# American Institute of Mathematical Sciences

January  2014, 13(1): 217-223. doi: 10.3934/cpaa.2014.13.217

## Diffusion effects in a superconductive model

 1 Univ. of Naples Federico II, Dept of Math and Appl, Via Claudio n. 21, 80125 Naples, Italy 2 Univ. of Naples Federico II, I.N.F.N., Sez. of Naples, Complesso MSA, V. Cintia, 80126 Naples, Italy

Received  November 2012 Revised  April 2013 Published  July 2013

A superconductive model characterized by a third order parabolic operator ${\mathcal L}_\varepsilon$ is analyzed. When the viscous terms, represented by higher-order derivatives, tend to zero, a hyperbolic operator ${\mathcal L}_0$ appears. Furthermore, if ${\mathcal P}_\varepsilon$ is the Dirichlet initial-boundary value problem for ${\mathcal L}_\varepsilon$, when ${\mathcal L} _\varepsilon$ turns into ${\mathcal L}_0 ,$ ${\mathcal P}_\varepsilon$ turns into a problem ${\mathcal P}_0$ with the same initial-boundary conditions of ${\mathcal P}_\varepsilon$. As long as the higher-order derivatives of the solution of ${\mathcal P}_0$ are bounded, an estimate of solution for the nonlinear problem related to the remainder term $r,$ is achieved. Moreover, some classes of explicit solutions related to ${\mathcal P}_0$ are determined, proving the existence of at least one motion whose derivatives are bounded. The estimate shows that the diffusion effects are bounded even when time tends to infinity.
Citation: Monica De Angelis, Gaetano Fiore. Diffusion effects in a superconductive model. Communications on Pure and Applied Analysis, 2014, 13 (1) : 217-223. doi: 10.3934/cpaa.2014.13.217
##### References:
 [1] T. Aktosun, F. Demontis, and C. van der Mee, Exact solutions to the sine-Gordon equation, Journal of Mathematical Physics, 51 (2010), 1-26. [2] A. Benabdallah, J. G. Caputo and A. C. Scott, Exponentially tapered josephson flux-flow oscillator, Phy. Rev. B, 54 (1996), 16139. [3] A. Benabdallah, J. G. Caputo and A. C. Scott, Laminar phase flow for an exponentially tapered josephson oscillator, J. Apl. Phys., 588 (2000), 3527. [4] S. Bondarenko and Nakagawa, SQUID-based magnetic microscope, in "Smart Materials for Ranking Systems," J. France et al (edition), Springer (2006), 195-201. [5] T. L. Boyadjiev, E. G. Semerdjieva and Yu. M. Shukrinov, Common features of vortex structure in long exponentially shaped Josephson junctions and Josephson junctions with inhomogeneities, Physica C, 460-462 (2007), 1317-1318. [6] G. Carapella, N. Martucciello and G. Costabile, Experimental investigation of flux motion in exponentially shaped Josephson junctions, PHYS REV B, 66 (2002), 134531. [7] J. Clarke, "SQUIDs for Everything," Nature Materials, 10 (2011). [8] J. Clarke, SQUIDs: Then and Now, chapter in BCS: 50 Years (eds. Leon N Cooper and Dmitri Feldman) World Scientific Publishing Co. Pte. Ltd., Singapore (2010). [9] S. A. Cybart et al., Dynes Series array of incommensurate superconducting quantum interference devices, Appl. Phys Lett, 93 (2008), 1-3. [10] A. D'Anna, M. De Angelis and G. Fiore, Existence and uniqueness for some 3rd order dissipative problems with various boundary conditions, Acta Appl. Math., 122 (2012), 255-267. [11] M. Dehghan and A. Shokri, A numerical method for solution of the two dimensional sine- Gordon equation using the radial basis functions, Mat Comp in Simulation, 79 (2008), 700-715. [12] M. De Angelis, On a model of superconductivity and biology, Advances and Applications in Mathematical Sciences, 7 (2010), 41-50. [13] M. De Angelis, Asymptotic analysis for the strip problem related to a parabolic third-order operator, Applied Mathematics Letters, 14 (2001), 425-430. [14] M. De Angelis, A priori estimates for excitable models, Meccanica (2013). doi: 10.1007/s11012-013-9763-2. [15] M. De Angelis, On exponentially shaped Josephson junctions, Acta appl. Math, 122 (2012), 179-189 [16] M. De Angelis, On a parabolic operator of dissipative systems,, submitted to Acta appl. Math, (). [17] M. De Angelis and G. Fiore, 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 (2013), 477-490. doi: 10.1016/j.jmaa.2013.03.029. [18] M. De Angelis, A. Maio and E. Mazziotti, Existence and uniqueness results for a class of non linear models, in "Mathematical Physics Models and Engineering Sciences" (ed. Liguori), Italy, (2008), 191-202. [19] M. De Angelis and E. Mazziotti, Non linear travelling waves with diffusion, Rend. Acc. Sc. Fis. Mat. Napoli, 73 (2006), 23-36. [20] De Angelis, A. M. Monte and P. M. Renno, On fast and slow times in models with diffusion, Math Models and Methods in Applied Sciences, 12 (2002), 1741-1749. [21] M. De Angelis and P. Renno, Asymptotic effects of boundary perturbations in excitable systems, (2013),, submitted to Discrete and Continuous Dynamical Systems - B. Available from: \url{http://arxiv.org/pdf/1304.3891v1.pdf}, (). [22] M. De Angelis and P. Renno, Existence, uniqueness and a priori estimates for a non linear integro-differential equation, Ric Mat, 57 (2008), 95-109. [23] De Angelis and P. M. Renno, Diffusion and wave behaviour in linear Voigt model, C. R. Mecanique, 330 (2002), 21-26 [24] Gutman S. Junhohg Ha, Identification problem for damped sine Gordon equation with point sources, J. Math. Anal. Appl., 375 (2011), 648-666. doi: 10.1016/j.jmaa.2010.10.006. [25] E. M. Izhikevich, "Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting," The MIT press, England, 2007. [26] M. Jaworski, "Fluxon Dynamics in Exponentially Shaped Josephson Junction," Phy. rev. B, 71 (2005), 1-6. [27] M. Jaworski, Exponentially tapered Josephson junction: some analytic results, Theor and Math Phys, 144 (2005), 1176-1180. [28] J. McCall and Lindsa, Superconductor cables: Advanced capabilities for the smart grid, Utility Automation Engineering TD, 13 (2008), 54. [29] J. D. Murray, "Mathematical Biology. I. An Introduction," Springer-Verlag, N.Y, 2002. [30] S. Rionero, Asymptotic behaviour of solutions to a nonlinear third order P.D.E modeling physical phenomena, Boll Unione Mat Ital, 9 (2012), 451-468. [31] H. Rogalla and P. H. Kes, "100 Years of Superconductivity," CRC Press, (2012). [32] A.C. Scott, "The Nonlinear Universe: Chaos, Emergence, Life," Springer-Verlag, 2007. [33] A.C. Scott, "Neuroscience A mathematical Primer," Springer-Verlag, 2002.

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##### References:
 [1] T. Aktosun, F. Demontis, and C. van der Mee, Exact solutions to the sine-Gordon equation, Journal of Mathematical Physics, 51 (2010), 1-26. [2] A. Benabdallah, J. G. Caputo and A. C. Scott, Exponentially tapered josephson flux-flow oscillator, Phy. Rev. B, 54 (1996), 16139. [3] A. Benabdallah, J. G. Caputo and A. C. Scott, Laminar phase flow for an exponentially tapered josephson oscillator, J. Apl. Phys., 588 (2000), 3527. [4] S. Bondarenko and Nakagawa, SQUID-based magnetic microscope, in "Smart Materials for Ranking Systems," J. France et al (edition), Springer (2006), 195-201. [5] T. L. Boyadjiev, E. G. Semerdjieva and Yu. M. Shukrinov, Common features of vortex structure in long exponentially shaped Josephson junctions and Josephson junctions with inhomogeneities, Physica C, 460-462 (2007), 1317-1318. [6] G. Carapella, N. Martucciello and G. Costabile, Experimental investigation of flux motion in exponentially shaped Josephson junctions, PHYS REV B, 66 (2002), 134531. [7] J. Clarke, "SQUIDs for Everything," Nature Materials, 10 (2011). [8] J. Clarke, SQUIDs: Then and Now, chapter in BCS: 50 Years (eds. Leon N Cooper and Dmitri Feldman) World Scientific Publishing Co. Pte. Ltd., Singapore (2010). [9] S. A. Cybart et al., Dynes Series array of incommensurate superconducting quantum interference devices, Appl. Phys Lett, 93 (2008), 1-3. [10] A. D'Anna, M. De Angelis and G. Fiore, Existence and uniqueness for some 3rd order dissipative problems with various boundary conditions, Acta Appl. Math., 122 (2012), 255-267. [11] M. Dehghan and A. Shokri, A numerical method for solution of the two dimensional sine- Gordon equation using the radial basis functions, Mat Comp in Simulation, 79 (2008), 700-715. [12] M. De Angelis, On a model of superconductivity and biology, Advances and Applications in Mathematical Sciences, 7 (2010), 41-50. [13] M. De Angelis, Asymptotic analysis for the strip problem related to a parabolic third-order operator, Applied Mathematics Letters, 14 (2001), 425-430. [14] M. De Angelis, A priori estimates for excitable models, Meccanica (2013). doi: 10.1007/s11012-013-9763-2. [15] M. De Angelis, On exponentially shaped Josephson junctions, Acta appl. Math, 122 (2012), 179-189 [16] M. De Angelis, On a parabolic operator of dissipative systems,, submitted to Acta appl. Math, (). [17] M. De Angelis and G. Fiore, 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 (2013), 477-490. doi: 10.1016/j.jmaa.2013.03.029. [18] M. De Angelis, A. Maio and E. Mazziotti, Existence and uniqueness results for a class of non linear models, in "Mathematical Physics Models and Engineering Sciences" (ed. Liguori), Italy, (2008), 191-202. [19] M. De Angelis and E. Mazziotti, Non linear travelling waves with diffusion, Rend. Acc. Sc. Fis. Mat. Napoli, 73 (2006), 23-36. [20] De Angelis, A. M. Monte and P. M. Renno, On fast and slow times in models with diffusion, Math Models and Methods in Applied Sciences, 12 (2002), 1741-1749. [21] M. De Angelis and P. Renno, Asymptotic effects of boundary perturbations in excitable systems, (2013),, submitted to Discrete and Continuous Dynamical Systems - B. Available from: \url{http://arxiv.org/pdf/1304.3891v1.pdf}, (). [22] M. De Angelis and P. Renno, Existence, uniqueness and a priori estimates for a non linear integro-differential equation, Ric Mat, 57 (2008), 95-109. [23] De Angelis and P. M. Renno, Diffusion and wave behaviour in linear Voigt model, C. R. Mecanique, 330 (2002), 21-26 [24] Gutman S. Junhohg Ha, Identification problem for damped sine Gordon equation with point sources, J. Math. Anal. Appl., 375 (2011), 648-666. doi: 10.1016/j.jmaa.2010.10.006. [25] E. M. Izhikevich, "Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting," The MIT press, England, 2007. [26] M. Jaworski, "Fluxon Dynamics in Exponentially Shaped Josephson Junction," Phy. rev. B, 71 (2005), 1-6. [27] M. Jaworski, Exponentially tapered Josephson junction: some analytic results, Theor and Math Phys, 144 (2005), 1176-1180. [28] J. McCall and Lindsa, Superconductor cables: Advanced capabilities for the smart grid, Utility Automation Engineering TD, 13 (2008), 54. [29] J. D. Murray, "Mathematical Biology. I. An Introduction," Springer-Verlag, N.Y, 2002. [30] S. Rionero, Asymptotic behaviour of solutions to a nonlinear third order P.D.E modeling physical phenomena, Boll Unione Mat Ital, 9 (2012), 451-468. [31] H. Rogalla and P. H. Kes, "100 Years of Superconductivity," CRC Press, (2012). [32] A.C. Scott, "The Nonlinear Universe: Chaos, Emergence, Life," Springer-Verlag, 2007. [33] A.C. Scott, "Neuroscience A mathematical Primer," Springer-Verlag, 2002.
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