Quenching Behavior of Parabolic Problems with Localized Reaction Term

Let p, q, T be positive real numbers, B = {x ∈ R : ∥x∥ < 1}, ∂B = {x ∈ R : ∥x∥ = 1}, x∗ ∈ B, △ be the Laplace operator in R. In this paper, the following the initial boundary value problem with localized reaction term is studied: ut(x, t) = ∆u(x, t) + 1 (1− u(x, t))p + 1 (1− u(x∗, t))q , (x, t) ∈ B × (0, T ), u(x, t) = 0, (x, t) ∈ ∂B × (0, T ), u(x, 0) = u0(x), x ∈ B, where u0 ≥ 0. The existence of the unique classical solution is established. When x∗ = 0, quenching criteria is given. Moreover, the rate of change of the solution at the quenching point near the quenching time is studied.


Introduction
Let p, q, T be positive real numbers, B = {x ∈ R n : ∥x∥ < 1}, ∂B = {x ∈ R n : ∥x∥ = 1}, x * ∈ B, △ be the Laplace operator in R n . In this paper, we consider the following the initial boundary value problem with localized reaction term: In 1975 that Kawarada [10] introduced the concept of quenching when he studied the following nonlinear parabolic boundary problem, , t > 0, −a < x < a, where a is a positive constant. He showed that if a is sufficiently large, then there exists a finite time T at which the solution ceases to exist as a classical solution, and max −a≤x≤a u(x, t) → 1 − as t → T − . The time T at which such a phenomenon occurs is called the quenching time, and the spatial point x where it occurs is referred to as quenching point. Furthermore, due to the symmetric property, the solution u quenches at the origin only. The quenching phenomenon was studied by many mathematicians ( [1,2,5,6,11,13,15]), and was extended to higher dimensional cases as (cf. [1,14] where Ω ⊂ R n is assumed to be a bounded domain with sufficiently smooth boundary, f have the following properties There have been a lot of papers on the quenching behavior of nonlinear parabolic equations. In ([4, 9, 12]) the authors have deal with the homogeneous equation with nonlinear boundary conditions. Others consider nonlinear equation with nonlinear boundary conditions. In [8], Deng and Xu consider a problem with nonlinear boundary outflux at one side: where β > 0, 0 < m < ∞. They show that u quenches in a finite time T and the only quenching point is x = 1, and they also give the quenching rate near the quenching time T , which in other word says that there exist constants C, C > 0, such that C ≤ u(1, t)(T − t) −1/(m+2β+1) ≤ C. In this paper, we will investigate the existence and non-existence property of the problem (1.1)-(1.3). We also give a criteria for the solution u of the problem quenches in a finite time T , and the rate of quenching is also given.

Existence of Solution
In this section, we will prove a local existence result of the solution of the problem (1.1)-(1.3). First of all, the following comparison result can be obtained by a similar argument as in the proof of the Theorem of Pao [16].
The following lemma states that if f is a C 1 -function in u, then u and u are necessarily ordered. Furthermore, by using the mean value theorem, for any functions u 1 , u 2 ∈< u, u >, we can obtain bounds for the function f (x, t, u).
We may assume that c 1 (x, t), c 2 (x, t), c 1 (x, t), and That implies f (x, t, u) satisfies the Lipschitz condition. By making use of the left-hand side Lipschitz condition, we are able to construct monotone sequences of upper and lower solutions of the problem. On the other hand, the right-hand side Lipschitz condition can be used to ensure the uniqueness of the solution.
Next, we are going to construct monotone sequences of functions which give the estimation of the solution u of the problem (1.1)-(1.3). Starting from initial iteration u (0) = u, and u (0) = u, we define two sequences of functions {u (k) } and {u (k) } for k = 1, 2, · · · respectively, where those functions satisfy the following linear problem, (2.5) with the initial and boundary conditions be given as where k = 1, 2, · · · . These sequences of functions satisfy the following estimations.

Proof.
Let w = u − u (1) . By the definition of upper solutions and the equation (2.5), we get Similarly, using the property of a lower solution, we obtain u (1) ≥ u. (1) . Then it follows from the equation (2.5), conditions (2.6), and the monotone property of f , we have w (1) satisfies Now assume that for some integer k > 1. Then by the equation (2.5), conditions (2.6), and the monotone property of f again, the function . Therefore, it follows from the mathematical induction, the lemma holds. Moreover, it follows from a direct comparison result, we have the functions u (k) and u (k) are ordered upper and lower solutions of the problem.
It follows from Lemma 2.6 that the sequence {u (k) } is monotone nonincreasing and is bounded from below; while the sequence {u (k) } is monotone nondecreasing and is bounded from above. Therefore the pointwise limits of these sequences exist.

Lemma 2.8. The pointwise limits
exist and satisfy the relation if u * is any other solution in the sector ⟨ u, u⟩, then by considering u * , u and u, u * as ordered upper and lower solutions the we can show that u * ≥ u and u * ≤ u. This implies that u = u * = u, and hence u * is the unique solution of problem (1.1)-(1.3).
It follows from an argument similar to the proof of the Theorem 3 of Chan and Liu [3] that u and u in (2.7) are solutions of the problem (1.1)-(1.3). Therefore we have the following local existence theorem.

Proof.
Let u(x, t) be the solution of the problem (1. Otherwise u(x, t) can be extend to a larger interval than (0, T ), and this contradicts with the definition of T . So it is suffice to prove that if λ 1 < (1 + p) ( We multiply φ 1 (x) on both sides of the equation(1.1) and integrate over B, this gives d dt (3.9) By Jensen's inequality, we have ∫ When t ∈ [0, T ), we have 0 < y(t) < 1 and Hence from (3.11), we have From the condition we have 1 − aλ 1 > 0. From (3.12), we obtain Note that when t * = 1 (1 − aλ 1 )(1 + p) , we have y(t * ) =

This gives
there exists x * * ∈B such that u(x * * , t * ) = 1. Hence the time T for the existence of the solution u satisfies We notes that the eigenvalue λ 1 decreases while the domain size increases. Hence larger the domain, more possible for the solution to quench.
Next, we would like to investigate some bounds for the solution u and the rate of quenching. Firstly, we assume that u quenches in a finite time T , x * = 0, and u 0 is a radial symmetric function satisfies: where r = ∥x∥. By the symmetric property of the domain B, the forcing term, and the initial datum, we have the solution u of the problem (1.1)-(1.3) is radial symmetric. Then the problem (1.1)-(1.3) becomes u t (r, t) =u rr (r, t) + n − 1 r u r (r, t) for (x, t) ∈ (0, 1) × (0, T ), and u(r, 0) = u 0 (r), r ∈ (0, 1), (3.14) Beside the assumption (A), we also assume that u 0 satisfies the following condition: There exists a positive constant µ such that The assumption (A1) implies that u(r, t) is an increasing function with respect to t for t > 0. We define the following functions u(0, s)) p ds, and Φ 0 ∈ C 2 ((0, 1)) ∩ C[0, 1] be a nonnegative function which satisfies Φ 0 (1) = 0, Φ ′ 0 (r) < 0 for r ∈ (0, 1), and max r∈ [0,1] | Φ 0 (r) |≤ 1. Now let Φ(r, t) be the radial symmetric solution of the homogenous heat equation with the initial value Φ 0 (r), and zero Dirichlet boundary condition. It follows from the maximum principle that The following lemmas are used in our discussion for the quenching behavior.
Lemma 3.2 shows that if the solution u quenches in a finite time, then the solution quenches at r = 0 only, that is u(0, t) → 1 − as t → T − . Lemma 3.3. Let u(r,t) be the solution of the problem(3.13)- (3.15). Then u(r,t) satisfies the inequality Proof. We first obtain the lower bound for the solution u(r, t). Let Since Φ is the solution of the homogenous heat equation, we get and U (r, 0) = u 0 (r) ≥ 0.
Next we obtain the upper bound of u. Let Then V (r, t) satisfies

Proof.We introduce a function
where η > 0 is a constant to be determined. By a direct computation, we get Since u t ≥ 0, let us take η such that 1 − ηΦ(r, t) ≥ 0. Then by Φ r ≤ 0, u r ≤ 0, we obtain .
By integrating the previous inequality with respect to t from t to T , we obtain Since u(0, t) → 1 − as t → T − , we have This gives the lower estimation of u(0, t) as where C 1 = [2(q + 1)] 1/q+1 .
Combing the equations (3.16) and (3.17), we have the quenching rate of u(0, t) as t near T .