Diffusion effects on a superconductive model

TL;DR

This paper analyzes the transition from a third order parabolic superconductive model to a hyperbolic model as viscous effects vanish, demonstrating bounded diffusion effects and explicit solutions with bounded derivatives.

Contribution

It introduces a detailed analysis of the diffusion effects in a superconductive model, including the transition to hyperbolic behavior and explicit solutions with bounded derivatives.

Findings

Diffusion effects remain bounded over time.

Existence of explicit solutions with bounded derivatives.

Transition from parabolic to hyperbolic operator as viscous terms vanish.

Abstract

A superconductive model characterized by a third order parabolic operator L" is analysed. When the viscous terms, represented by higher - order deriva- tives, tend to zero, a hyperbolic operator L0 appears. Furthermore, if P" is the Dirichlet initial boundary - value problem for L", when L" turns into L0; P" turns into a problem P0 with the same initial - boundary conditions as P". The solution of the nonlinear problem related to the remainder term r is achieved, as long as the higher-order derivatives of the solution of P0 are bounded. More- over, some classes of explicit solutions related to P0 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.