doi: 10.3934/dcdss.2022140
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Reconstruction of a convolution kernel in an integrodifferential problem with a fractional time derivative

Dipartimento di matematica, Piazza di Porta S. Donato 5, 40127 Bologna, Italy

Dedicated to Professor Jerome A. Goldstein in occasion of his eightieth birthday
The author is a member of GNAMPA of Istituto Nazionale di Alta Matematica.

Received  April 2022 Early access July 2022

We consider the problem of reconstruction of a convolution kernel (together with the solution) for a linear abstract evolution equation with a fractional time derivative.

Citation: Davide Guidetti. Reconstruction of a convolution kernel in an integrodifferential problem with a fractional time derivative. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022140
References:
[1]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, vol. 96, Birkhäuser Verlag, 2001. doi: 10.1007/978-3-0348-5075-9.

[2]

J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equations, Inverse Problems, 25 (2009), 115002, 16 pp. doi: 10.1088/0266-5611/25/11/115002.

[3]

P. ClémentG. Gripenberg and S.-O. Londen, Schauder estimates for equations with fractional derivatives, Trans. Am. Math. Soc., 352 (2000), 2239-2260.  doi: 10.1090/S0002-9947-00-02507-1.

[4]

P. Clément, G. Gripenberg and S.-O. Londen, Regularity properties of solutions of fractional evolution equations, Lecture Notes in Pure and Applied Mathematics, 215 (2001), 235–246, Dekker, New York.

[5]

P. ClémentS.-O. Londen and G. Simonett, Quasilinear evolution equations and continuous interpolation spaces, J. Diff. Eq., 196 (2004), 418-447.  doi: 10.1016/j.jde.2003.07.014.

[6]

F. Colombo and D. Guidetti, A global in time existence and uniqueness result for a semilinear integrodifferential parabolic inverse problem in Sobolev spaces, Math. Models Methods Appl. Sci., 17 (2007), 1-29. 

[7]

F. Colombo and D. Guidetti, Some results in the identification of memory kernels, Operator Theory: Advances and Applications, 216 (2011), 121-138.  doi: 10.1007/978-3-0348-0069-3_7.

[8]

G. Da Prato and P. Grisvard, Sommes d'opérateurs linéaires et équations differentielles opérationelles, J. Math. Pures Appliquees, 54 (1975), 305-387. 

[9]

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

[10]

P. Feng and E. T. Karimov, Inverse source problems for time fractional mixed parabolic-hyperbolic-type equations, J. Inverse Ill-Posed Problems, 23 (2015), 339-353.  doi: 10.1515/jiip-2014-0022.

[11]

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

[12]

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

[13]

D. Guidetti, Optimal regularity for mixed parabolic problems in spaces of functions which are Hölder continuous with respect to space variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 763-790. 

[14]

D. Guidetti, On linear elliptic and parabolic problems in Nikol'skij spaces, Progress in nonlinear Differential Equations and their Applications, 80 (2011), 275-300.  doi: 10.1007/978-3-0348-0075-4_15.

[15]

D. Guidetti, On maximal regularity for abstract parabolic problems with fractional time derivative, Mediterr. J. Math., 16 (2019), Paper No. 40, 26 pp. doi: 10.1007/s00009-019-1309-y.

[16]

J. Janno, Determination of the order of fractional derivative and a kernel in an inverse problem for a generalized time fractional diffusion equation, Electronic Journal of Differential Equations, (2016), Paper No. 199, 28 pp.

[17]

J. Janno, Determination of time-dependent sources and parameters of nonlocal diffusion and wave equations from final data, Frac. Calc. Appl. Anal., 23 (2020), 1678-1701.  doi: 10.1515/fca-2020-0083.

[18]

J. Janno and K. Kasemets, Identification of a kernel in an evolutionary integral equation occurring in subdiffusion, J. Inverse Ill-Posed Problems, 25 (2017), 777-798.  doi: 10.1515/jiip-2016-0082.

[19]

N. Kinash and J. Janno, Inverse problems for a generalized subdiffusion equation with final overdetermination, Math. Model. Anal., 24 (2019), 236-262. 

[20]

A. Lorenzi and E. Sinestrari, An inverse problem in the theory of materials with memory, Nonlinear Anal., 12 (1988), 1317-1335.  doi: 10.1016/0362-546X(88)90080-6.

[21]

A. Lunardi, Interpolation Theory, Scuola Normale Superiore, 2009.

[22]

R. Metzler and J. Klafter, Boundary value problems for fractional diffusion equations, Physica A, 278 (2000), 107-125.  doi: 10.1016/S0378-4371(99)00503-8.

[23]

H. Tanabe, Equations of Evolution, Pitman, 1979.

[24]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978.

show all references

References:
[1]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, vol. 96, Birkhäuser Verlag, 2001. doi: 10.1007/978-3-0348-5075-9.

[2]

J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equations, Inverse Problems, 25 (2009), 115002, 16 pp. doi: 10.1088/0266-5611/25/11/115002.

[3]

P. ClémentG. Gripenberg and S.-O. Londen, Schauder estimates for equations with fractional derivatives, Trans. Am. Math. Soc., 352 (2000), 2239-2260.  doi: 10.1090/S0002-9947-00-02507-1.

[4]

P. Clément, G. Gripenberg and S.-O. Londen, Regularity properties of solutions of fractional evolution equations, Lecture Notes in Pure and Applied Mathematics, 215 (2001), 235–246, Dekker, New York.

[5]

P. ClémentS.-O. Londen and G. Simonett, Quasilinear evolution equations and continuous interpolation spaces, J. Diff. Eq., 196 (2004), 418-447.  doi: 10.1016/j.jde.2003.07.014.

[6]

F. Colombo and D. Guidetti, A global in time existence and uniqueness result for a semilinear integrodifferential parabolic inverse problem in Sobolev spaces, Math. Models Methods Appl. Sci., 17 (2007), 1-29. 

[7]

F. Colombo and D. Guidetti, Some results in the identification of memory kernels, Operator Theory: Advances and Applications, 216 (2011), 121-138.  doi: 10.1007/978-3-0348-0069-3_7.

[8]

G. Da Prato and P. Grisvard, Sommes d'opérateurs linéaires et équations differentielles opérationelles, J. Math. Pures Appliquees, 54 (1975), 305-387. 

[9]

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

[10]

P. Feng and E. T. Karimov, Inverse source problems for time fractional mixed parabolic-hyperbolic-type equations, J. Inverse Ill-Posed Problems, 23 (2015), 339-353.  doi: 10.1515/jiip-2014-0022.

[11]

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

[12]

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

[13]

D. Guidetti, Optimal regularity for mixed parabolic problems in spaces of functions which are Hölder continuous with respect to space variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 763-790. 

[14]

D. Guidetti, On linear elliptic and parabolic problems in Nikol'skij spaces, Progress in nonlinear Differential Equations and their Applications, 80 (2011), 275-300.  doi: 10.1007/978-3-0348-0075-4_15.

[15]

D. Guidetti, On maximal regularity for abstract parabolic problems with fractional time derivative, Mediterr. J. Math., 16 (2019), Paper No. 40, 26 pp. doi: 10.1007/s00009-019-1309-y.

[16]

J. Janno, Determination of the order of fractional derivative and a kernel in an inverse problem for a generalized time fractional diffusion equation, Electronic Journal of Differential Equations, (2016), Paper No. 199, 28 pp.

[17]

J. Janno, Determination of time-dependent sources and parameters of nonlocal diffusion and wave equations from final data, Frac. Calc. Appl. Anal., 23 (2020), 1678-1701.  doi: 10.1515/fca-2020-0083.

[18]

J. Janno and K. Kasemets, Identification of a kernel in an evolutionary integral equation occurring in subdiffusion, J. Inverse Ill-Posed Problems, 25 (2017), 777-798.  doi: 10.1515/jiip-2016-0082.

[19]

N. Kinash and J. Janno, Inverse problems for a generalized subdiffusion equation with final overdetermination, Math. Model. Anal., 24 (2019), 236-262. 

[20]

A. Lorenzi and E. Sinestrari, An inverse problem in the theory of materials with memory, Nonlinear Anal., 12 (1988), 1317-1335.  doi: 10.1016/0362-546X(88)90080-6.

[21]

A. Lunardi, Interpolation Theory, Scuola Normale Superiore, 2009.

[22]

R. Metzler and J. Klafter, Boundary value problems for fractional diffusion equations, Physica A, 278 (2000), 107-125.  doi: 10.1016/S0378-4371(99)00503-8.

[23]

H. Tanabe, Equations of Evolution, Pitman, 1979.

[24]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978.

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