The Quenching Solutions of a Singular Parabolic Equation

ARIJ BOUZELMATE, FATIMA SENNOUNI, ABDELILAH GMIRA

Department of Mathematics,

Abdelmalek Essaadi University,

LaR2A Laboratory, Faculty of Sciences,

Tetouan,

MOROCCO

Abstract: - This article is dedicated to the study of the self-similar solutions of a nonlinear parabolic equation.

More precisely, we consider the following uni-dimensional equation:

(E) : ut(x, t) = (um)xx(x, t)− |x|qu−p(x, t), x ∈R, t > 0,

where m > 1, q > 1and p > 0.

Initially, we employed a fixed point theorem and an associated energy function to establish the existence of solu-

tions. Subsequently, we derived some important results on the asymptotic behavior of solutions near the origin.

Key-Words: - Singular parabolic equation, Self-similar solutions, Quenching solutions, Existence of solutions,

Asymptotic behavior.

Received: July 21, 2023. Revised: October 22, 2023. Accepted: November 4, 2023. Published: November 29, 2023.

1 Introduction

In this article, we are studying the semi-linear

parabolic equation (E)with a strictly positive initial

datum

ut(x, t) = (um)xx(x, t)− |x|qu−p(x, t),

(x, t)∈R×(0,∞),

u(x0, t0)>0, x0∈R, t0>0.

(1)

The case m= 1 has been widely studied in re-

cent years, due to its multiple applications in micro-

electro-mechanical system (MEMS), see for example

works, [1], [2], [3], [4], [5], [6], [7]. The main aim

of our paper is to generalize these results to the case

m > 1. For additional details on the context and

derivation of the MEMS model, we recommend the

reader to see papers, [8], [9], [10].

Physically, the function uis the distance between

an elastic membrane in the interior of a micro-electro-

mechanical system, and the fixed bottom plate. So

when ubecomes zero, i.e., the membrane touches the

bottom plate, the system then breaks down. This phe-

nomenon is called "Quenching" or "Touchdown" or

deactivating phenomenon.

Using standard parabolic equations techniques,

the problem constituted by equation (E)and a strictly

positive initial datum, admits a unique local solution.

Such a solution can be vanish in a finite time, and

therefore the term u−pwill not be defined. This is the

"Quenching" phenomenon in Mathematics, for more

details on this phenomenon, see the articles, [11],

[12], [13], [14]. In other words, there exists a finite

time Tso that:







u(x, t)>0in R×(0, T ),

lim

t→T−

inf inf

x∈Ru(x, t)= 0.(2)

A point x0is said to be a quenching point if

there exists a sequence of points (xn, tn)n∈Nsuch that

xn→x0, tn→Tand u(xn, tn)→0when ntends

to infinity.

Now, we will study the problem given by (1), using

"self-similar" solutions. These solutions are obtained

by looking for transformations that leave the equation

invariant see, [15], [16], [17], [18].

Afterwards, inspired by the works, [1], [19], [20],

the equation (E)possesses special self-similar solu-

tions of the form:

u(x, t) = (T−t)αU(y), y = (T−t)β|x|,(3)

where x∈R,0≤t≤T, and the two constants

αand βare given by:

α=q+ 2

2(p+ 1) −q(m−1)

and

β=−m+p

2(p+ 1) −q(m−1).

So if 2(p+ 1) −q(m−1) = 0,the function

Umust verify the following ordinary differential

equation:

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(Um)′′ (y) + βy U′(y) + αU (y)−yqU−p(y) = 0,

(4)

with y > 0.

Note that the equation (4) is not necessarily ver-

ified to the point y= 0. Also, if Uis a solution of

the equation (4) and it is well defined at the point

y= 0,then lim

t→T−u(0, t) = u(0, T −) = 0.So for

the point zero to be the only quenching point it is nec-

essary and sufficient that U(y)>0 for all y > 0and

that lim

y→+∞U(y)= 0. Moreover, in this case the de-

activating behavior is of the range (T−t)α.Indeed,

to obtain information on the deactivating behavior of

the solution of the parabolic equation (E), we need to

study the ordinary differential equation (4).

The other sections of this article are divided as fol-

lows. In Section 2, we provide a result on the exis-

tence of global positive solutions to (4) using a fixed-

point theorem and an associated energy functional,

where αand βare two reals verifying the following

condition:

α > 0, β < 0,and Γ := mα

β(1 + mα

β)>0.(5)

Then we assume that:

0≤p−m

m< q < 2(p+ 1)

m−1.(6)

In Section 3, we study the asymptotic behavior of

solutions of equation (4) near the origin. First, we use

the monotonicity argument of the transformed solu-

tion using the associated energy function. Then, we

derive a result on the asymptotic behavior.

Finally, in Section 4, we summarize what we have

been able to prove about this equation in the previous

sections, and also we give some ideas that we will

develop in our future work.

2 Existence of global solutions

In this section, we use a fixed-point theorem and an

energy functional to demonstrate the existence of

global solutions of equation (4).

Note that if l= Γ−1/(m+p), then the function:

U0(y) = Γ−1/(m+p)y−α/βfor y > 0,(7)

is a particular solution of equation (4) on [0,+∞]and

satisfies U0(0) = 0.

For this reason, we will restrict our study to the

case 0< l = Γ−1/(m+p)and we will be interested in

the solutions Uof equation (4) which satisfy U(0) =

0and behave near to infinity as l y −α/βwhere lis a

strictly positive constant. More precisely, we need to

study the following problem:

(P)









(Um)′′(y) + βyU′(y) + αU(y)−yqU−p(y) = 0,

for y > 0,

U(0) = 0 and U′(0) = 0,

lim

y→+∞yα/βU(y) = l > 0.

It must be said that the equation (4) is not veri-

fied at the point y= 0. We will prove that there is a

positive solution of problem (P)that vanishes, and its

derivative at the point y= 0. Due to this singular con-

dition, the standard ODE theory cannot be applied.

Theorem 2.1. Assume that 0< l = Γ−1/(m+p).

Then the problem (P)has a positive solution Ude-

fined on [0,+∞]and having the following asymptotic

behavior near to +∞:

U(y) = ly−α/β+1

βµf(l)y−µ−α/β+O(y−2µ−α/β),

as y→+∞.

(8)

where

f(y) = Γym−y−p,for all y > 0,(9)

and

µ= 2 + α(m−1)

β>0.(10)

The proof of this theorem is divided into three

steps. The first step concerns the existence of a so-

lution near to infinity. The second step involves to

extend the solution on [0,+∞],and the last step con-

sists to prove that the solution U(y)>0for all y > 0.

Proof. We will proceed in three steps.

Step1: There exists a constant M > 0,such that the

problem (P)admits a solution on [M, +∞[.

We Set

V(x) = ymα/βUm(y),(11)

where y=ex, y> 0 and x∈R.

Hence the function Vverifies for all x∈Rthe fol-

lowing equation:

V′′(x)−aV ′(x) + β

meµxV1−m

m(x)V′(x)+

f(V1/m)(x) = 0,(12)

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where

a= 1 + 2mα

β<0.(13)

Note that f(y)can only vanish at the point

y= Γ−1/(m+p),therefore Γ−m/(m+p)is the

unique constant that is a solution to the equation

(12).

Now we will express the equation (12) as the sys-

tem below:

dV (x)

dx =W(x),

dW (x)

dx = (a−β

meµxV1−m

m(x))W(x)−f(V1/m(x)).

Because of the term eµx which poses a problem near

infinity, we will introduce a new derivation variable,

t=eµx which transforms the system above to:

dV (t)

dt =W(t)

µt ,(14)

dW (t)

dt = ( a

µt−β

mV1−m

m(t))W(t)−f(V1/m)(t)

µt .

(15)

Hence

(e−A(t)W(t))′=−e−A(t)f(V1/m)(t)

µt ,(16)

where A(t)is a primitive of a

µt −β

mV1−m

m(t).

Assuming that lim

t→+∞W(t)=0and integrating

(16) over the interval (t, ∞), we obtain:

W(t) = 1

µeA(t)+∞

se−A(s)f(V1/m)(s)ds.

(17)

From the expression of the function A, the integral

(17) becomes:

W(t) = 1

µ+∞

s(t

s)a/µf(V1/m)(s)×(18)

exp(−t

V1−m

m(τ)dτ)ds.

Note that the condition at infinity of Uin the prob-

lem (P)requires the function Vto verify:

lim

t→+∞V(t) = lm.(19)

Hence

V(t) = lm−+∞

W(s)

µs ds. (20)

In what follows, we'll prove the existence of a func-

tion Wnear infinity that verifies (18) and (20) and

the limit:

lim

t→+∞tW(t) + mf(l)

βt lm−1= 0.

For that, we define on the Banach space:

X=φ∈ C0([M, +∞[); sup |t φ(t)|

t∈[M,+∞[

≤K,

the functional H:∀t∈[M, +∞[and ∀φ∈X,

H(φ)(t) = t

µ+∞

s(t

s)a/µf(Ψ(φ)(s))×

exp(−t

Ψ1−m(φ)(τ)dτ)ds +mf(l)

βlm−1,(21)

where

Ψ(φ)(t) = lm−+∞

φ(s)

µs2ds+mf(l)

µβt lm−11/m

(22)

By cutting the integral of the formula (21) into two

parts: the main part using a Taylor Expansion around

l(with a remainder term), we obtain estimations on

each part, then we prove that His a contraction of X

in Xfor certain constants.

In this case, according to a fixed-point argument,

there will exist a function φ∈Xsuch that:

H(φ)(t) = φ(t),for all t≥M.

So the function t→φ(t)

t−mf(l)

βt lm−1

is a solution of the equation (18) in the set

[M, +∞[.Consequently the function:

V(t) = lm+mf(l)

µβt lm−1−

+∞



φ(s)

µs2ds, ∀t≥M

(23)

is a solution of the equation (12).Or

+∞

φ(s)

µs2ds

≤K+∞

µs3ds

4µt2=O(1

t2)as t→ ∞.

(24)

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Hence, for tsufficiently large

V1/m(t) = l+f(l)

µβt +O(1

t2).(25)

In consequence, the function:

U(y) = y−α/βV1/m(t),(26)

with t=eµx is a solution to the problem (P)on

[M, +∞[.Moreover replacing V1/m(t)with its ex-

pression (25) in (26), we obtain:

U(y) = ly−α/β+f(l)

µβeµx y−α/β+O(y−α/β

(eµx)2),

as y→ ∞.

Using the fact that y=exfrom the expression (11),

we obtain:

U(y) = ly−α/β+1

µβ f(l)y−µ−α/β+O(y−2µ−α/β),

as y→ ∞.

We've achieved the desired result, so the first step

has been accomplished.

Step2: Ucan be extended to [0,+∞].

Let Vbe the function defined by the relation (11)

on an interval of the type [X, +∞[(with X > 0).

Using a symmetrization, we put:

ω(x) = V(−x),∀x∈]−∞,−X](27)

Thus ωverifies on ]−∞,−X]the following equa-

tion:

ω′′(x) + aω′(x)−β

me−µxω1−m

m(x)ω′(x)+

f(ω1/m) = 0.(28)

We consider the energy functional:

E(ω)(x) = 1

2ω′2(x) + F(ω(x)),(29)

where

F(ω(x)) = ω(x)

Γm

f(s1/m)ds (30)

=ω(x)

Γm

(µs −s−p/m)ds > 0.(31)

Hence

E′(ω)(x) = −a+β

me−µxω1−m

m(x)ω′2(x).

Since β < 0and a < 0, then

E′(ω)(x)<−2a E(ω)(x)for all x∈]−∞,−X].

Therefore the energy E(ω)is finite for any finite

xand increases as an exponential, so ωcan be

extended to x= +∞. In addition, we have by (19),

lim

x→−∞ω(x) = lm>0. Which completes the proof

of the second step .

Step3: U(y)>0for all y > 0.

From the expression of the energy functional E,

the integral F(ω)(x)exists, and as p≥mthen

necessarily ω(x)>0, hence U(y)>0for all y > 0.

Finally, we deduce the existence of a positive so-

lution Uof the problem (P)defined on [0,+∞].

This completes the proof of Theorem 2.1.

3 Asymptotic behavior near the

origin

In this section, we propose to study the asymptotic

behavior of solutions of problem (P)near the origin,

which is the same as studying the asymptotic behav-

ior of solutions Vof equation (12) near to −∞.

Now, we introduce the following lemma.

Lemma 3.1. Let Vbe a monotone solution of equa-

tion (12). Then

(i) lim

x→−∞E(V)(x) = +∞.

where E(V)is given by the relation (29).

(ii) Vis decreasing near to (−∞),moreover

lim

x→−∞V(x) = +∞.

Proof. For (i), we suppose that lim

x→−∞E(V)(x)ex-

ists and it is finite, i.e. lim

x→−∞E(V)(x) = L≥

0.Then, since Vis monotone, lim

x→−∞V(x) :=

d∈[0,+∞].As Lis finite so lim

x→−∞F(V)(x)and

lim

x→−∞ |V′(x)|are finite, and from the expression (30)

we deduce that dis finite. But Vconverges, then

lim

x→−∞V′(x)=0.Considering that µ > 0and that

d= Γ−m/(m+p), as x→ −∞ in the equation (12),

then we obtain:

V′′(−∞) = −f(d1/m)= 0.

Which is impossible since V′converges. This contra-

diction gives that lim

x→−∞E(V)(x) = +∞.

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For (ii), assuming that lim

x→−∞V(x) = dis finite.

Then

+∞=lim

x→−∞E(V)(x)

=lim

x→−∞ 1

2V′2(x) + F(V)(x)

=lim

x→−∞F(V)(x) = F(d).

Hence necessarily d= 0.Otherwise, by integrating

the equation (12) over ]x, x1[,we obtain:

V′(x1)−V′(x)−a[V(x1)−V(x)]

+β

mx1

eµsV1−m

m(s)V′(s)ds

=−x1

f(V1/m)(s)ds.

Note that

mx1

eµsV1−m

m(s)V′(s)ds =

βeµx1V1

m(x1)−βeµxV1

m(x)−βµ x1

eµsV1

m(s)ds.

Since µ > 0, V is bounded and monotone then

V′(−∞) = 0 and thus the first member is bounded

whereas p≥mthe second member will not be

bounded, which is contradictory. In conclusion

lim

y→−∞V(y) = +∞and Vis decreasing near to

(−∞). The proof is complete.

The following theorem gives the asymptotic be-

havior of the solution Uof problem (P)near the ori-

gin which is equivalent to giving the asymptotic be-

havior of the solution Vof equation (12) near to −∞,

through the change of variable y=exand the ex-

pression of Vgiven by (11). Let us assume that Vis

monotone. Then we have the following result.

Theorem 3.1. Assume that 0< l = Γ−1/(m+p).Let

Ube a solution of problem (P)on [0,+∞[. Then

lim

y→0+

y U′(y)

U(y)∈ {0,1/m}.(32)

Proof. Using expression (11) and the change of vari-

able y=ex, then proving (32) is equivalent to prove:

lim

x→−∞

V′(x)

V(x)∈mα

β,1 + mα

β.

According to the previous Lemma, we have

lim

x→−∞

V′(x)

V(x)=Land lim

x→−∞V′(x) = +∞.

Dividing the equation (12) by V′, we obtain:

V′′ (y)

V′(y)−a+β

meµyV1−m

m(y)

+µV′(y)

V(y)−V−p/m(y)

V′(y)= 0.

Applying the Hopital's rule, we obtain:

L2−aL +µ= 0,

i.e.

L2−(1 + 2mα

β)L+mα

β(1 + mα

β) = 0.(33)

Resolving this equation, we obtain:

L=mα

βor eslse L= 1 + mα

β.

Hence

lim

y→0+

y(Um)′(y)

Um(y)∈ {0,1}.

Thus we proved the required result.

4 Conclusion

In the present paper, we have proven the global exis-

tence of positive solutions Uof problem (P). Under

some assumptions, These solutions behave like y−α/β

near +∞. We have given also their asymptotic behav-

ior near the origin.

The results obtained can be more developed by

finding the exact limit of y U ′(y)

U(y)at the origin and

also treating the case when the function yα/βU(y)is

not monotone. This will be part of our future work.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

Arij Bouzelmate and Abdelilah Gmira proposed

the subject of the article to their Ph.D. student

Fatima Sennouni. This paper is an extension

of the work carried out by Arij Bouzelmate and

Abdelilah Gmira. It brings together the tech-

niques of Nonlinear Analysis. All the results

were carried out by the three authors Arij Bouzel-

mate, Abdelilah Gmira and Fatima Sennouni.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

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