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

0 f(u)rdr) 2, for 0 < r < 1, t > 0, u(1, t) = u󸀠(0, t) = 0, for t > 0, u(r, 0) = u 0 (r), 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.

The original motivation for studying such problems comes from the plasma Ohmic heating process.The plasma is an electrical axis-symmetric conductor and so it could be heated by passing a current through it.This is called Ohmic heating and it is the same kind of heating that occurs in thermistors.We consider that the axis-symmetric conductor  is a part of simple circuit in series with another constant one,  and a constant voltage  is applied (see Figure 1).Let (, ) and () be the temperature and the electrical resistivity of the conductor , respectively.
Here, the conductor  ⊂ R 3 is a prismatic one with the length  and the cross-sectional area , and the length of conductor  is   .Assume that the diameter of the crosssection  is much less than  and the temperature (, ) is independent of the variable  1 .Suppose that the curved surface of the conductor , Γ 0 is well thermal, and we can specify Based on the derivation in [1] (see also [2,3]), we get the temperature  of the material which satisfies the following: where () =  −1 (),  = /, and  = ( 0 /)  .The initial data  0 () is a positive, smooth function and satisfies  0 () = 0 on .
In this paper, we focus on the problem (3) in radially symmetric case.So, we assume additionally that cross-section  is a unit disk and the initial data be radially and decreasing, that is, where  = || ∈ (0, 1).Thus, the problem in axis-symmetric case can be formulated into the problem (1).Furthermore, it is easy to see that the axis-symmetric solution to the problem (1) is radially decreasing (see [4]).
Here, we would like to address the works on the Ohmic heating model with one conductor .The problem with only one conductor can be formulated into the following problem with different boundary conditions: where Ω ⊂ R 2 is an open, bounded domain, () is the electrical conductivity () = 1/(), and the parameter  is a positive constant, which is depending upon the electric current or potential difference and also upon the "size" of the conductor (see [1,[5][6][7][8][9]).
For problem (5), Lacey et al. have proved that if  is an increasing function, then the blow up cannot take place (see [1,10]).If  is a decreasing function, Lacey proved that comparison techniques was valid, by which he studied the asymptotic behavior of the solutions to (5) for special  (see [1]).Taking the advantage of this fact, Lacey [1,6] and Tzanetis [7] proved the occurrence of blow up for onedimensional model (5) and for the two-dimensional radially symmetric model (5), respectively.On the other hand, they proved that the global solution of (5) for some special (), such as () =  − , asymptotically converges to its unique steady state.
In [2], Du and Fan considered the nonlinear diffusion model for two conductors with one of the conductors remains constant.When  is decreasing, they proved that comparison principle is valid and the solution of the model was always global in time.Furthermore, if  is a decreasing exponential function, they proved that the solution of the problem converges asymptotically to the unique steady state.See also [3] for some results on asymptotic behavior of the global solution in one-dimensional case.
Inspired by these works, modified by the methods in [2], one can easily prove that the comparison principle for model ( 1) is valid and the solution of ( 1) is global in time.Finally, the main purpose of this paper is to give the asymptotic behavior and to show the asymptotic stability of the problem (1) with generally deceasing function ().
Remark 2. The equations in models (1), (3), and ( 5) are semilinear parabolic equations with nonlocal sources.For the works on the global existence and blow up of nonlocal parabolic equations, the authors would like to refer to [11][12][13][14] and the references therein.
The analysis and techniques in this paper is based on the analysis for the ordinary differential equations and comparison arguments.(1) In this section, we will consider the asymptotic stability for the problem (1), and give the proof of Theorem 1.

Figure 1 :
Figure 1: Electric current flows through two conductors.