\(^\ast \)Université d’Abobo-Adjamé, UFR-SFA, Département de Mathématiques et Informatiques, 02 BP 801 Abidjan 02, (Côte d’Ivoire), International University of Grand-Bassam Route de Bonoua Grand-Bassam BP 564 Grand-Bassam, (Côte d’Ivoire) and Laboratoire de Modélisation Mathématique et de Calcul Économique LM2CE settat, (Maroc), e-mail: nachidhalima@yahoo.fr.

1 Introduction

Consider the following boundary value problem

where \(f:(0,\infty )\rightarrow (0,\infty )\) is a \(C^{1}\) nondecreasing function, \(\int _0^{\alpha }\tfrac {{\rm d}s}{f(s)}{\lt}\infty ,\) for any positive real \(\alpha \). \(\lim _{s \rightarrow 0^{+}}f(0)=\infty ,\) \(a \in C^{1}([0,1]),\) \(a(x){\gt}0,\) \(x \in (0,1),\) \(a'(0)=0,\) \(a'(1)=0,\) \(u'_{0}(1)=0\) and \(u'_{0}(0)=0\). The initial data \(u_{0}\in C^{2}([0,1])\), \(u_{0}(x){\gt}0,\) \(x \in (0,1),\) \(u_{0}(x){\lt}1,\) \(x \in (0,1),\)

Here \([0,T]\) is the maximal time interval on which the solution \(u\) of (1)–(3) exists. The time \(T \) may be finite or infinite. When \(T\) is infinite, then we say that the solution \(u\) of (1)–(3) exists globally. When \(T\) is finite, then the solution \(u\) of (1)–(3) develops a singularity in a finite time, namely, \(\lim _{t\rightarrow T}u_{min}(t)=0,\) where \(u_{min}(t)={\rm min}_{0\leq x \leq 1}u(x,t).\) In this last case,we say that the solution \(u\) of (1)–(3) quenches in a finite time and the time \(T\) is called the quenching time of the solution \(u.\)

The theoretical study of solutions for semilinear parabolic equations which quench in a finite time has been the subject of investigations of many authors (see [2], [4]–[7], [11], [12], [16] and the references cited therein). Local in time existence of a classical solution has been proved and this solution is unique. In addition, it is shown that if the initial data at (3) satisfies \(u''_{0}(x)-a(x)u^{-p}_{0}(x)\leq -Au^{-p}_{0}(x)\) in \([0,1]\) where \(A\in (0,1]\) and \(p{\gt}0\), then the classical solution \(u\) of (1)–(3) quenches in a finite time \(T\) and we have the following estimates

for \(t\in (0,T)\), (see, for instance [4]–[6]).

In this paper, we are interested in the numerical study of the phenomenon of quenching. Our aim is to build a semidiscrete scheme where solution obeys the property of the continuous one. In order to facilitate our discussion, let us notice that the first condition in (4) allows the solution \(u\) to attain its minimum at the point \(x=0,\) and the second one permits the solution \(u\) to decrease with respect to the second variable. The hypotheses (5) are compatibility condition which ensure the regularity of the solution \(u.\)

This paper is organized as follows. In the next section, we give some results about the discrete maximum principle. In the third section, under some conditions, we prove that the solution of a semidiscrete form of (1)–(3) quenches in a finite time and estimate its semidiscrete quenching time. In the fourth section, we prove the convergence of the semidiscrete quenching time. In the fifth section, we study the results of sections 3 and 4 taking a discrete form of (1)–(3). Finally, in the last section, we give some numerical results to illustrate our analysis.

2 Properties of a semidiscrete problem

We start our study by the construction of a semidiscrete scheme as follows. Let \(I\) be a positive integer, and let \(h=\tfrac {1}{I}\). Define the grid \(x_{i}=ih\), \(0\leq i \leq I,\) and approximate the solution \(u\) of the problem (1)–(3) by the solution \(U_{h}(t)=(U _{0}(t),U_{1}(t),\ldots ,U_{I}(t))^{T}\) of the following semidiscrete equations

where

where \(\beta _i {\gt}0,\) \(\varphi _{i}{\gt}0\)

\(\beta _i \) and \(\varphi _{i}\) are approximations of \(a(x_{i})\) and \(u_{0}(x_{i})\), respectively. There is another reason which has motivated our choice, namely we want to know the behavior of the quenching time when one perturbs slightly either the potential or the initial datum.

Here \((0,T^{h}_{q})\) is the maximal time interval on which \(\| U_{h}(t)\| _{\inf }{\gt}0\) where

When the time \(T^{h}_{q}\) is finite, we say that the solution \(U_{h}(t)\) of (7)–(9) quenches in a finite time and the time \(T^{h}_{q}\) is called the quenching time of the solution \(U_{h}(t)\). The following lemma is a semidiscrete form of the maximum principle.

Let \(T_{0}\) be any quantity satisfying the inequality \(T_{0}{\lt}T\) and define the vector \(Z_{h}(t)=e^{\lambda t}V_{h}(t)\) where \(\lambda \) is such that

Set \(m=\min _{0\leq t\leq T_{0}}\| Z_{h}(t)\| _{\inf }\). Since \(Z_{h}(t)\) is a continuous vector on the compact \([0,T_0]\), there exist \(i_0\in \{ 0,...,I\} \) and \(t_0\in [0,T_0]\) such that \(m=Z_{i_0}(t_0)\). We observe that

From (11), we obtain the following inequality

We deduce from (13)–(15) that \((\alpha _{i_{0}}(t_{0})-\lambda )Z_{i_{0}}(t_{0})\geq 0\), which implies that \(Z_{i_{0}}(t_{0})\geq 0\). Therefore, \(V_{h}(t)\geq 0\) for \(t\in [0,T_{0}]\) and the proof is complete.

Another form of the maximum principle for semidiscrete equations is the following comparison lemma.

Let \(Z_{h}(t)=W_{h}(t)-V_{h}(t)\) and let \(t_{0}\) be the first \(t\in (0,T)\) such that \(Z_{h}(t){\gt}0\) for \(t \in [0,t_{0})\) but \(Z_{i_{0}}(t_{0})=0\) for a certain \(i_{0}\in \{ 0,...,I\} \). We see that

Therefore, we have

which contradicts the first strict inequality of the lemma and this ends the proof.

3 Quenching problem

In this section, under some assumptions, we show that the solution \(U_{h}\) of (7)–(9) quenches in a finite time and estimate its semidiscrete quenching time. We need the following result about the operator \(\delta ^{2}\).

Applying Taylor’s expansion, we find that

where \(\theta _{i}\) is an intermediate value between \(U_{i}\) and \(U_{i+1}\), \(\eta _{i}\) the one between \(U_{i-1}\) and \(U_{i}\), \(U_{-1}=U_{1}\), \(U_{I+1}=U_{I-1}\), \(\eta _{0}=\theta _{0}\), \(\eta _{I}=\theta _{I}\). Use the fact that \(U_{h}{\gt}0\) to complete the rest of the proof.

The statement of the result about solutions which quench in a finite time is the following.

Since \((0,T_{q}^{h})\) is the maximal time interval on which \(\| U_{h}(t)\| _{\inf }{\gt}0\), our aim is to show that \(T_{q}^{h}\) is finite and satisfies the above inequality. Introduce the vector \(J_{h}(t)\) defined as follows

A straightforward calculation gives

From Lemma 3.1, we have \(\delta ^{2}(f(U))_i\geq f'(U_{i})\delta ^2U_{i},\) \( 0\leq i\leq I,\) which implies that

Using (7), we arrive at

From (16), we observe that \(J_{h}(0)\leq 0\). We deduce from Lemma 2.1 that \(J_{h}(t)\leq 0\) for \(t\in (0,T_{q}^{h})\), which implies that

These estimates may be rewritten in the following form \(\tfrac {{\rm d}U_{i}}{f(U_{i})}\leq -A{\rm d}t\), \(0\leq i\leq I\). Integrating the above inequalities over the interval \((t,T_{q}^{h})\), we get

Using the fact that \(\| \varphi _{h}\| _{\inf }=U_{i_{0}}(0)\) for a certain \(i_{0}\in \{ 0,...,I\} \) and taking \(t=0\) in (18), we obtain the desired result.

4 Convergence of the semidiscrete quenching time

In this section, under some assumptions, we show that the solution of the semidiscrete problem quenches in a finite time and its semidiscrete quenching time converges to the real one when the mesh size goes to zero. We denote

In order to obtain the convergence of the semidiscrete quenching time, we firstly prove the following theorem about the convergence of the semidiscrete scheme.

Let \(K{\gt}0\) and \(L{\gt}0\) be such that

The problem (7)–(9) has for each \(h\), a unique solution \(U_{h}\in C^{1}([0,T_{q}^{h}),\mathbb {R}^{I+1})\). Let \(t(h)\leq \min \{ T,T_{q}^{h}\} \) be the greatest value of \(t{\gt}0\) such that

The relation (19) implies that \(t(h){\gt}0\) for \(h\) sufficiently small. By the triangle inequality, we obtain

which implies that

Since \(u\in C^{4,1}\), taking the derivative in \(x\) on both sides of (1) and due to the fact that \(u_{x}\), \(u_{xt}\) vanish at \(x=0\) and \(x=1\), we observe that \(u_{xxx}\) also vanishes at \(x=0\) and \(x=1\). Applying Taylor’s expansion, we discover that

To establish the above equalities for \(i=0\) and \(i=I\), we have used the fact that \(u_{x}\) and \(u_{xxx}\) vanish at \(x=0\) and \(x=1\). A direct calculation yields

for \(1\leq i\leq I-1\). Let \(e_{h}(t)=U_{h}(t)-u_{h}(t)\) be the error of discretization. From the mean value theorem, we have

for \(0\leq i\leq I,\ t\in (0,t(h))\), where \(\theta _{i}\) is an intermediate value between \(U_{i}(t)\) and \(u(x_{i},t)\). Using (21), (22), we arrive at

Introduce the vector \(z_{h}(t)\) defined as follows

A straightforward computation reveals that

It follows from Comparison Lemma 2.2 that

In the same way, we also prove that

which implies that

for \(t\in (0,t(h))\). Let us show that \(t(h)=\min \{ T,T^{h}_{q}\} \). Suppose that \(t(h){\lt}\min \{ T,T^{h}_{q}\} \). From (22), we obtain

Let us notice that both last formulas for \(t(h)\) are valid for sufficiently small \(h\). Since the term on the right hand side of the above inequality goes to zero as \(h\) goes to zero, we deduce that \(\tfrac {\varrho }{2}\leq 0\), which is impossible. Consequently \(t(h)=\min \{ T,T^{h}_{q}\} \).

Now, let us show that \(t(h)=T\). Suppose that \(t(h)=T^{h}_{q}{\lt}T\). Reasoning as above, we prove that we have a contradiction and the proof is complete.

Now, we are in a position to prove the main theorem of this section.

Let \(0{\lt}\varepsilon {\lt}T_{q}/2\). There exists \(\varrho \in (0,1)\) such that

Since \(u\) quenches in a finite time \(T_{q}\), there exist \(h_{0}(\varepsilon ){\gt}0\) and a time \(T_{0}\in (T_{q}-\tfrac {\varepsilon }{2},T_{q})\) such that \(0{\lt}u_{\min }(t){\lt}\tfrac {\varrho }{2}\) \(\quad \) for \(t\in [T_{0},T_{q})\), \(h\leq h_{0}(\varepsilon )\). It is not hard to see that \(u_{\min }(t){\gt}0\) for \(t\in [0,T_{0}]\), \(h\leq h_{0}(\varepsilon )\). From Theorem 4.1, the problem (7)–(9) has a solution \(U_{h}(t)\) and we get \(\| U_{h}(t)-u_{h}(t)\| _{\infty }\leq \tfrac {\varrho }{2}\) for \(t\in [0,T_{0}]\), \(h\leq h_{0}(\varepsilon )\), which implies that \(\| U_{h}(T_{0})-u_{h}(T_{0})\| _{\infty }\leq \tfrac {\varrho }{2}\) for \(h\leq h_{0}(\varepsilon )\). Applying the triangle inequality, we find that

From Theorem 3.1, \(U_{h}(t)\) quenches at the time \(T_{q}^{h}\). We deduce from Remark 3.1 and (22) that for \(h\leq h_{0}(\varepsilon )\),

which leads us to the desired result.

5 Full discretizations

In this section, we study the phenomenon of quenching using a full discrete explicit scheme of (1)–(3). Approximate the solution \(u(x,t)\) of the problem (1)–(3) by the solution \(U^{{(n)}}_{h}=(U_{0}^{(n)},U_{1}^{(n)},\ldots ,U_{I}^{(n)})^{T}\) of the following explicit scheme

where \(n\geq 0\),

If \(U_h^{(n)}{\gt}0\), then \(-\tfrac {f(U_i^{(n)})}{U_i^{(n)}}\geq -\tfrac {f(\| U_h^{(n)}\| _{\inf })}{\| U_h^{(n)}\| _{\inf }}\), \(0\leq i\leq I\), and a straightforward computation reveals that

In order to permit the discrete solution to reproduce the properties of the continuous one when the time \(t\) approaches the quenching time \(T_{q}\), we need to adapt the size of the time step so that we choose
\(\Delta t_n= \min \{ \tfrac {(1-\tau )h^2}{2},\tau \tfrac {f(\| U_h^{(n)}\| _{\inf })}{\| U_h^{(n)}\| _{\inf }}\} \) with \(0{\lt}\tau {\lt}1\). We observe that \(1-2\tfrac {\Delta t_n}{h^2}-\| \beta _{h}\| _\infty \Delta t_n \tfrac {f(\| U_h^{(n)}\| _{\inf })}{\| U_h^{(n)}\| _{\inf }}\geq 0\), which implies that \(U_{h}^{(n+1)}{\gt}0\). Thus, since by hypothesis \(U_{h}^{(0)}=\varphi _{h}{\gt}0\), if we take \(\Delta t_n\) as defined above, then using a recursion argument, we see that the positivity of the discrete solution is guaranteed. Here, \(\tau \) is a parameter which will be chosen later to allow the discrete solution \(U_{h}^{(n)}\) to satisfy certain properties useful to get the convergence of the numerical quenching time defined below.

If necessary, we may take \(\Delta t_n= \min \{ \tfrac {(1-\tau )h^2}{K},\tau \tfrac {f(\| U_h^{(n)}\| _{\inf })}{\| U_h^{(n)}\| _{\inf }}\} \) with \(K{\gt}2\) because in this case, the positivity of the discrete solution is also guaranteed.
The following lemma is a discrete form of the maximum principle.

If \(V_{h}^{(n)}\geq 0\), then a routine computation yields

Since \(\Delta t_n\leq \tfrac {h^2}{2+\| a_{h}^{(n)}\| _{\infty }h^2}\), we see that \(1-2\tfrac {\Delta t_n}{h^2}-\Delta t_n\| a_{h}^{(n)}\| _{\infty }\) is nonnegative. From (27), we deduce by induction that \(V_h^{(n)}\geq 0\) which ends the proof.

A direct consequence of the above result is the following comparison lemma. Its proof is straightforward.

Now, let us give a property of the operator \(\delta _{t}\) stated in the following lemma. Its proof is quite similar to that of Lemma 3.1, so we omit it here.

We have \(\int _{0}^{\infty }\tfrac {ab^{x}}{f(ab^{x})}= \sum _{n=0}^{\infty }\int _{n}^{n+1}\tfrac {ab^{x}{\rm d}x}{f(ab^{x})}.\) We observe that \(ab^{x}\geq ab^{n+1}\) for \(n\leq x\leq n+1,\) which that \( \int _{n}^{n+1}\tfrac {ab^{x}{\rm d}x}{f(ab^{x})}\geq \tfrac {ab^{n+1}}{f(ab^{n+1})}.\) Consequently, we get

Use the fact that \(\int _{0}^{\infty }\tfrac {ab^{x}}{f(ab^{x})}=-\tfrac {1}{\ln (b)}\int _{0}^{a}\tfrac {{\rm d}\sigma }{f(\sigma )}\) to complete the rest of the proof.

The theorem below is the discrete version of Theorem 4.1.

For each \(h\), the problem (25)–(26) has a solution \(U_{h}^{(n)}\). Let \(N\leq J\) be the greatest value of \(n\) such that

We know that \(N\geq 1\) because of (16). Applying the triangle inequality, we have

As in the proof of Theorem 4.1, using Taylor’s expansion, we find that for \(n{\lt}N\), \(0\leq i\leq I\),

Let \(e_h^{(n)}=U_h^{(n)}-u_h(t_n)\) be the error of discretization. From the mean value theorem, we get for \(n{\lt}N\), \(0\leq i\leq I\),

where \(\xi _{i}^{(n)}\) is an intermediate value between \(u(x_i,t_n)\) and \(U_{i}^{(n)}\). Since
\(u_{xxxx}(x,t)\), \(u_{tt}(x,t)\) are bounded,\(u(x,t)\geq \rho \) and \(\Delta t_{n}=O(h^{2})\), then there exists a positive constant \(M\) such that

Set \(L=-(\| a_h\| \infty +1)f'(\tfrac {\rho }{2})\) and introduce the vector \(V_h^{(n)}\) defined as follows

A straightforward computation gives

We observe from (29) that \(-\beta _{i}f'(\xi _{i}^{(n)})\) is bounded from above by \(L\). It follows from Comparison Lemma 5.2 that \(V_{h}^{(n)}\geq e_{h}^{(n)}\). By the same way, we also prove that \(V_{h}^{(n)}\geq -e_{h}^{(n)}\), which implies that

Let us show that \(N=J\). Suppose that \(N{\lt}J\). If we replace \(n\) by \(N\) in (29) and use (30), we find that

Since the term on the right hand side of the second inequality goes to zero as \(h\) goes to zero, we deduce that \(\tfrac {\rho }{2}\leq 0\), which is a contradiction and the proof is complete.

To handle the phenomenon of quenching for discrete equations, we need the following definition.

The following theorem reveals that the discrete solution \(U_h^{(n)}\) of (25)-(26) quenches in a finite time under some hypotheses.

Introduce the vector \(J_h^{(n)}\) defined as follows

A straightforward computation yields for \(0\leq i\leq I\), \(n\geq 0\),

Using (25), we arrive at

It follows from Lemmas 5.3 and 3.1 that for \(0\leq i\leq I\), \(n\geq 0\),

We deduce from (25) that

Obviously, the inequalities (31) ensure that \(J_h^{(0)}\leq 0\). Applying Lemma 5.1, we get \(J_h^{(n)}\leq 0\) for \(n\geq 0\), which implies that

These estimates reveal that the sequence \(U_h^{(n)}\) is nonincreasing. By induction, we obtain \(U_{h}^{(n)}\leq U_{h}^{(0)}=\varphi _{h}\). Thus, the following holds

Let \(i_{0}\) be such that \(\| U^{(n)}_{h}\| _{\inf }=U^{(n)}_{i_{0}}\). Replacing \(i\) by \(i_{0}\) in (38), we obtain

and by iteration, we arrive at

Since the term on the right hand side of the above equality goes to zero as \(n\) approaches infinity, we conclude that \(\| U_h^{(n)}\| _{\inf }\) tends to zero as \(n\) approaches infinity. Now, let us estimate the numerical quenching time. Due to (33) and the restriction \(\Delta t_n\leq \tfrac {\tau \| U_h^{(n)}\| _{\inf }}{f(\| U_h^{(n)}\| _{\inf })}\), it is not hard to see that

because \(\tfrac {s}{f(s)}\) is nondecreasing for \(s{\gt}0.\) It follows from Lemma 5.4 that

Use the fact that the quantity on the right hand side of the above inequality converges towards is finite to complete the rest of the proof.

In the sequel, we take \(\tau =h^{2}\).

Now, we are in a position to state the main theorem of this section.

We know from Remark 5.1 that \(\tfrac {\tau }{\ln (1-\tau ^{'})}\) is bounded. Letting \(0{\lt}\varepsilon {\lt}T_{q}/2\), there exists a constant \(R\in (0,1)\) such that

Since \(u\) quenches at the time \(T_{q}\), there exist \(T_{1}\in (T_{q}-\tfrac {\varepsilon }{2},T_{q})\) and \(h_{0}(\varepsilon ){\gt}0\) such that \(0{\lt}u_{\min }(t){\lt}\tfrac {R}{2}\) for \(t\in [T_1,T_q)\), \(h\leq h_{0}(\varepsilon )\). Let \(q\) be a positive integer such that \(t_q=\sum _{n=0}^{q-1}\Delta t_n\in [T_1,T_q)\) for \(h\leq h_{0}(\varepsilon )\). It follows from Theorem 5.1 that the problem (25)–(26) has a solution \(U_h^{(n)}\) which obeys \(\| U_h^{(n)}-u_h(t_n)\| _{\infty }{\lt}\tfrac {R}{2}\) for \(n\leq q\), \(h\leq h_{0}(\varepsilon )\), which implies that

From Theorem 5.2, \(U_h^{(n)}\) quenches at the time \(T_{h}^{\Delta t}\). It follows from Remark 5.1 and (36) that \(|T_{h}^{\Delta t}-t_q|\leq \tfrac {\tau \| U_h^{(q)}\| _{\inf }}{f(\| U_h^{(q)}\| _{\inf })}-\tfrac {\tau }{\ln (1-\tau ’)}\int _{0}^{\| U_h^{(q)}\| _{\inf }}\tfrac {{\rm d}\sigma }{f(\sigma )}{\lt}\tfrac {\varepsilon }{2}\) because \(\| U_h^{(q)}\| _{\inf }{\lt}R\) for \(h\leq h_{0}(\varepsilon )\). We deduce that for \(h\leq h_{0}(\varepsilon )\),

which leads us to the result.

6 Numerical results

In this section, we present some numerical approximations to the quenching time for the solution of the problem (1)–(3) in the case where \(p=1\) and \(u_{0}(x)=\tfrac {2+\varepsilon \cos ({\pi }x)}{4}\) with \(0{\lt}\varepsilon \leq 1\). Firstly, we take the explicit scheme in (25)–(26). Secondly, we use the following implicit scheme

where \(n\geq 0\), \(\Delta t_n=K\| U_h^{(n)}\| _{\inf }^{p+1}\) with \(K=10^{-3}\).

In both cases, \(\varphi _{i}=\tfrac {2+\varepsilon \cos ({\pi }ih)}{4}\), \(0\leq i\leq I\). For the above implicit scheme, the existence and positivity of the discrete solution \(U_{h}^{(n)}\) is guaranteed using standard methods (see [3]). In the tables 1–8, in rows, we present the numerical quenching times, the numbers of iterations and the CPU times corresponding to meshes of 16, 32, 64, 128. We take for the numerical quenching time \(t_{n}=\sum _{j=0}^{n-1}\Delta t_{j}\) which is computed at the first time when