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+2𝜋𝑏∫¹₀𝑓(𝑢)𝑟𝑑𝑟)², for0<𝑟<1,𝑡>0,𝑢(1,𝑡)=𝑢^′(0,𝑡)=0, for𝑡>0,𝑢(𝑟,0)=𝑢₀(𝑟),for0≤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.

1. Introduction

In this paper, we consider the asymptotic behavior for the following initial boundary value problem: 𝑢_𝑡=𝑢_𝑟𝑟+𝑢_𝑟𝑟+𝑓(𝑢)𝑎+2𝜋𝑏∫¹₀𝑓(𝑢)𝑟𝑑𝑟²,𝑢()for0<𝑟<1,𝑡>0,1,𝑡=𝑢^′()𝑢()0,𝑡=0,for𝑡>0,𝑟,0=𝑢₀(𝑟),for0≤𝑟≤1, (1) where𝑎 and 𝑏are some positive parameters.𝑓is a continuous, positive, and decreasing function, and the initial data𝑢₀(𝑟)is a decreasing smooth positive function.

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 conductor𝐴is a part of simple circuit in series with another constant one, 𝐵and a constant voltage𝐸is applied (see Figure 1). Let𝑢(𝑥,𝑡)and𝜌(𝑢)be the temperature and the electrical resistivity of the conductor𝐴, respectively.

Figure 1 

Electric current flows through two conductors.

Here, the conductor𝐴⊂ℝ³is a prismatic one with the length𝐿and the cross-sectional area 𝑆, and the length of conductor𝐵is𝐿^′. Assume that the diameter of the cross-section𝐷is much less than𝐿and the temperature𝑢(𝑥,𝑡)is independent of the variable𝑥₁. Suppose that the curved surface of the conductor𝐴,Γ₀is well thermal, and we can specify 𝑢=0onΓ₀. (2)

Based on the derivation in [1] (see also [2, 3]), we get the temperature𝑢of the material which satisfies the following: 𝑢_𝑡𝑓(𝑢)−Δ𝑢=𝑎+𝑏∫_𝐷𝑓(𝑢)𝑑𝑥²𝑢()𝑢(),𝑥∈𝐷,𝑡>0,𝑥,𝑡=0,𝑥∈𝜕𝐷,𝑡>0,𝑥,0=𝑢₀(𝑥),𝑥∈𝐷, (3) where𝑓(𝑢)=𝜌⁻¹(𝑢),  𝑎=𝐿/𝐸,  and 𝑏=(𝜌₀/𝐸)𝐿^′. The initial data𝑢₀(𝑥)is a positive, smooth function and satisfies𝑢₀(𝑥)=0on𝜕𝐷.

In this paper, we focus on the problem (3) in radially symmetric case. So, we assume additionally that cross-section 𝐷 is a unit disk and the initial data be radially and decreasing, that is, 𝑢₀(𝑥)=𝑢₀(𝑟),𝑢^′₀(𝑟)<0, (4) where𝑟=|𝑥|∈(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 conductor𝐴. The problem with only one conductor can be formulated into the following problem with different boundary conditions: 𝑢_𝑡(𝑢)=Δ𝑢+𝜆𝑓∫_Ω𝑓(𝑢)𝑑𝑥²,for𝑥∈Ω,𝑡>0, (5) whereΩ⊂ℝ²is an open, bounded domain,𝑓(𝑢)is the electrical conductivity𝜎(𝑢)=1/𝜌(𝑢), and the parameter𝜆is a positive constant, which is depending upon the electric current or potential difference and also upon the “size” of the conductor (see [1, 5–9]).

For problem (5), Lacey et al. have proved that if𝑓is an increasing function, then the blow up cannot take place (see [1, 10]). If𝑓is a decreasing function, Lacey proved that comparison techniques was valid, by which he studied the asymptotic behavior of the solutions to (5) for special 𝑓 (see [1]). Taking the advantage of this fact, Lacey [1, 6] and Tzanetis [7] proved the occurrence of blow up for one-dimensional 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 special𝑓(𝑢), such as𝑓(𝑢)=𝑒^−𝑢, 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. When𝑓is decreasing, they proved that comparison principle is valid and the solution of the model was always global in time. Furthermore, if𝑓is 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 function𝑓(𝑢).

Theorem 1. Assume that𝑓satisfies −𝑠𝑓^′(𝑠)<𝑓(𝑠),𝑓𝑜𝑟𝑠>0, (6) the solution𝑢(𝑟,𝑡)of the problem (1) converges asymptotically to the unique steady state𝜔(𝑟), namely, 𝑢(𝑟,𝑡)⟶𝜔(𝑟),𝑎𝑠𝑡⟶+∞, (7) for0<𝑟<1.

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 [11–14] and the references therein.

The analysis and techniques in this paper is based on the analysis for the ordinary differential equations and comparison arguments.

2. The Asymptotic Stability for Problem (1)

In this section, we will consider the asymptotic stability for the problem (1), and give the proof of Theorem 1.

Proof of Theorem 1. Firstly, we deal with the local steady solution𝜔(𝑟;𝜇)corresponding to the problem (1), consider 𝜔^′′+1𝑟;𝜇𝑟𝜔^′𝜔𝜔𝑟;𝜇+𝜇𝑓𝑟;𝜇=0,0<𝑟<1,1;𝜇=𝜔^′0;𝜇=0, (8) with the positive parameter 1𝜇=𝑎+2𝜋𝑏∫¹₀𝑓(𝜔(𝑟;𝜇))𝑟𝑑𝑟². (9)

Moreover, multiplying𝑟on both sides of the equation in (8), and integrating over[0,1]yield ¹₀𝑓𝜔𝜔𝑟;𝜇𝑟𝑑𝑟=−^′1;𝜇𝜇. (10) Multiplying𝜔^′(𝑟;𝜇)on both sides of the equation in (8), and integrating over[0,1]yields 𝜔^′(1;𝜇)²=2𝜇₀^𝑀(𝜇)𝑓(𝑠)𝑑𝑠−2¹₀𝜔^′(𝑟;𝜇)²𝑟𝑑𝑟, (11) where𝑀(𝜇)=𝜔(0;𝜇)=max_𝑟∈[0,1]𝜔(𝑟;𝜇).

In view of𝜔^′(𝑟;𝜇)<0, for any𝑟∈(0,1), it follows from (11) that 𝜔^′1;𝜇=−2𝜇₀^𝑀(𝜇)𝑓(𝑠)𝑑𝑠−2¹₀𝜔^′𝑟;𝜇²𝑟𝑑𝑟. (12)

Combining this with (9) and (10) yields 𝜇𝑎+2𝜋𝑏¹₀𝑓(𝜔(𝑟;𝜇))𝑟𝑑𝑟²=𝑎²𝜇+4𝜋²𝑏²𝜔^′(1;𝜇)²𝜇−4𝜋𝑎𝑏𝜔^′1;𝜇=1. (13)

It is easily seen that the problem (8) does not possess nontrivial solution with the parameter𝜇=0. Namely,𝜔≡0and𝑀=0for𝜇=0. Therefore, it follows from (11) that lim_𝜇→0¹₀𝜔^′(𝑟;𝜇)²𝑟𝑑𝑟⟶0, (14) since(𝜔^′(1;𝜇))²=2𝜇∫₀^𝑀(𝜇)𝑓(𝑠)𝑑𝑠−2∫¹₀((𝜔^′(𝑟;𝜇))²/𝑟)𝑑𝑟>0and∫¹₀((𝜔^′(𝑟;𝜇))²/𝑟)𝑑𝑟>0.

Define the following: 𝐹𝜇∶=𝑎²𝜇+4𝜋²𝑏²𝜔^′(1;𝜇)²𝜇−4𝜋𝑎𝑏𝜔^′1;𝜇−1,for𝜇>0. (15) Then, it follows from a direct computation that𝐹(0+)<0and𝐹(1/𝑎²)>0. Then, there exists at least one root𝜇∈(0,1/𝑎²)to (13).

Making odd extension to the problem (8), we get 𝜔^′′+1𝑟;𝜇𝑟𝜔^′𝜔𝜔𝑟;𝜇+𝜇𝑓𝑟;𝜇=0,−1<𝑟<1,±1;𝜇=0. (16)

We claim that the differentiation of𝜔(𝑟;𝜇)with respect to the parameter𝜇 is always positive, namely,𝜔_𝜇(𝑟;𝜇)>0, for any0≤𝑟≤1. In fact, by differentiating on both sides of (16) respect to the parameter𝜇, one gets −𝜔_𝜇^′′−1𝑟;𝜇𝑟𝜔^′_𝜇𝑟;𝜇−𝜇𝑓^′𝜔𝑟;𝜇𝜔_𝜇𝜔𝑟;𝜇=𝑓𝑟;𝜇>0. (17)

In view of𝜔_𝜇(±1;𝜇)=0, by comparison principle, we have𝜔_𝜇(𝑟;𝜇)>0, for any𝑟∈[−1,1]. Then,𝑀_𝜇(𝜇)=𝜔_𝜇(0;𝜇)>0.

Set 𝜑𝜇𝜔∶=^′(1;𝜇)²𝜇=2𝜇∫₀^𝑀(𝜇)𝑓(𝑠)𝑑𝑠−2∫¹₀𝜔^′𝑟;𝜇²/𝑟𝑑𝑟𝜇. (18) Let𝑧(𝑟;𝜇)=𝜔(𝑟;𝜇)/𝛼,  𝛼=√𝜇, we can rewrite 𝜑𝜇=𝑧^′(1;𝜇)². (19) Then, it follows from (8) that𝑧_𝛼(1;𝜇)=𝑧^′_𝛼(0;𝜇)=𝑧(1;𝜇)=𝑧^′(0;𝜇)and 𝑧^′′+𝑧𝑟;𝜇^′𝑟;𝜇𝑟+𝛼𝑓𝜔𝑟;𝜇=0. (20)

Differentiating on both sides of (20) with respect to the parameter𝛼, we have 𝑧_𝛼^′′+𝑧𝑟;𝜇^′𝛼𝑟;𝜇𝑟+𝛼²𝑓^′𝜔𝑟;𝜇𝑧_𝛼𝜔𝑟;𝜇=−𝑓𝑟;𝜇−𝑓^′𝜔𝑟;𝜇𝜔𝑟;𝜇. (21)

If the condition (6) holds, then it follows from maximum principle and Hopf’s boundary lemma that 𝑧_𝛼𝑟;𝜇>0,𝑧^′_𝛼1;𝜇<0. (22) Thus, 𝜑^′𝜇=1𝑑2𝛼𝑧𝑑𝛼^′(1;𝜇)²<0, (23) since𝑧^′(1;𝜇)=𝜔^′(1;𝜇)/𝛼<0. Therefore, we can conclude that𝐹^′(𝜇)>0, for any𝜇∈(0,1/𝑎²). Furthermore, (13) possesses a unique root𝜇^∗in(0,1/𝑎²), which shows that the problem possesses unique steady state𝜔(𝑟,𝜇^∗).

Next, we will show that the global solution of the problem (1) converges to its steady state. Inspired by the form of steady state𝜔(𝑟;𝜇^∗), we seek for a decreasing in time, upper solution of a form similar to the steady state𝑣(𝑟,𝑡)=𝜔(𝑟,𝜇(𝑡))=𝜔, where𝜇(𝑡)will be determined later.

Then, one has 𝑣_𝑡−𝑣_𝑟𝑟−1𝑟𝑣_𝑟−𝑓(𝑣)𝑎+2𝜋𝑏∫¹₀𝑟𝑓(𝑣)𝑑𝑟²=𝜔_𝜇⋅𝜇^′(𝑡)−𝜔_𝑟𝑟−1𝑟𝜔_𝑟−𝑓(𝜔)𝑎+2𝜋𝑏∫¹₀𝑟𝑓(𝜔)𝑑𝑟²=𝜔_𝜇⋅𝜇^′(𝑡)−𝑓(𝜔)𝑎+2𝜋𝑏∫¹₀𝑟𝑓(𝜔)𝑑𝑟²×1−𝜇(𝑡)𝑎+2𝜋𝑏¹₀𝑟𝑓(𝜔)𝑑𝑟²=𝜔_𝜇𝜇^′(𝑡)−𝑓(𝜔)𝜔_𝜇𝑎+2𝜋𝑏∫¹₀𝑟𝑓(𝜔)𝑑𝑟²×1−𝜇(𝑡)𝑎+2𝜋𝑏¹₀𝑟𝑓(𝜔)𝑑𝑟². (24)

Since𝑢₀(𝑟)and 𝑢^′₀(𝑟)are bounded in[0,1], we can choose𝜇(0), such that 𝑣(𝑟,0)=𝜔𝑟,𝜇(0)≥𝑢₀(𝑟). (25)

Set 𝑔𝜇(𝑡)=inf_𝑟∈[0,1]𝑓(𝜔)𝜔_𝜇1𝑎+2𝜋𝑏∫¹₀𝑟𝑓(𝜔)𝑑𝑟²×𝜇(𝑡)𝑎+2𝜋𝑏¹₀𝑟𝑓(𝜔)𝑑𝑟²−1. (26) Note that𝑔(𝜇(𝑡))≥0, provided that𝜇(𝑡)≥𝜇^∗, where𝜇^∗is the unique root of (13), since𝐹(𝜇)is increasing with respect to𝜇. Thus, we can choose a decreasing function𝜇(𝑡)such that 0≥𝜇^′(𝑡)≥−𝑔𝜇(𝑡),lim_𝑡→∞𝜇(𝑡)=𝜇^∗. (27) Then we have established an upper solution𝑣(𝑟,𝑡)=𝜔(𝑟;𝜇(𝑡))to the problem (1), which satisfies 𝑣(𝑟,𝑡)⟶𝜔𝑟;𝜇^∗=𝜔(𝑟),as𝑡⟶∞. (28)

Similarly, we can construct a lower solution as𝑣(𝑟,𝑡)=𝜔(𝑟;𝜇(𝑡)), where𝜇(𝑡)is increasing in time and tends to𝜇^∗. Finally, by the comparison principle, we obtain a pair of upper-lower solutions(𝑣(𝑟,𝑡),𝑣(𝑟,𝑡)), such that 𝑣(𝑟,𝑡)≤𝑢(𝑟,𝑡)≤𝑣(𝑟,𝑡),for𝑟∈[0,1],𝑡>0, (29) and it completes the proof of Theorem 1.

Acknowledgments

This work is supported in part by NSFC Grant (11171236), SRFDP (no. 20100181120031), the Fundamental Research Funds for the Central Universities (0082604132187, skqy201224), and the fund of Key Disciplinary of Computer Software and Theory, Sichuan Grant (no. SZD0802-09-1).