill-posed integrodifferential singular parabolic problem in the unit cube of $\mathbb{R}^n$. Evolution Equations and Control Theory, 2014, 3 (3) : 499-524. doi: 10.3934/eect.2014.3.499
##### References:
 [1] D. Bainov and P. Simeonov, Integral Inequalities and Applications, Translated by R. A. M. Hoksbergen and V. Covachev [V. Khr. Kovachev], Mathematics and its Applications (East European Series), 57, Kluwer Academic Publishers Group, Dordrecht, 1992. doi: 10.1007/978-94-015-8034-2. [2] P. Cannarsa, J. Tort and M. Yamamoto, Determination of source terms in a degenerate parabolic equation, Inverse Problems, 26 (2010), 105003, 20 pp. doi: 10.1088/0266-5611/26/10/105003. [3] M. Choulli, Une Introduction aux Problèms Inverses Elliptiques et Paraboliques, Mathematiques and Applications, 65, Springer-Verlag, Berlin Heidelberg, 2009. doi: 10.1007/978-3-642-02460-3. [4] P. Lax, Functional Analysis, Wiley-Interscience, 2002. [5] A. Lorenzi, Two strongly ill-posed problems, AIP Conference Proceedings, Melville, New York, 1329 (2011), 150-169. [6] A. Lorenzi, Recovering a constant in a strongly ill-posed parabolic problem, J. Abstr. Differ. Equ. Appl., 2 (2012), 72-92. [7] A. Lorenzi, Linear integro-differential Schrödinger and plate problems without initial conditions, Appl. Math. Optim., 67 (2013), 391-418. doi: 10.1007/s00245-013-9192-6. [8] A. Lorenzi, Severely ill-posed linear parabolic integrodifferential problems, J. Inverse Ill-Posed Probl., (2012). [9] A. Lorenzi, Recovering a t-function in a strongly ill-posed integro-differential parabolic problem with integral boundary conditions, to appear in Mathematical Modelling and Analysis. [10] A. Lorenzi and L. Lorenzi, A strongly ill-posed problem for a degenerate parabolic equation with unbounded coefficients in an unbounded domain $\Omega\times \mathcal O$ of $\mathbb R^{M+N}$, Inverse Problems, 29 (2013), 025007, 22 pp. doi: 10.1088/0266-5611/29/2/025007. [11] A. Lorenzi and F. Messina, Unique continuation and continuous dependence results for a strongly ill-posed integro-differential parabolic problem, J. Inverse Ill-Posed Probl., 20 (2012), 615-636. doi: 10.1515/jip-2012-0032. [12] A. Lorenzi and I. Munteanu, Recovering a constant in the two-dimensional Navier-Stokes system with no initial condition, to appear in Applied Mathematics and Optimization. doi: 10.1007/s00245-014-9261-5. [13] A. Lorenzi and M. Yamamoto, Continuous dependence and uniqueness for a strongly ill-posed problem for linear integrodifferential parabolic equations, in progress.

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##### References:
 [1] D. Bainov and P. Simeonov, Integral Inequalities and Applications, Translated by R. A. M. Hoksbergen and V. Covachev [V. Khr. Kovachev], Mathematics and its Applications (East European Series), 57, Kluwer Academic Publishers Group, Dordrecht, 1992. doi: 10.1007/978-94-015-8034-2. [2] P. Cannarsa, J. Tort and M. Yamamoto, Determination of source terms in a degenerate parabolic equation, Inverse Problems, 26 (2010), 105003, 20 pp. doi: 10.1088/0266-5611/26/10/105003. [3] M. Choulli, Une Introduction aux Problèms Inverses Elliptiques et Paraboliques, Mathematiques and Applications, 65, Springer-Verlag, Berlin Heidelberg, 2009. doi: 10.1007/978-3-642-02460-3. [4] P. Lax, Functional Analysis, Wiley-Interscience, 2002. [5] A. Lorenzi, Two strongly ill-posed problems, AIP Conference Proceedings, Melville, New York, 1329 (2011), 150-169. [6] A. Lorenzi, Recovering a constant in a strongly ill-posed parabolic problem, J. Abstr. Differ. Equ. Appl., 2 (2012), 72-92. [7] A. Lorenzi, Linear integro-differential Schrödinger and plate problems without initial conditions, Appl. Math. Optim., 67 (2013), 391-418. doi: 10.1007/s00245-013-9192-6. [8] A. Lorenzi, Severely ill-posed linear parabolic integrodifferential problems, J. Inverse Ill-Posed Probl., (2012). [9] A. Lorenzi, Recovering a t-function in a strongly ill-posed integro-differential parabolic problem with integral boundary conditions, to appear in Mathematical Modelling and Analysis. [10] A. Lorenzi and L. Lorenzi, A strongly ill-posed problem for a degenerate parabolic equation with unbounded coefficients in an unbounded domain $\Omega\times \mathcal O$ of $\mathbb R^{M+N}$, Inverse Problems, 29 (2013), 025007, 22 pp. doi: 10.1088/0266-5611/29/2/025007. [11] A. Lorenzi and F. Messina, Unique continuation and continuous dependence results for a strongly ill-posed integro-differential parabolic problem, J. Inverse Ill-Posed Probl., 20 (2012), 615-636. doi: 10.1515/jip-2012-0032. [12] A. Lorenzi and I. Munteanu, Recovering a constant in the two-dimensional Navier-Stokes system with no initial condition, to appear in Applied Mathematics and Optimization. doi: 10.1007/s00245-014-9261-5. [13] A. Lorenzi and M. Yamamoto, Continuous dependence and uniqueness for a strongly ill-posed problem for linear integrodifferential parabolic equations, in progress.
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