\(^\ast \)Université d’Abobo-Adjamé, UFR-SFA, Département de Mathématiques et Informatiques, 02 BP 801 Abidjan 02, (Côte d’Ivoire), e-mail: firmingoh@yahoo.fr.

\(^\S \)Institut National Polytechnique Houphouet-Boigny de Yamoussoukro, BP 1093 Yamoussoukro, (Côte d’Ivoire), e-mail: theokboni@yahoo.fr.

1 Introduction

Let \(\Omega \) be a bounded domain in \(\mathbb {R}^N\) with smooth boundary \(\partial \Omega .\) Consider the following initial-boundary value problem

where \(\Delta \) is the Laplacian, \(\nu \) is the exterior normal unit vector on \(\partial \Omega .\) For the nonlinear term \(f(u),\) our standing assumptions are the following
(A1) \(f:(0,\infty )\rightarrow (0,\infty )\) is a \(C^1\) convex, nonincreasing function satisfying \(\lim _{s\rightarrow 0^+}f(s)=\infty ,\) \(\int _0^{\alpha }\tfrac {{\rm d}\sigma }{f(\sigma )}{\lt}\infty \) for any positive real \(\alpha ,\) and \(\int _0^{\infty }\tfrac {{\rm d}\sigma }{f(\sigma )}=\infty .\)
(A2) There exists a positive constant \(C_0\) such that

where \(H(s)\) is the inverse of the function \(F(s)\) defined as follows

For the initial datum, we make the following hypotheses
(A3): \(u_0\in C^2(\overline{\Omega }),\) \(u_0(x){\gt}0\) in \(\overline{\Omega },\) and there exists a positive constant \(B\) such that

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 \(F(s)=\tfrac {s^{p+1}}{p+1},\) \(H(s)=((p+1)s)^{1/(p+1)},\) and \(-sf'(H(s))=\tfrac {p}{p+1}.\)
Here \((0,T)\) is the maximal time interval of existence of the solution \(u,\) and by a solution we mean the following.

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,

where \(u_{\min }(t)=\min _{x\in \overline{\Omega }}u(x,t).\) 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 \(u_0^h\in C^2(\overline{\Omega }),\) \(u^h_0(x)\geq u_0(x)\) in \(\Omega ,\) and \(\lim _{h\rightarrow 0}\| u_0^h-u_0\| _{\infty }=0.\) 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 \(\| u_0^h-u_0\| _\infty .\) 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.

2 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 ] .

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 \(\tfrac {F(u^h_{0min})}{A}\) 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

which implies that \(\Delta f(v)\geq f'(v)\Delta v\) in \(\Omega \times (0,T_h).\) 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

and due to (9), we discover that

We infer from the maximum principle that

or equivalently

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 \(0\leq v(x,T_h)\leq v(x,0)\) in \(\Omega .\) It is easy to see that

We deduce from (14) that \(T_h\leq \tfrac {1}{A} \int _{0}^{u^h_{0min}} \tfrac {{\rm d}\sigma }{f(\sigma )}\) or equivalently

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.

We also need the following result which shows an upper bound of \(u_{\min }(t)\) for \(t\in (0,T).\) To prove the above estimate, we proceed as follows. Introduce the function \(w(t)\) defined as follows \(w(t)=u_{\min }(t)\) for \(t\in [0,T)\) and let \(t_1,\) \(t_2\in [0,T).\) Then, there exist \(x_1,\) \(x_2\in \Omega \) 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

which implies that \(w(t)\) is Lipschitz continuous. Further, if \(t_2{\gt}t_1,\) then

Obviously, it is not hard to see that \(\Delta u(x_2,t_2)\geq 0.\) Letting \(t_1\rightarrow t_2,\) we obtain \(w'(t)\geq -f(w(t))\) for a.e. \(t\in (0,T)\) or equivalently \(\tfrac {{\rm d}w}{f(w)}\geq -{\rm d}t\) for a.e \(t\in (0,T).\) Integrate the above inequality over \((t,T)\) to obtain \(T-t\geq \int _0^{w(t)}\tfrac {{\rm d}\sigma }{f(\sigma )}\) for \(t\in (0,T).\) Since \(w(t)=u_{\min }(t),\) we arrive at \(u_{\min }(t)\leq H(T-t)\) for \(t\in (0,T)\) and the proof is complete.

3 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 \(\| u_0^h-u_0\| _\infty .\) Our result regarding the continuity of the quenching time is stated in the following theorem.

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 \(T_h\geq T.\) In order to obtain this result, we proceed as follows. Since \(u_0^h(x)\geq u_0(x)\) in \(\Omega ,\) we know from the maximum principle that \(v\geq u\) as long as all of them are defined. This implies that \(T_h\geq T,\) and consequently, we have \(T_h-T=|T_h-T|.\) 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 \(t_0{\lt}T.\) A routine computation reveals that

where \(\theta \) is an intermediate value between \(u\) and \(v.\) Due to the fact that \(v(x,t)\geq u(x,t)\) in \(\Omega \times (0,t_0),\) then making use of Remark 4, it is easy to check that

which implies that

In view of the condition (4), we observe that there exists a positive constant \(C_1\) such that

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 \(t_1\in (0,T)\) be a time such that \(\| e(\cdot ,t_1)\| _\infty \leq C_2\| u_0^h-u_0\| _\infty (T-t_1)^{-C_1}=a\) for \(h\) small enough. This implies that

Making use of Remark 3 and the triangle inequality, it is easy to see that

Since \(\| e(\cdot ,t_1)\| _\infty \leq a\) and due to the fact that the function \(F: [0,\infty )\rightarrow [0,\infty )\) is increasing, we infer from Theorem 5 that

Having in mind that \(H\) is the inverse of \(F,\) we deduce that \(H: [0,\infty ) \rightarrow [0,\infty )\) is also increasing. We recall that \(\lim _{s\rightarrow \infty } F(s)=\infty ,\) which implies that \(\lim _{s\rightarrow \infty } H(s)=\infty .\) Introduce the function \(\varphi \) defined as follows

It is clear that \(\varphi (x)\) is increasing for \(x\in [0,\infty ).\) In addition \(\lim _{x\rightarrow \infty } \varphi (x)=\infty .\) According to the fact that \(\varphi (1)+a\) belongs to \((0,\infty ),\) we conclude that there exists a positive constant \(C_3\) such that \( \varphi (1)+a\leq \varphi (C_3),\) which implies that \(H(T-t_1)+a\leq H(C_3(T-t_1)).\) Recalling that \(F(x)\) is increasing for \(x\in [0,\infty ),\) we deduce that \(\tfrac {1}{A}F(H(T-t_1)+a)\leq \tfrac {1}{A}F(H(C_3(T-t_1))).\) Exploiting the above inequality and (17), we find that

We deduce from (18) and the triangle inequality that

which leads us to

Use the equality (16) to complete the rest of the proof.

4 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, \(u_0(x)=4+3\cos (\pi \| x\| )+\tfrac {\varepsilon }{2+\cos (\pi \| x\| )},\) with \(\varepsilon \) a nonnegative parameter, \(B=\{ x \in \mathbb {R}^{N}; \; \| x\| {\lt}1\} ,\) \(S=\{ x \in \mathbb {R}^{N}; \| x\| =1\} ,\) \(1\leq N \leq 3.\) The above problem may be rewritten in the following form

where we take \(\varphi (r)=4+3\cos (\pi r)+\tfrac {\varepsilon }{2+cos(\pi r)}.\) 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 \(x_i=ih,\) \(0\leq i \leq I,\) and approximate the solution \(u\) of (19)–(21) by the solution \(U_h^{(n)}=(U_0^{(n)}, ..., U_I^{(n)})^T\) of the following explicit scheme

After a little transformation, the above equations become

with \(U_{hmin}^{(n)}=\min _{0\leq i \leq I}U_i^{(n)}.\) Let us notice that, if \(\Delta t_n\leq \tfrac {h^2(U_{hmin}^{(n)})^{p+1}}{h^2+2N(U_{hmin}^{(n)})^{p+1}},\) then one easily sees by induction that \(U_{hmin}^{(n)}\) is positive for \(n\geq 0.\) Thus, the above condition is the CFL condition that ensures the stability of our scheme. It is important to note that if one chooses \(\Delta t_n=\min \{ \tfrac {(1-h^2)h^2}{2N}, h^2(U_{hmin}^{(n)})^{p+1}\} ,\) 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 \(U_h^{(n)}\) of the implicit scheme below

As in the case of the explicit scheme, here again, we transform our scheme to an adaptive scheme by choosing \(\Delta t_n= h^2 (U_{hmin}^{(n)})^{p+1}.\) The implicit scheme gives the following equations

The above scheme leads us to the following tridiagonal linear system

where \(A_{h}^{(n)}\) is a \( (I+1)\times (I+1)\) tridiagonal matrix defined as follows

with

It is not hard to see that

These inequalities imply that the matrix \(A_h^{(n)}\) is invertible and its inverse \((A_h^{(n)})^{-1}\) is a positive matrix. Consequently, it is easy to see that, if \(U_h^{(n)}\) is positive, then \(U_h^{(n+1)}\) exists and is also positive. Thus, since \(U_h^{(0)}=\varphi _h\) is positive, we show by induction that \(U_h^{(n)}\) exists and is positive. It is not hard to see that \(u_{rr}(0,t)=\lim _{r\rightarrow 0} \tfrac {u_r(r,t)}{r}.\) Hence, if \(r=0,\) then we see that

These observations have been taken into account in the construction of our schemes at the first node. We need the following definition.

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 \(t_{n}=\sum _{j=0}^{n-1}\Delta t_{j}\) which is computed at the first time when \(U_{hmin}^{(n)}\leq 10^{-10}.\) The order \((s)\) of the method is computed from

Numerical experiments for \(p=1, N=2\)

First case: \(\varepsilon =0\)

Second case: \(\varepsilon =1/50\)

Third case: \(\varepsilon =1/100\)

In what follows, we give some plots to illustrate our analysis. In Figures 1, 2 and 3 we can appreciate that the discrete solution quenches in a finite time. We also remark that the representation of the discrete solution when \(\varepsilon =0\) is practically the same that the one when \(\varepsilon =1/50\) or \(\varepsilon =1/100.\)