A class of inverse problems for viscoelastic material with dominating Newtonian viscosity

Janno, Jaan; von Wolfersdorf, Lothar

doi:10.1090/qam/1704447

Authors: Jaan Janno and Lothar von Wolfersdorf
Journal: Quart. Appl. Math. 57 (1999), 465-474
MSC: Primary 35R30; Secondary 35Q72, 45K05, 45Q05, 74D05, 76A10
DOI: https://doi.org/10.1090/qam/1704447
MathSciNet review: MR1704447
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Memory kernels in linear stress-strain relations involving a Newtonian viscosity are identified by solving a class of inverse problems. The inverse problems are reduced to nonlinear Volterra integral equations of the first kind which in turn lead to corresponding Volterra equations of the second kind by differentiation. Applying the contraction principle with weighted norms we derive global (in time) existence, uniqueness and stability of the solution to the inverse problems under similar assumptions as for related inverse problems in heat flow.

A. L. Bukhgeĭm, Inverse problems of memory reconstruction, J. Inverse Ill-Posed Probl. 1 (1993), no. 3, 193–205. MR 1256117, DOI https://doi.org/10.1515/jiip.1993.1.3.193
Maurizio Grasselli, An identification problem for an abstract linear hyperbolic integrodifferential equation with applications, J. Math. Anal. Appl. 171 (1992), no. 1, 27–60. MR 1192492, DOI https://doi.org/10.1016/0022-247X%2892%2990375-N
Maurizio Grasselli, On an inverse problem for a linear hyperbolic integrodifferential equation, Forum Math. 6 (1994), no. 1, 83–110. MR 1253179, DOI https://doi.org/10.1515/form.1994.6.83
M. Grasselli, S. I. Kabanikhin, and A. Lorentsi, An inverse problem for an integro-differential equation, Sibirsk. Mat. Zh. 33 (1992), no. 3, 58–68, 218 (Russian, with Russian summary); English transl., Siberian Math. J. 33 (1992), no. 3, 415–426 (1993). MR 1178458, DOI https://doi.org/10.1007/BF00970889
M. Grasselli, S. I. Kabanikhin, and A. Lorenzi, An inverse hyperbolic integrodifferential problem arising in geophysics. II, Nonlinear Anal. 15 (1990), no. 3, 283–298. MR 1065260, DOI https://doi.org/10.1016/0362-546X%2890%2990165-D
Gustaf Gripenberg, Unimodality and viscoelastic pulse propagation, Quart. Appl. Math. 51 (1993), no. 1, 183–189. MR 1205945, DOI https://doi.org/10.1090/qam/1205945

J. Janno, On an inverse problem for a model of radial wave propagation in the media with memory. In: Numerical Methods and Optimization, Vol. 2 (eds.: G. Vainikko et al), Estonian Acad. Sci., Tallinn 1990, pp. 4–19

J. Janno and L. Von Wolfersdorf, Inverse problems for identification of memory kernels in viscoelasticity, Math. Methods Appl. Sci. 20 (1997), no. 4, 291–314. MR 1432135, DOI https://doi.org/10.1002/%28SICI%291099-1476%2819970310%2920%3A4%3C291%3A%3AAID-MMA860%3E3.3.CO%3B2-N
J. Janno and L. von Wolfersdorf, Identification of weakly singular memory kernels in viscoelasticity, ZAMM Z. Angew. Math. Mech. 78 (1998), no. 6, 391–403 (English, with English and German summaries). MR 1632697, DOI https://doi.org/10.1002/%28SICI%291521-4001%28199806%2978%3A6%3C391%3A%3AAID-ZAMM391%3E3.3.CO%3B2-A
J. Janno and L. von Wolfersdorf, Identification of weakly singular memory kernels in heat conduction, Z. Angew. Math. Mech. 77 (1997), no. 4, 243–257 (English, with English and German summaries). MR 1449128, DOI https://doi.org/10.1002/zamm.19970770403
J. Janno and L. v. Wolfersdorf, Inverse problems for identification of memory kernels in heat flow, J. Inverse Ill-Posed Probl. 4 (1996), no. 1, 39–66. MR 1393354, DOI https://doi.org/10.1515/jiip.1996.4.1.39
A. Lorenzi, Identification problems for integrodifferential equations, Ill-posed problems in natural sciences (Moscow, 1991) VSP, Utrecht, 1992, pp. 342–366. MR 1219995

A. C. Pipkin, Lectures on Viscoelasticity Theory, 2nd ed., Springer, New York, 1986

Jan Prüss, Evolutionary integral equations and applications, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1993. [2012] reprint of the 1993 edition. MR 2964432
Michael Renardy, William J. Hrusa, and John A. Nohel, Mathematical problems in viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 35, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. MR 919738