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Diffusion effects in a superconductive model

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  • 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.
    Mathematics Subject Classification: Primary: 82D55,74K30; Secondary: 35K35, 35E05.


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