Asymptotic Stability for an Axis-Symmetric Ohmic Heating Model in Thermal Electricity - Anyin Xia; Mingshu Fan; Shan Li (Journal of Applied Mathematics)
The asymptotic behavior of the solution for the Dirichlet problem of the parabolic equation with nonlocal term ut=urr+ur/r+f(u)/(a+2πb∫01‍f(u)rdr)2, for  0<r<1,  t>0,u1,t=u′(0,t)=0, for  t>0,  ur,0=u0r,  for  0≤r≤1. The model prescribes the dimensionless temperature when the electric current flows through two conductors, subject to a fixed potential difference. One of the electrical resistivity of the axis-symmetric conductor depends on the temperature and the other one remains constant. The main results show that the temperature remains uniformly bounded for the generally decreasing function f(s), and the global solution of the problem converges asymptotically to the unique equilibrium.