Convergence to a stationary state of solutions to inverse problems of parabolic type

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June 2013, 6(3): 711-722. doi: 10.3934/dcdss.2013.6.711

Convergence to a stationary state of solutions to inverse problems of parabolic type

Davide Guidetti ^1,

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

Received April 2010 Revised October 2010 Published December 2012

Abstract
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We illustrate some results of existence and uniqueness of solutions to inverse parabolic problems of partial recostruction of the forcing term. In particular, we look for conditions assuring that the solution and the unknown part of the forcing term converge to a stationary state.

Keywords: inverse problems, Parabolic problems, asymptotic expansion of solutions..

Mathematics Subject Classification: Primary: 35K25, 35R30; Secondary: 35C2.

Citation: Davide Guidetti. Convergence to a stationary state of solutions to inverse problems of parabolic type. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 711-722. doi: 10.3934/dcdss.2013.6.711

References:

[1]	Y. Y. Belov, "Inverse Problems for Partial Differential Equations," Inverse and Ill-posed Problems Series, VSP, 2002. doi: 10.1515/9783110944631.
[2]	D. Guidetti, On elliptic problems in Besov spaces, Math. Nachr., 152 (1991), 247-275. doi: 10.1002/mana.19911520120.
[3]	D. Guidetti, On interpolation with boundary conditions, Math. Z., 207 (1991), 439-460. doi: 10.1007/BF02571401.
[4]	D. Guidetti, Convergence to a stationary state for solutions to parabolic inverse problems of reconstruction of convolution kernels, Diff. Int. Eq., 20 (2007), 961-990.
[5]	D. Guidetti, Asymptotic expansion of solutions to an inverse problem of parabolic type, Adv. Diff. Eq., 13 (2008), 399-426.
[6]	D. Guidetti, On linear elliptic and parabolic problems in Nikol'skij spaces, in "Parabolic Problems: the Herbert Amann Festschrift," 275-300, Birkhäuser (2011).
[7]	D. Guidetti, Asymptotic expansion of solutions to an inverse problem of parabolic type with nonhomogeneous boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 777-817. doi: 10.1017/S0308210510000788.
[8]	A. F. Güvenilir and V. K. Kalantarov, The asymptotic behavior of solutions to an inverse problem for differential operator equations, Mathematical and Computer Modelling, 37 (2003), 907-914. doi: 10.1016/S0895-7177(03)00106-7.
[9]	V. Kamynin and E. Francini, Asymptotic behavior of solutions of some inverse problems for higher order parabolic equations, Russ. Jour. Math. Phys., 6 (1999), 394-408.
[10]	H. Triebel, "Theory of Functions Spaces," Monogra. Math., Birkhäuser, 1983.
[11]	I. A. Vasin and V. L. Kamynin, On the asymptotic behavior of solutions to inverse problems for parabolic equations, Siberian Math. Journ., 38 (1997), 647-662. doi: 10.1007/BF02674572.
[12]	I. A. Vasin and V. L. Kamynin, Asymptotic behavior of the solutions of inverse problems for parabolic equations with irregular coefficients, Mat. Sbornik, 188 (1997), 49-64. doi: 10.1070/SM1997v188n03ABEH000210.

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References:

[1]	Y. Y. Belov, "Inverse Problems for Partial Differential Equations," Inverse and Ill-posed Problems Series, VSP, 2002. doi: 10.1515/9783110944631.
[2]	D. Guidetti, On elliptic problems in Besov spaces, Math. Nachr., 152 (1991), 247-275. doi: 10.1002/mana.19911520120.
[3]	D. Guidetti, On interpolation with boundary conditions, Math. Z., 207 (1991), 439-460. doi: 10.1007/BF02571401.
[4]	D. Guidetti, Convergence to a stationary state for solutions to parabolic inverse problems of reconstruction of convolution kernels, Diff. Int. Eq., 20 (2007), 961-990.
[5]	D. Guidetti, Asymptotic expansion of solutions to an inverse problem of parabolic type, Adv. Diff. Eq., 13 (2008), 399-426.
[6]	D. Guidetti, On linear elliptic and parabolic problems in Nikol'skij spaces, in "Parabolic Problems: the Herbert Amann Festschrift," 275-300, Birkhäuser (2011).
[7]	D. Guidetti, Asymptotic expansion of solutions to an inverse problem of parabolic type with nonhomogeneous boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 777-817. doi: 10.1017/S0308210510000788.
[8]	A. F. Güvenilir and V. K. Kalantarov, The asymptotic behavior of solutions to an inverse problem for differential operator equations, Mathematical and Computer Modelling, 37 (2003), 907-914. doi: 10.1016/S0895-7177(03)00106-7.
[9]	V. Kamynin and E. Francini, Asymptotic behavior of solutions of some inverse problems for higher order parabolic equations, Russ. Jour. Math. Phys., 6 (1999), 394-408.
[10]	H. Triebel, "Theory of Functions Spaces," Monogra. Math., Birkhäuser, 1983.
[11]	I. A. Vasin and V. L. Kamynin, On the asymptotic behavior of solutions to inverse problems for parabolic equations, Siberian Math. Journ., 38 (1997), 647-662. doi: 10.1007/BF02674572.
[12]	I. A. Vasin and V. L. Kamynin, Asymptotic behavior of the solutions of inverse problems for parabolic equations with irregular coefficients, Mat. Sbornik, 188 (1997), 49-64. doi: 10.1070/SM1997v188n03ABEH000210.

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