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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

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 & 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.  Google Scholar

[2]

D. Guidetti, On elliptic problems in Besov spaces, Math. Nachr., 152 (1991), 247-275. doi: 10.1002/mana.19911520120.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

D. Guidetti, Asymptotic expansion of solutions to an inverse problem of parabolic type, Adv. Diff. Eq., 13 (2008), 399-426.  Google Scholar

[6]

D. Guidetti, On linear elliptic and parabolic problems in Nikol'skij spaces, in "Parabolic Problems: the Herbert Amann Festschrift," 275-300, Birkhäuser (2011). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

H. Triebel, "Theory of Functions Spaces," Monogra. Math., Birkhäuser, 1983. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

Y. Y. Belov, "Inverse Problems for Partial Differential Equations," Inverse and Ill-posed Problems Series, VSP, 2002. doi: 10.1515/9783110944631.  Google Scholar

[2]

D. Guidetti, On elliptic problems in Besov spaces, Math. Nachr., 152 (1991), 247-275. doi: 10.1002/mana.19911520120.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

D. Guidetti, Asymptotic expansion of solutions to an inverse problem of parabolic type, Adv. Diff. Eq., 13 (2008), 399-426.  Google Scholar

[6]

D. Guidetti, On linear elliptic and parabolic problems in Nikol'skij spaces, in "Parabolic Problems: the Herbert Amann Festschrift," 275-300, Birkhäuser (2011). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

H. Triebel, "Theory of Functions Spaces," Monogra. Math., Birkhäuser, 1983. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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