CONTINUITY OF THE QUENCHING TIME IN A SEMILINEAR HEAT EQUATION WITH NEUMANN BOUNDARY CONDITION

. This paper concerns the study of a semilinear parabolic equation subject to Neumann boundary conditions and positive initial datum. Under some assumptions, we show that the solution of the above problem quenches in a ﬁnite time and estimate its quenching time. We also prove the continuity of the quenching time as a function of the initial datum. Finally, we give some numerical results to illustrate our analysis.

For the initial datum, we make the following hypotheses (A3): u 0 ∈ C 2 (Ω), u 0 (x) > 0 in Ω, and there exists a positive constant B such that ∆u 0 (x) − f (u 0 (x)) ≤ −Bf (u 0 (x)) in Ω. (5) It is worth noting that, if we choose f (s) = s −p with p a positive real, then it is not hard to see that f satisfies the different assumptions listed in the introduction of the paper.Also, in this case, we note that F (s) = s p+1 p+1 , H(s) = ((p + 1)s) 1/(p+1) , and −sf (H(s)) = p p+1 .Here (0, T ) is the maximal time interval of existence of the solution u, and by a solution we mean the following.Definition 1.A solution of (1)-( 3) is a function u(x, t) continuous verifying (1)-(3), u(x, t) > 0 in Ω × [0, T ), and twice continuously differentiable in x and once in t in Ω × (0, T ).
The time T may be finite or infinite.When T is infinite, then we say that the solution u exists globally.When T is finite, then the solution u develops a quenching in a finite time, namely, lim t→T u min (t) = 0, where u min (t) = min x∈Ω u(x, t).In this last case, we say that the solution u quenches in a finite time, and the time T is called the quenching time of the solution u.Since the pioneering work of Kawarada in [25] regarding the phenomenon of quenching, solutions of semilinear parabolic equations which quench in a finite time have been the subject of investigation of many authors (see, [2]- [4], [7], [8], [12], [14], [15], [22], [26], [28]- [30], [33], [37], and the references cited therein).In particular, in [7], the problem (1)-(3) has been studied.By standard methods, it is well known that, making use of the assumptions made on the paper, one easily proves the existence and uniqueness of solutions (see, [7, Theorems 2.1 and 2.2], [16, Chapter 7, Theorem 13 (and its corollary)], [27,Theorem 7.4]).It is also shown that the solution of (1)-(3) quenches in a finite time and its quenching time has been estimated (see, [7]).In this paper, we are interested in the continuity of the quenching time as a function of the initial datum.More precisely, we consider the following initial-boundary value problem where in Ω, and lim h→0 u h 0 − u 0 ∞ = 0.Here (0, T h ) is the maximal time interval on which the solution v of ( 6)-( 8) exists.Definition 1 is valid for the solution v of ( 6)-( 8) and we assume that this solution is sufficiently regular.It is worth noting that the regularity of solutions increases with respect to the regularity of initial data, and one may apply without difficulties the maximum principle (see, [16,Chapter 7, Theorem 13 (and its corollary)], [27, Chapter 5, Theorem 53], [35,Chapter 4]).In the present paper, we prove that if h is small enough, then the solution v of ( 6)- (8) quenches in a finite time and its quenching time T h goes to T as h goes to zero, where T is the quenching time of the solution u of (1)-( 3).In addition we provide an upper bound of |T h − T | in terms of u h 0 − u 0 ∞ .This work is in parts the continuity of our earlier study in [8] where we considered a particular case of the problem (1)-(3) choosing f (s) = s −p with p a positive constant.Under some hypotheses, we showed that the time T h goes to T as h tends to zero.Let us point out that in [8], we have not found an upper bound of |T h − T |.Similar results have been obtained in [5], [10], [18]- [20], [23], [24], [32], [36], where the authors considered analogous problems within the framework of the phenomenon of blow-up (we say that a solution blows up in a finite time if it reaches the value infinity in a finite time).The remainder of the paper is written in the following manner.In the next section, we present some results concerning quenching solutions for our subsequent use.In the third section, we analyze the continuity of the quenching time as a function of the initial datum and finally, in the last section, we show some computational results to illustrate the theory given in the paper.

QUENCHING TIME
In this section, under some assumptions, we show that the solution v of ( 6)- (8) quenches in a finite time and estimate its quenching time.
We begin by proving the following result which is inspired of an idea of Friedman and McLeod in [17].
Theorem 2. Suppose that there exists a constant A ∈ (0, 1] which is independent of h such that the initial datum at (8) satisfies Then, the solution v of (6)- (8) quenches in a finite time, and an upper bound of its quenching time is Proof.Let T h be a time up to which v remains strictly positive everywhere.Our aim is to show that T h is finite and is one of its upper bound.Introduce the function J(x, t) defined as follows A straightforward computation yields Again by a direct calculation, we observe that Using this estimate and (10), we arrive at It follows from ( 6) and ( 11) that Taking into account the expression of J, we find that Employing the condition (7), it is not hard to see that ∂J ∂ν = ∂v ∂ν t + Af (v) ∂v ∂ν = 0 on ∂Ω × (0, T h ), and due to (9), we discover that We infer from the maximum principle that This estimate may be rewritten in the following manner Integrate the above inequality over (0, T h ) to obtain From ( 12), we observe that v is nonincreasing with respect to the second variable, which implies 0 Consequently, v quenches in a finite time because the quantity on the right hand side of the above inequality is finite.This finishes the proof.
Remark 3. Let t ∈ (0, T h ).Integrating the inequality in (12) from t to T h , we get We deduce that Remark 4. In view of the condition (5) and reasoning as in the proof of Theorem 2, it is not hard to see that there exists a positive constant C such that u min (t) ≥ H(C(T − t)) for t ∈ (0, T ).
We also need the following result which shows an upper bound of u min (t) for t ∈ (0, T ).
Theorem 5. Let u be solution of (1)-( 3).Then, the following estimate holds Proof.To prove the above estimate, we proceed as follows.Introduce the function w(t) defined as follows w(t) = u min (t) for t ∈ [0, T ) and let t 1 , t 2 ∈ [0, T ).Then, there exist x 1 , x 2 ∈ Ω such that w(t 1 ) = u(x 1 , t 1 ) and w(t 2 ) = u(x 2 , t 2 ).Applying Taylor's expansion, we observe that Obviously, it is not hard to see that ∆u(x 2 , t 2 ) ≥ 0. Letting t 1 → t 2 , we obtain w (t) ≥ −f (w(t)) for a.e.t ∈ (0, T ) or equivalently dw f (w) ≥ −dt for a.e t ∈ (0, T ).Integrate the above inequality over (t, T ) to obtain T −t ≥ for t ∈ (0, T ).Since w(t) = u min (t), we arrive at u min (t) ≤ H(T − t) for t ∈ (0, T ) and the proof is complete.Remark 6. Regarding the last part of the proof of Theorem 5, one sees that T ≥ u 0min 0 dσ f (σ) .Thus, we have a lower bound of the quenching time of the solution u of (1)-(3).In the same way, it is not hard to see that is a lower bound of the quenching time of the solution v of ( 4)-( 6).

CONTINUITY OF THE QUENCHING TIME
This section is dedicated to our main result.Our aim consists in proving that, if h is small enough, then the solution v of ( 6)- (8) quenches in a finite time and its quenching time T h goes to T as h tends to zero.We also provide an upper bound of |T h − T | in terms of u h 0 − u 0 ∞ .Our result regarding the continuity of the quenching time is stated in the following theorem.
Theorem 7. Suppose that the problem (1)-(3) has a solution u which quenches at the time T.Then, under the assumption of Theorem 2, the solution v of (6)- (8) quenches in a finite time T h , and there exist positive constants α and γ such that for h small enough, the following estimate holds Proof.We know from Theorem 2 that the solution v quenches in a finite time T h .Now, to achieve our objective, it remains to demonstrate the above estimate.We begin by proving that T h ≥ T. In order to obtain this result, we proceed as follows.Since u h 0 (x) ≥ u 0 (x) in Ω, we know from the maximum principle that v ≥ u as long as all of them are defined.This implies that T h ≥ T, and consequently, we have In order to show the remaining part of the proof, we proceed by introducing the error function e(x, t) defined as follows Let t 0 be any positive quantity satisfying t 0 < T. A routine computation reveals that in Ω, where θ is an intermediate value between u and v. Due to the fact that v(x, t) ≥ u(x, t) in Ω × (0, t 0 ), then making use of Remark 6, it is easy to check that θ(x, t) ≥ u min (t) ≥ H(C(T − t)) in Ω × (0, t 0 ), (15) which implies that In view of the condition (4), we observe that there exists a positive constant C 1 such that e t ≤ ∆e + C 1 T −t e in Ω × (0, t 0 ).Let Z(t) be the solution of the following ODE When we solve the above ODE, we observe that its solution Z(t) is given explicitly by ).On the other hand, an application of the maximum principle renders where C 2 = T C 1 .Fix a a positive constant and let t 1 ∈ (0, T ) be a time such that e(•, t 1 ) ∞ ≤ C 2 u h 0 − u 0 ∞ (T − t 1 ) −C 1 = a for h small enough.This implies that Making use of Remark 3 and the triangle inequality, it is easy to see that Since e(•, t 1 ) ∞ ≤ a and due to the fact that the function F : [0, ∞) → [0, ∞) is increasing, we infer from Theorem 5 that Having in mind that H is the inverse of F, we deduce that H : [0, ∞) → [0, ∞) is also increasing.We recall that lim s→∞ F (s) = ∞, which implies that lim s→∞ H(s) = ∞.Introduce the function ϕ defined as follows According to the fact that ϕ(1) + a belongs to (0, ∞), we conclude that there exists a positive constant C 3 such that ϕ(1) + a ≤ ϕ(C 3 ), which implies that Exploiting the above inequality and (17), we find that We deduce from (18) and the triangle inequality that Use the equality (16) to complete the rest of the proof.

NUMERICAL RESULTS
In this section, we give some computational experiments to confirm the theory given in the previous section.We consider the radial symmetric solution of the following initial-boundary value problem where p is a positive constant, u 0 (x) = 4 + 3 cos(π x ) + ε 2+cos(π x ) , with ε a nonnegative parameter, B = {x ∈ R N ; x < 1}, S = {x ∈ R N ; x = 1}, 1 ≤ N ≤ 3. The above problem may be rewritten in the following form u r (0, t) = 0, u r (1, t) = 0, t ∈ (0, T ), ( 20) where we take ϕ(r) = 4 + 3 cos(πr) + ε 2+cos(πr) .We start by the construction of an adaptive scheme as follows.Let I be a positive integer and let h = 1/I.Define the grid x i = ih, 0 ≤ i ≤ I, and approximate the solution u of ( 19)-( 21) by the solution After a little transformation, the above equations become hmin is positive for n ≥ 0. Thus, the above condition is the CFL condition that ensures the stability of our scheme.

It is important to note that if one chooses ∆t
hmin ) p+1 }, then the above CFL condition is fulfilled.Consequently, in the sequel, we shall pick the above time step for our explicit scheme.An important fact concerning the phenomenon of quenching is that, if the solution u quenches at the time T, then when the time t approaches the quenching time T, the solution u decreases to zero rapidly.Thus, in order to permit the discrete solution to reproduce the properties of the continuous one when the time t approaches the quenching time T, we need to adapt the size of the time step.This is the reason why we have chosen the above time step.For this time step, our explicit scheme becomes an adaptive scheme which is one of suitable schemes for problems whose solutions quench in a finite time.We also approximate the solution u of ( 19)-( 21) by the solution As in the case of the explicit scheme, here again, we transform our scheme to an adaptive scheme by choosing ∆t n = h 2 (U (n) hmin ) p+1 .The implicit scheme gives the following equations (22) ( The above scheme leads us to the following tridiagonal linear system h is a (I + 1) × (I + 1) tridiagonal matrix defined as follows These inequalities imply that the matrix exists and is also positive.Thus, since U (0) h = ϕ h is positive, we show by induction that U (n) h exists and is positive.It is not hard to see that u rr (0, t) = lim r→0 ur(r,t) r .Hence, if r = 0, then we see that u t (0, t) = N u rr (0, t) − (u(0, t)) −p , t ∈ (0, T ), These observations have been taken into account in the construction of our schemes at the first node.We need the following definition.Definition 8. We say that the discrete solution U (n) h of the explicit scheme or the implicit scheme quenches in a finite time if lim n→∞ U (n) hmin = 0, and the series ∞ n=0 ∆t n converges.The quantity ∞ n=0 ∆t n is called the numerical quenching time of the discrete solution U (n) h .In the following tables, in rows, we present the numerical quenching times, the numbers of iterations, the CPU times and the orders of the approximations corresponding to meshes of 16, 32, 64, 128.We take for the numerical quenching time t n = n−1 j=0 ∆t j which is computed at the first time when U (n) hmin ≤ 10 −10 .The order (s) of the method is computed from s .
Numerical experiments for p = 1, N = 2 First case: ε = 0 Third case: ε = 1/100 Remark 9.If we consider the problem ( 19)-( 21) in the case where p = 1 and the initial datum ϕ(r) = 4 + 3 cos(πr) + ε 2+cos(πr) with ε = 0, then we see that the numerical quenching time of the discrete solution for the explicit scheme or the implicit scheme is approximately equal to that in which the initial datum increases slightly, that is when ε is a small positive real (see, Tables 1-6 for an illustration).This result confirms the theory regarding the continuity of the quenching time as a function of the initial datum.
In what follows, we give some plots to illustrate our analysis.In Figures 1 We also remark that the representation of the discrete solution when ε = 0 is practically the same that the one when ε = 1/50 or ε = 1/100.

Table 1 .
Numerical quenching times, numbers of iterations, CPU times (seconds) and orders of the approximations obtained with the explicit Euler method.

Table 2 .
Numerical quenching times, numbers of iterations, CPU times (seconds) and orders of the approximations obtained with the implicit Euler method.

Table 3 .
Numerical quenching times, numbers of iterations, CPU times (seconds) and orders of the approximations obtained with the explicit Euler method.

Table 4 .
Numerical quenching times, numbers of iterations, CPU times (seconds) and orders of the approximations obtained with the implicit Euler method.

Table 5 .
, 2 and 3 we can appreciate that the discrete solution quenches in a finite time.Numerical quenching times, numbers of iterations, CPU times (seconds) and orders of the approximations obtained with the explicit Euler method.

Table 6 .
Numerical quenching times, numbers of iterations, CPU times (seconds) and orders of the approximations obtained with the implicit Euler method.