Abstract
An integro differential equation which is able to describe the evolution of a large class of dissipative models, is considered. By means of an equivalence, the focus shifts to the perturbed sine-Gordon equation that in superconductivity finds interesting applications in multiple engineering areas. The Neumann boundary problem is considered, and the behaviour of a viscous term, defined by a higher-order derivative with small diffusion coefficient \(\varepsilon ,\) is investigated. The Green function, expressed by means of Fourier series, is considered, and an estimate is achieved. Furthermore, some classes of solutions of the hyperbolic equation are determined, proving that there exists at least one solution with bounded derivatives. Results obtained prove that diffusion effects are bounded and tend to zero when \(\varepsilon\) tends to zero.
Similar content being viewed by others
References
Oquendo HP (2017) Frictional versus Kelvin Voigt damping in a transmission problem. Math Meth Appl Sci 40:7026–7032
Renardy M (2004) On localized Kelvin-Voigt damping. ZAMM Z Angew Math Mech 84:280–283
Rackea R (2004) Nonlinear thermoelastic plate equations Global existence and decay rates for the Cauchy problem. J Differ Equ 263(2017):8138–8177
Straughan B (2011) Heat waves 318, vol 177. Springer series in applied mathematical sciences. Springer, New York
Shamaev AS, Shumilova VV (2017) Plane acoustic wave propagation through a composite of elastic and Kelvin Voigt viscoelastic material layers. Mech Solids 52(1):25–34
De Angelis M, Renno P (2008) On the FitzHugh- Nagumo model In: Proceedings WASCOM 2007 XIV international conference on waves and stability in continuous media. World Scientific, Singapore, pp 193–198
Angelis M (2013) A priori estimates for excitable models. Meccanica 48(10):2491–2496
De Angelis M, Renno P (2014) Asymptotic effects of boundary perturbations in excitable systems, discrete and continuous dynamical systems. Ser B 19(7):2039–2045
Murray JD (2003) Mathematical biology, I, II. Springer, New York
Carillo S (2015) Singular kernel problems in materials with memory. Meccanica 50:603–615
Fabrizio M, Lazzari B, Nibbi R (2017) Existence and stability for a visco-plastic material with a fractional constitutive equation. Math Methods Appl Sci 40:6306–6315
Fabrizio M, Lazzari B, Nibbi R (2015) Asymptotic stability in linear viscoelasticity with supplies. J Math Anal Appl 427:629–645
Carillo S, Chipot M, Valente V, Vergara Caffarelli G (2017) A magneto-viscoelasticity problem with a singular memory kernel. Nonlinear Anal Real World Appl 35:200–210
De Angelis M (2010) On a model of superconductivity and biology. Adv Appl Math Sci 7(1):41–50
Scott AC (2002) Neuroscience a mathematical primer. Springer, Berlin, p 352
Angelis M (2015) Mathematical contributions to the dynamics of the Josephson junctions: state of the art and open problems. Nonlinear Dyn Syst Theory 15(3):231–241
De Angelis M, Renno P (2014) On asymptotic effects of boundary perturbations in exponentially shaped Josephson junctions. Acta Appl Math 132(1):251–259
Scott A (1970) Active and nonlinear wave propagation in electronics. Wiley, Hoboken
Barone A, Paterno G (1982) Physics and application of the Josephson effect. Wiley, Hoboken, p 530
Pankratov AL, Fedorov KG, Salerno M, Shitov SV, Ustinov AV (2015) Nonreciprocal transmission of microwaves through a long Josephson junction. Phys Rev B 92:104501
Carelli P (2002) DC squid systems. In: International symposium on high critical temperatures superconductors devices
Feldman MJ (1999) Digital applicatons of Josephson Junction. In: Ohta H, Ishii C (eds) Phsisics and applications of mesoscopic Josephson Junctions. Physical Society of Japan, Tokyo, pp 289–304
Barone A, Pagano S (2010) Josephson devices chapter. In: Cooper LN, Feldman D (eds) BCS: 50 years. World Scientific Publishing Co. Pte. Ltd., Singapore
Clarke J (2011) SQUIDs for everything. Nat Mater 10:262–263
Clarke J (2010) SQUIDs : Then an now, chapter. In: Cooper LN, Feldman D (eds) BCS 50 years. World Scientific Publishing Co Pte. Ltd., Singapore
Jung P, Butz S, Marthaler M, Fistul MV, Leppa Kangas J, Koshelets VP, Ustinov AV (2014) Multistability and switching in a superconducting metamaterial. Nat Commun 5:3730
Gourlay S (2017) Powering the field forward. Cern Courier. Int J High-Energy Phys 57(7):17–22
De Angelis M (2013) Asymptotic estimates related to an integro differential equation. Nonlinear Dyn Syst Theory 13(3):217–228
De Angelis M, Fiore G (2013) Existence and uniqueness of solutions of a class of third order dissipative problems with various boundary conditions describing the Josephson effect. J Math Anal Appl 404:477–490
Keener JP, Sneyd J (1998) Mathematical physiology. Springer, New York
Nekhamkina O, Sheintuch M (2006) Boundary-induced spatiotemporal complex patterns in excitable systems. Phys Rev E 73:1–4
Forest MG, Christiansen PL, Pagano S, Parmentier RD, Soerensen MP, Sheu SP (1990) Numerical evidence for global bifurcations leading to switching phenomena in long Josephson junctions. Wave Motion 12:213–226
Jaworski M (2005) Exponentially tapered Josephson junction: some analytic results. Theor Math Phys 144(2):1176–1180
De Angelis M, Monte AM, Renno P (2002) On fast and slow times in models with diffusion. Math Models Methods Appl Sci 12(12):1741–1749
De Angelis M, Fiore G (2014) Diffusion effects in a superconductive model. Commun Pure Appl Anal 13(1):217–223
Benabdallah A, Caputo JG, Scott AC (2000) Laminar phase flow for an exponentially tapered josephson oscillator. J Appl Phys 588(6):3527
Pankratov AL, Sobolev KVP, Mygind J (2007) Influence of surface losses and the self-pumping effect on current-voltage characteristics of a long Josephson junction. Phys Rev B 75:184516
De Angelis M (2018) A wave equation perturbed by viscous terms: fast and slow times diffusion effects in a Neumann problem Ricerche mat. https://doi.org/10.1007/s11587-018-0400-1
De Angelis M (2012) On exponentially shaped Josephson junctions. Acta Apl Math 122:179–189
De Angelis M, Renno P (2008) Existence, uniqueness and a priori estimates for a nonlinear integro-differential equation. Ricerche Mat 57(1):95–109
D’Anna A, De Angelis M, Fiore G (2012) Existence and uniqueness for some 3rd order dissipative problems with various boundary conditions. Acta Appl Math 122:225–267
De Angelis M (2001) Asymptotic analysis for the strip problem related to a parabolic third-order operator. Appl Math Lett 14(4):425–430
Cannon JR (1984) The one-dimensional heat equation. Addison-Wesley Publishing Company, Boston
De Angelis M, Maio Mazziotti AE (2008) Existence and uniqueness results for a class of non linear models. In: Mathematical physics models and engineering sciences. Liguori Editore, Naples, Italy, pp 191–202
Johnson S, Suarez P, Biswas A (2012) New exact solutions for the SineGordon equation in 2+1 dimensions. Comput Math Math Phys 52(1):98–104
Chen W-X, Lin J (2014) Some new exact solutions of (1+2) dimensional sine-Gordon equation. Abstr Appl Anal, 2014, article ID 645456
Aktosun T, Demontis F, Van der Mee C (2010) Exact solutions for the sine-Gordon equation. J Math Phys 51:123521
Fiore G, Guerriero G, Maio A, Mazziotti E (2015) On kinks and other travelling-wave solutions of a modified sine-Gordon equation. Meccanica 50:1989–2006
Alvis-Zuniga J-L, Marin-Ramirez A-M, Ortiz-Ortiz R-D (2016) Solutions for the Perturbed Sine-Gordon Equation. J Eng Appl Sci 11:2294–2297
Mitinovic S, Pecaric J, Fink AM (1991) Inequalities involving functions and their integrals and derivatives. Kluwer Academic Publisher, Dordrecht
Acknowledgements
The author is grateful to anonymous referees for their helpful comments and suggestions. This paper has been performed under the auspices of G.N.F.M. of INdAM.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
De Angelis, M. On the transition from parabolicity to hyperbolicity for a nonlinear equation under Neumann boundary conditions. Meccanica 53, 3651–3659 (2018). https://doi.org/10.1007/s11012-018-0906-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11012-018-0906-3