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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 $\mathbb{R}^n$

Alfredo Lorenzi 1, and  Luca Lorenzi 2,

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.
Keywords: continuous dependence results., Carleman estimates, uniqueness, linear singular differential and integro-differential parabolic equations, Ill-posed problems.
Mathematics Subject Classification: Primary: 30R35; Secondary: 35K20, 34G10, 45N05, 45Q0.
Citation: Alfredo Lorenzi, Luca Lorenzi. A strongly ill-posed integrodifferential singular parabolic problem in the unit cube of $\mathbb{R}^n$. Evolution Equations & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

P. Lax, Functional Analysis, Wiley-Interscience, 2002.  Google Scholar

[5]

A. Lorenzi, Two strongly ill-posed problems, AIP Conference Proceedings, Melville, New York, 1329 (2011), 150-169.  Google Scholar

[6]

A. Lorenzi, Recovering a constant in a strongly ill-posed parabolic problem, J. Abstr. Differ. Equ. Appl., 2 (2012), 72-92.  Google Scholar

[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.  Google Scholar

[8]

A. Lorenzi, Severely ill-posed linear parabolic integrodifferential problems, J. Inverse Ill-Posed Probl., (2012). Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

A. Lorenzi and M. Yamamoto, Continuous dependence and uniqueness for a strongly ill-posed problem for linear integrodifferential parabolic equations,, in progress., ().   Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

P. Lax, Functional Analysis, Wiley-Interscience, 2002.  Google Scholar

[5]

A. Lorenzi, Two strongly ill-posed problems, AIP Conference Proceedings, Melville, New York, 1329 (2011), 150-169.  Google Scholar

[6]

A. Lorenzi, Recovering a constant in a strongly ill-posed parabolic problem, J. Abstr. Differ. Equ. Appl., 2 (2012), 72-92.  Google Scholar

[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.  Google Scholar

[8]

A. Lorenzi, Severely ill-posed linear parabolic integrodifferential problems, J. Inverse Ill-Posed Probl., (2012). Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

A. Lorenzi and M. Yamamoto, Continuous dependence and uniqueness for a strongly ill-posed problem for linear integrodifferential parabolic equations,, in progress., ().   Google Scholar

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