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