Asymptotic Stability for an Axis-Symmetric Ohmic Heating Model in Thermal Electricity

Abstract

The asymptotic behavior of the solution for the Dirichlet problem of the parabolic equation with nonlocal term $u_t=u_{rr}+u_r/r+f(u)/(a+\mathrm{2}\pi b{\int }_{\mathrm{0}}^{\mathrm{1}}\mathrm{‍}f(u)rdr{)}^{\mathrm{2}}$, $u\left(\mathrm{1},t\right)=u^{\prime }(\mathrm{0},t)=\mathrm{0}$, $\text{for}t>\mathrm{0},u\left(r,\mathrm{0}\right)=u_{\mathrm{0}}\left(r\right),\mathrm{}\text{for}\mathrm{0}\le r\le \mathrm{1}$. 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.