Exponential decay for a neutral one-dimensional viscoelastic equation

Year 2018, Volume: 47 Issue: 3, 625 - 635, 01.06.2018

Nasser-eddine Tatar

Abstract

In this work we consider a viscoelastic string subject to a delay of neutral type. The delay occurs in the second time derivative. Although delays are known by their destructive nature, here we prove an exponential decay result. We shall use the multiplier method and modify the classical energy by judicious choices of other functionals. This would lead to an appropriate differential inequality which allows us to conclude. It seems that this issue has never been discussed before in the literature.

Keywords

Exponential decay, modified energy, multiplier technique, neutral delay, stability

References

• Boltzmann L. Zur theorie der elastischen nachwirkung, Ann. Phys. Chem 7, 624-654, 1876.
• Bontsema J. and De Vries S. A. Robustness of exible systems against small time delays, in Proc. 27th Conference on Decision and Control, Austin, Texas, Dec. 1988.
• Christensen C. M., Theory of Viscoelasticity, Academic Press 1971.
• Coleman B. D. and Gurtin M. E., Waves in materials with memory II, Arch. Rational Mech. Anal. 19, 239-265, 1965.
• Dafermos C. M. Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37, 297- 308, 1970.
• Datko R. Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26, 697-713, 1988.
• Datko R., Lagnese J. and Polis M. P. An example of the eect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim. 24, 152-156, 1986.
• Day, N. A. Thermodynamics of Simple Materials with Fading Memory, Springer-Verlag, Berlin 1972.
• Desch W., Hannsgen K. B., Renardy Y. and Wheeler R. L. Boundary stabilization of an Euler-Bernoulli beam with viscoelastic damping, in Proc. 26th Conference on Decision and Control, Los Angeles, CA, Dec. 1987.
• Eduardo Hernández M. and Henriquez H. R. Existence results for second order partial neutral functional dierential equations, Dyn. Cont. Discrete Impul. Syst. Series A: Math. Anal. 15, 645-670, 2008.
• Eduardo Hernández M., Henriquez H. R. and McKibben M. A. Existence of solutions for second order partial neutral functional dierential equations, Integr. Equ. Oper. Theory 62, 191-217, 2008.
• Grimmer R., Lenczewski R. and Schappacher W. Well-posedness of hyperbolic equations with delay in the boundary conditions, in Semigroup Theory and Applications, P. Clement, S. Invernizzi, E. Mitidieri, and I. Vrabie, eds., Lecture Notes in Pure and Appl. Math. 116 (Marcel Dekker, New York and Basel, 1989).
• Guesmia A. Asymptotic stability of abstract dissipative systems with innite memory, J. Math. Anal. Appl. 382, 748-760, 2011.
• Hale J. K. Theory of Functional Dierential Equations, (Springer-Verlag, New York, 1997).
• Hale J. K. and Verduyn Lunel S.M. Introduction to Functional Dierential Equations, in: Applied Mathematical Sciences, 99 (Springer Verlag, New York, 2003).
• Hannsgen K. B., Renardy Y. and Wheeler R. L. Eectiveness and robustness with respect to time delays of boundary feedback stabilization in one-dimensional viscoelasticity, SIAM J. Control Optim. 26 (1988), 1200-1234.
• Kolmanovskii V. and Myshkis A. Introduction to the Theory and Applications of Functional Dierential Equations, (Kluwer Academic Publishers, Dordrecht, 1999).
• Liu G. and Yan J. Global asymptotic stability of nonlinear neutral dierential equation, Commun Nonlinear Sci. Numer. Simulat. 19, 1035-1041, 2014.
• MacCamy R. C. A model for one-dimensional nonlinear viscoelasticity, Quart. Appl. Math. 35, 21-33, 1977.
• Medjden M. and Tatar N.-e. Asymptotic behavior for a viscoelastic problem with not neces- sarily decreasing kernel, Appl. Math. Comput. 167 (2), 1221-1235, 2005.
• Medjden M. and Tatar N.-e. On the wave equation with a temporal nonlocal term, Dynamic Systems and Applications 16, 665-672, 2007.
• Messaoudi S. General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl. 341 (2), 1457-1467, 2008.
• Pata V. Exponential stability in linear viscoelasticity, Quart. Appl. Math. LXIV (3), 499- 513, 2006.
• Renardy M, HrusaW. J. and Nohel J. A., Mathematical Problems in Viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, 35, Longman Scientic and Technical, Hohn Wiley and Sons, New York, 1987.
• Rojsiraphisal T. and Niamsup P. Exponential stability of certain neutral dierential equa- tions, Appl. Math. Comput. 217, 3875-3880, 2010.
• Sánchez J. and Vergara V. Long-time behavior of nonlinear integro-dierential evolution equations, Nonlin. Anal.: T. M. A. 91, 20-31, 2013.
• Tatar N.-e. On a problem arising in isothermal viscoelasticity, Int. J. Pure and Appl. Math. 3 (1), 1-12, 2003.
• Tatar N.-e. Exponential decay for a viscoelastic problem with singular kernel, Zeit. Angew. Math. Phys. 60 (4), 640-650, 2009.
• Tatar N.-e. On a large class of kernels yielding exponential stability in viscoelasticity, Appl. Math. Comp. 215 (6), 2298-2306, 2009.
• Tatar N.-e. Arbitrary decays in linear viscoelasticity, Journal Math. Physics 52 (1), 013502, 2011.
• Tatar N.-e. A new class of kernels leading to an arbitrary decay in viscoelasticity, Mediterr. J. Math. 10 (1), 213-226, 2013.
• Wang J. Existence and stability of solutions for neutral dierential equations with delay, International Conference on Multimedia Technology (ICMT), 2011, 2462-2465, 10.1109/ICMT.2011.6002527.
• Wang W., Fan Q., Zhang Y. and Li S. Asymptotic stability of solution to nonlinear neutral and Volterra functional dierential equations in Banach spaces, Appl. Math. Comput. 237, 217-226, 2014.
• Wang W.-S., Li S.-F. and Yang R.-S. Contractivity and exponential stability of solutions to nonlinear neutral functional dierential equations in Banach spaces, Acta Math. Appl. Sinica, English Series 28 (2), 289-304, 2012.
• Wen L., Wang W. and Yu Y. Dissipativity and asymptotic stability of nonlinear neutral delay integro-dierential equations, Nonlin. Anal. T. M. A. 72, 1746-1754, 2010.
• Wu J. Theory and Applications of Partial Functional-Dierential Equations, in: Applied Mathematical Sciences, 119 (Springer-Verlag, New York, 1996).
• Wu J. and Xia H. Self-sustained oscillations in a ring array of coupled lossless transmission lines, J. Di. Eqs. 124 (1), 247-278, 1996.
• Wu J. and Xia H. Rotating waves in neutral partial functional-dierential equations, J. Dynam. Di. Eqs. 11 (2), 209-238, 1999.
• Xie S. Solvability of impulsive partial neutral second-order functional integro-dierential equations with innite delay, Boundary Value Prob. 2013, 203, 2013.
• Ye R. and Zhang G. Neutral functional dierential equations of second-order with innite delays, Electron. J. Di. Eqs. 2010 (36), 1-12, 2010.