Asymptotic stability for a symmetric parabolic problem modeling Ohmic heating

• Published in: Boundary value problems Vol. 2014; no. 1; pp. 1 - 8
• Main Authors: Fan, Mingshu, Xia, Anyin, Zhang, Lei
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 14.02.2014; Hindawi Limited
• Subjects: Analysis; Approximations and Expansions; Asymptotic properties; Boundary value problems; Conductors (devices); Difference and Functional Equations; Dirichlet problem; Electric potential; Heating; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Stability; steady state; nonlocal parabolic equation; axis-symmetric; asymptotic stability; Ohmic heating model
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/1687-2770-2014-40

We consider the asymptotic behavior of the solution of the non-local parabolic equation u t = ( κ ( u ) ) r r + ( κ ( u ) ) r r + f ( u ) ( a + 2 π b ∫ 0 1 f ( u ) r d r ) 2 , for 0 < r < 1 , t > 0 , with a homogeneous Dirichlet boundary condition. The equation is the so-called Ohmic-heating model, which comes from thermal electricity in this paper, u and f ( u ) represent the temperature of the conductor and the electrical conductivity. The model prescribes the dimensionless temperature when the electric current flows through two axis-symmetric conductors, subject to a fixed electric potential difference. The global existence and uniform boundedness of the solution to the problem indicate that the temperature of the conductor remains uniformly bounded. Furthermore, the asymptotic stability of the global solution is obtained. MSC: 35K20, 35K55, 35K65, 80M35.