Quenching Phenomena Due to a Concentrated Nonlinear Source in an Infinitely Long Cylinder

This article studies a semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source in an infinitely long cylinder. We study the effects of the strength of the source on quenching. Criteria for global existence of the solution and for quenching are investigated.


Introduction
The quenching phenomena have been studied since 1975 [1]. The quenching models have many applications in science and engineering. For example, the models may include combustion, ignition, thermal explosion and damage of material.
In some situation, the model involves a concentrated source, such as in a chemical reaction process due to the effect of a catalyst, or in the ignition of a combustible medium through the use of either a heated wire or a pair of small electrodes to supply a large amount of energy to a very confined area [2]. In 2006, the quenching problem with a concentrated source has been posted correctly in a multi-dimensional bounded domain by Chan [3]. In this paper, we study the quenching problem with a concentrated nonlinear source in the unbounded domain, the infinitely long cylinder.
Let R and b be positive real numbers such that b < R, D be a two-dimensional ball centered at the origin with a radius R or , B be a two-dimensional ball centered at the origin with a radius b or , and . We also let Ω and ω be the infinitely long cylinders centered at the origin with the radii R and b respectively. Namely, ( ) , B ω ∂ = ∂ × −∞ ∞ . For 3 x ∈  , let υ(x) denote the unit outward normal at x ω ∈ ∂ , and ( ) x ω χ be the characteristic function which is 1 for x ω ∈ and 0 for x ω ∉ . We study the following problem in the infinitely long cylinder Ω with a concentrated nonlinear source on ∂ω: , 0 0 on , , 0 on 0, , where α and T are positive real numbers, Ω is the closure of Ω , f is a given function such that   [3] in the following theorem.
Theorem 1.1. There exists some q t such that for 0 q t t ≤ < , the integral equation has a unique continuous nonnegative solution u. Furthermore, u is a nondecreasing function of t. If q t is finite, then at q t , u quenches everywhere on ∂B only.
Note that D and B are both the balls centered at the origin. By radial symmetry, the problem (1.2) can be written in the polar form: where J₀ and J₁ are the Bessel functions of the first kind of order 0 and 1 respectively, and x n is the n th root of the Bessel function J₀(x). The integral equation A solution u of (1.5) is said to quench in a finite time if there exists a number We note from Theorem 1.1 that ( ) , u r t attains its maximum at r b = . If t q is finite, then at t q , u quenches only at r b = .
In Section 2, we show that there exists a unique positive number * α such that u exists globally for * α α ≤ and quenches in a finite time for * α α > . We also derive a formula for computing * α . In Section 3, we study the effects of b and R on quenching.

The Critical Value α *
In this section, we study the effect of α on quenching. We modify the techniques used in proving Lemma 1 of Chan and Wong [7] for a singular parabolic problem to establish the following lemma.
Proof. The Green function from (1.4) can be written as Since each term under the summation is positive and R is positive, ( ) , ; , also positive. Next, we will prove that the Green's function (2.1) is bounded.
By using the asymptotic formula of ( ) ( ) for large positive value z. Since cos 1 z ≤ and 1 0 z − → as z → ∞ , there exists a positive constant k₁ and z₀ such that for For large n, it follows from (2.2) that Journal of Applied Mathematics and Physics Hence, there exists an integer N such that for n N > , It follows from ( which converges uniformly for t in any compact subset of ( ] Then, there exists a constant k₃ such that , it follows from Theorem 1.1 that u attains its absolute maximum at r b for some constant k₄. By choosing α sufficiently small, namely,

of Applied Mathematics and Physics
Since each term of the series is positive, we have 1 e 1.
x R t − − → Thus, there exists some By choosing α sufficiently large, say is Green's function corresponding to problem (2.9).
The next result shows that there exists a critical value * α for α . The proof of the following theorem is similar to that for Theorem 2.5 of Chan and Tragoonsirisak [10].
Theorem 2.5. There exists a unique * α ,     The critical value * α is determined as the supremum of all positive values α for which a solution U of (2.9) exists. Since U(r) attains its maximum at r b = , it follows from (2.10) and (2.11) that Hence, we have (2.12).
To show that u exists globally when Hence, for given R and b, we have ( )

Critical Location of the Source b * and Radius R *
In this section, we fix α, and study the effect of the location of the source b and the effect of the radius R on quenching.
then u exists globally. Proof. We have

2) If
by the second derivative test, ϕ(b) attains its maximum at b R e = .  Theorem 3.3. For given α and R, then u exists globally for any b.
2) if there exist * b and ** b such that u exists globally for We note from the assumption that If R is smaller than or equal to e/4, then u exists globally (for any location of the source b). If R is larger than e/4, then the quenching may occur (depend on the location of the source b). Let 1 R = . By using a graphing calculator to solve (3.4), namely, it follows from Theorem 3.3 (2) that we have * 0.1161 b ≈ and ** 0.6995 b ≈ (round to four decimal places). The fractional diffusion problem with a concentrated source in one-dimensional domain was recently studied ([12] [13]). Many problems in the real world involve more than one dimension. Based on the current study here, the fractional diffusion problem with a concentrated source in multi-dimensional domain would be the future work.