Blowup Analysis for a Semilinear Parabolic System with Nonlocal Boundary Condition

Volume 2009, Article ID 516390,14pages doi:10.1155/2009/516390

Research Article

Blowup Analysis for a Semilinear Parabolic System with Nonlocal Boundary Condition

Yulan Wang

and Zhaoyin Xiang

1School of Mathematics and Computer Engineering, Xihua University, Chengdu 610039, China

2School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu 610054, China

Correspondence should be addressed to Zhaoyin Xiang,[email protected] Received 23 July 2009; Accepted 26 October 2009

Recommended by Gary Lieberman

This paper deals with the properties of positive solutions to a semilinear parabolic system with nonlocal boundary condition. We first give the criteria for finite time blowup or global existence, which shows the important influence of nonlocal boundary. And then we establish the precise blowup rate estimate for small weighted nonlocal boundary.

Copyrightq2009 Y. Wang and Z. Xiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In this paper, we devote our attention to the singularity analysis of the following semilinear parabolic system:

ut−Δuv^p, vt−Δvu^q, x∈Ω, t >0 1.1 with nonlocal boundary condition

ux, t

Ωf x, y

u y, t

dy, vx, t

Ωg x, y

v y, t

dy, x∈∂Ω, t >0, 1.2

and initial data

ux,0 u0x, vx,0 v0x, x∈Ω, 1.3

where Ω ⊂ R^N is a bounded connected domain with smooth boundary ∂Ω, p and qare positive parameters. Most physical settings lead to the default assumption that the functions fx, y, gx, y defined forx ∈ ∂Ω, y ∈ Ω are nonnegative and continuous, and that the initial data u₀x, v₀x ∈ C¹Ω are nonnegative, which are mathematically convenient and currently followed throughout this paper. We also assume that u0, v₀ satisfies the compatibility condition on∂Ω, and thatfx,·/≡0 andgx,·/≡0 for anyx∈∂Ωfor the sake of the meaning of nonlocal boundary.

Over the past few years, a considerable effort has been devoted to studying the blowup properties of solutions to parabolic equations with local boundary conditions, say Dirichlet, Neumann, or Robin boundary condition, which can be used to describe heat propagation on the boundary of containersee the survey papers1,2 . For example, the system1.1and 1.3with homogeneous Dirichlet boundary condition

ux, t vx, t 0, x∈∂Ω, t >0 1.4

has been studied extensivelysee3–5 and references therein, and the following proposition was proved.

Proposition 1.1. iAll solutions are global ifpq ≤ 1, while there exist both global solutions and finite time blowup solutions depending on the size of initial data whenpq > 1 (See [4]).iiThe asymptotic behavior near the blowup time is characterized by

C⁻¹₁ ≤max

x∈Ω ux, tT −t^p1/pq−1≤C₁, C⁻¹₂ ≤max

x∈Ω vx, tT−t^q1/pq−1≤C₂ 1.5 for someC₁, C₂>0 (See [3,5]).

For the more parabolic problems related to the local boundary, we refer to the recent works6–9 and references therein.

On the other hand, there are a number of important phenomena modeled by parabolic equations coupled with nonlocal boundary condition of form1.2. In this case, the solution could be used to describe the entropy per volume of the material 10–12 . Over the past decades, some basic results such as the global existence and decay property have been obtained for the nonlocal boundary problem 1.1–1.3 in the case of scalar equationsee 13–16 . In particular, for the blowup solutionuof the single equation

u_t−Δuu^p, x∈Ω, t >0, ux, t

Ωf x, y

u y, t

dy, x∈∂Ω, t >0, ux,0 u₀x, x∈Ω,

1.6

under the assumption that

Ωfx, ydy 1, Seo15 established the following blowup rate estimate

p−1_−1/p−1

T−t^−1/p−1≤max

x∈Ω ux, t≤C1T−t^−1/γ−1 1.7

for anyγ ∈ 1, p. For the more nonlocal boundary problems, we also mention the recent works17–22 . In particular, Kong and Wang in 17 , by using some ideas of Souplet23 , obtained the blowup conditions and blowup profile of the following system:

ut Δu

Ωu^mx, tvⁿx, tdx, vt Δv

Ωu^px, tv^qx, tdx, x∈Ω, t >0 1.8

subject to nonlocal boundary1.2, and Zheng and Kong in22 gave the condition for global existence or nonexistence of solutions to the following similar system:

u_t Δuu^m

Ωvⁿ y, t

dy, v_t Δvv^q

Ωu^p y, t

dy, x∈Ω, t >0 1.9

with nonlocal boundary condition 1.2. The typical characterization of systems 1.8 and 1.9 is the complete couple of the nonlocal sources, which leads to the analysis of simultaneous blowup.

To our surprise, however, it seems that there is no work dealing with singularity analysis of the parabolic system1.1with nonlocal boundary condition1.2except for the single equation case, although this is a very classical model. Therefore, the basic motivation for the work under consideration was our desire to understand the role of weight function in the blowup properties of that nonlinear system. We first remark by the standard theory 4,13 that there exist local nonnegative classical solutions to this system.

Our main results read as follows.

Theorem 1.2. Suppose that 0< pq≤1. All solutions to1.1–1.3exist globally.

It follows fromTheorem 1.2andProposition 1.1ithat any weight perturbation on the boundary has no influence on the global existence whenpq≤1, while the following theorem shows that it plays an important role whenpq >1. In particular,Theorem 1.3iiis completely different from the case of the local boundary1.4 by comparing withProposition 1.1i.

Theorem 1.3. Suppose thatpq >1.

iFor any nonnegative fx, yandgx, y, solutions to1.1–1.3blow up in finite time provided that the initial data are large enough.

iiIf

Ωfx, ydy ≥1,

Ωgx, ydy ≥1 for anyx∈∂Ω, then any solutions to1.1–1.3 with positive initial data blow up in finite time.

iiiIf

Ωfx, ydy <1,

Ωgx, ydy <1 for anyx∈∂Ω, then solutions to1.1–1.3with small initial data exist globally in time.

Once we have characterized for which exponents and weights the solution to problem 1.1–1.3can or cannot blow up, we want to study the way the blowing up solutions behave

as approaching the blowup time. To this purpose, the first step usually consists in deriving a bound for the blowup rate. For this bound estimate, we will use the classical method initially proposed in Friedman and McLeod24 . The use of the maximum principle in that process forces us to give the following hypothesis technically.

HThere exists a constant 0< δ <1, such thatΔu0 1−δv^p₀ ≥0,Δv0 1−δu^q₀ ≥0.

However, it seems that such an assumption is necessary to obtain the estimates of type1.5 or1.10unless some additional restrictions on parametersp, qare imposedfor the related problem, we refer to the recent work of Matano and Merle25 .

Here to obtain the precise blowup rates, we shall devote to establishing some relationship between the two componentsuandvas our problem involves a system, but we encounter the typical difficulties arising from the integral boundary condition. The following theorem shows that we have partially succeeded in this precise blowup characterization.

Theorem 1.4. Suppose thatpq >1,p, q≥1,fx, y gx, y,

Ωfx, ydy ≤1,and assumption (H) holds. If the solutionu, vof1.1–1.3with positive initial datau0, v₀blows up in finite time T, then

C₁⁻¹≤max

x∈Ω ux, tT−t^p1/pq−1≤C₁, C₂⁻¹≤max

x∈Ω vx, tT−t^q1/pq−1≤C₂, 1.10

whereC₁, C₂are both positive constants.

Remark 1.5. Ifq pandu₀ v₀, thenTheorem 1.4implies that for the blowup solution of problem1.6, we have the following precise blowup rate estimate:

C⁻¹₁ T−t^−1/p−1≤max

x∈Ω ux, t≤C1T−t^−1/p−1, 1.11

which improves the estimate1.7. Moreover, we relax the restriction onf.

Remark 1.6. By comparing with Proposition 1.1ii,Theorem 1.4 could be explained as the small perturbation of homogeneous Dirichlet boundary, which leads to the appearance of blowup, does not influence the precise asymptotic behavior of solutions near the blowup time and the blowup rate exponentsp1/pq−1andq1/pq−1are just determined by the corresponding ODE systemu_t v^p, v_t u^q. Similar phenomena are also noticed in our previous work18 , where the single porous medium equation is studied.

The rest of this paper is organized as follows. Section 2 is devoted to some preliminaries, which include the comparison principle related to system 1.1–1.3. In Section 3, we will study the conditions for the solution to blow up and exist globally and hence prove Theorems1.2and1.3. Proof ofTheorem 1.4is given inSection 4.

2. Preliminaries

In this section, we give some basic preliminaries. For convenience, we denote Q_T Ω × 0, T, ST ∂Ω×0, T, QT Ω×0, T.We begin with the definition of the super- and subsolution of system1.1–1.3.

Definition 2.1. A pair of functionsu, v∈C^2,1QT

CQTis called a subsolution of1.1–1.3 if

u_t−Δu≤v^p, v_t−Δv≤u^q, x, t∈Q_T, ux, t≤

Ωf x, y

u y, t

dy, vx, t≤

Ωg x, y

v y, t

dy, x, t∈S_T, ux,0≤u₀x, vx,0≤v₀x, x∈Ω.

2.1

A supersolution is defined with each inequality reversed.

Lemma 2.2. Suppose thatc₁, c₂, f,andgare nonnegative functions. Ifw₁, w₂∈C^2,1QT CQT satisfy

w_1t−Δw1≥c₁x, tw2, w_2t−Δw2≥c₂x, tw1, x, t∈Q_T, w₁x, t≥

Ωf x, y

w₁ y, t

dy, w₂x, t≥

Ωg x, y

w₂ y, t

dy, x, t∈S_T, w₁x,0>0, w₂x,0>0, x∈Ω,

2.2

thenw₁, w₂>0 onQ_T.

Proof. Set t₁ : sup{t ∈ 0, T : w_ix, t > 0, i 1,2}. Since w₁x,0, w2x,0 > 0, by continuity, there existsδ > 0 such thatw1x, t, w2x, t > 0 for allx, t ∈ Ω×0, δ. Thus t₁∈δ, T .

We claim that t1 < T will lead to a contradiction. Indeed, t1 < T suggests that w₁x1, t₁ 0 or w₂x1, t₁ 0 for somex₁ ∈ Ω. Without loss of generality, we suppose thatw₁x1, t₁ 0inf_Q

t1w₁. Ifx₁∈Ω, we first notice that

w1t−Δw1≥c1w2≥0, x, t∈Ω×0, t1 . 2.3

In addition, it is clear thatw1≥0 on boundary∂Ωand at the initial statet0. Then it follows from the strong maximum principle thatw₁≡0 inQ_t1, which contradicts tow₁x,0>0.

Ifx₁∈∂Ω, we shall have a contradiction:

0w1x1, t1≥

Ωf x1, y

w1

y, t1

dy >0. 2.4

In the last inequality, we have used the facts thatfx,·/≡0 for anyx∈∂Ωandw1y, t1>0 for anyy∈Ω, which is a direct result of the previous case.

Therefore, the claim is true and thust1T, which implies thatw1, w2>0 onQT. Remark 2.3. If

Ωfx, ydy ≤ 1 and

Ωgx, ydy ≤ 1 for anyx ∈ ∂ΩinLemma 2.2, we can obtainw1, w₂≥0,0inQ_Tunder the assumption thatw1x,0, w2x,0≥0,0forx∈Ω.

Indeed, for any >0, we can conclude thatw1x, t e^t, w₂x, t e^t>0,0inQ_Tas the proof ofLemma 2.2. Then the desired result follows from the limit procedure → 0.

From the above lemma, we can obtain the following comparison principle by the standard argument.

Proposition 2.4. Let u, v) and u, v be a subsolution and supersolution of 1.1–1.3 in Q_T, respectively. Ifux,0, vx,0<ux,0, vx,0forx∈Ω, thenu, v<u, vinQT.

3. Global Existence and Blowup in Finite Time

In this section, we will use the super and subsolution technique to get the global existence or finite time blowup of the solution to1.1–1.3.

Proof ofTheorem 1.2. As 0< pq≤1, there exists, l∈0,1such that 1

p ≥ l s, 1

q≥ s

l. 3.1

Then we let φx, y x ∈ ∂Ω, y ∈ Ω be a continuous function satisfying φx, y ≥ max{fx, y, gx, y}and set

ax

Ωφx, ydy _1−s/s

, bx

Ωφx, ydy _1−l/l

, x∈∂Ω. 3.2

We consider the following auxiliary problem:

w_t Δwkw, x∈Ω, t >0,

wx, t ax bx 1

Ω

φ

x, y 1

|Ω|

w

y, t dy

, x∈∂Ω, wx,0 1u^1/s₀ x v^1/l₀ x, t >0,

3.3

where|Ω|is the measure ofΩandk:1/s1/l. It follows from13, Theorem 4.2 thatwx, t exists globally, and indeedwx, t>1,x, t∈Ω×0,∞ see13, Theorem 2.1 .

Our intention is to show thatu, v: w^s, w^lis a global supersolution of1.1–1.3.

Indeed, a direct computation yields

u_tsw^s−1Δwkw ≥ sw^s−1Δww^s,

Δusw^s−1Δwss−1w^s−2|∇w|² ≤ sw^s−1Δw, 3.4

and thus

u_t−Δu≥w^s w^l_s/l

≥v^p. 3.5

Here we have used the conclusionw > 1 and inequality3.1. We still have to consider the boundary and initial conditions. Whenx∈∂Ω, in view of H ¨older’s inequality, we have

ux, t≥ax^s

Ωφx, ywy, tdy _s

Ωφx, ydy _1−s

Ωφx, ywy, tdy s

≥

Ωfx, ydy _1−s

Ωfx, ywy, tdy _s

Ω f^1−s

x, y^1/1−s dy

_1−s

Ωf^sx, yw^sy, t^1/sdy s

≥

Ωf^1−s x, y

f x, y

w y, t_s

dy

Ωf x, y

w^s y, t

dy

Ωf x, y

u y, t

dy.

3.6

Similarly, we have also forvthat

v_t−Δv≥u^q, x∈Ω, t >0, v≥

Ωg x, y

v y, t

dy, x∈∂Ω, t >0. 3.7

It is clear that u0x < ux,0 and v0x < vx,0. Therefore, we get u, v is a global supersolution of 1.1–1.3 and hence the solution to 1.1–1.3exists globally by Proposition 2.4.

Proof ofTheorem 1.3. i Let u, v be the solution to the homogeneous Dirichlet boundary problem1.1,1.4, and1.3. Then it is well known that for sufficiently large initial data the

solutionu, vblows up in finite time whenpq >1see4 . On the other hand, it is obvious thatu, vis a subsolution of problem1.1–1.3. Henceforth, the solution of1.1–1.3with large initial data blows up in finite time provided thatpq >1.

iiWe consider the ODE system:

ft h^pt, ht f^qt, t >0,

f0 a >0, h0 b >0, 3.8

wherea 1/2min_Ωu0x, b 1/2min_Ωv0x. Thenpq >1 implies thatf, hblows up in finite timeT see26 . Under the assumption that

Ωfx, ydy≥1 and

Ωgx, ydy≥1 for anyx∈ ∂Ω,f, his a subsolution of problem1.1–1.3. Therefore, byProposition 2.4, we see that the solutionu, vof problem1.1–1.3satisfiesu, v≥f, hand thenu, vblows up in finite time.

iiiLetψ₁xbe the positive solution of the linear elliptic problem:

−Δψ1x ₀, x∈Ω, ψ₁x

Ωf x, y

dy, x∈∂Ω, 3.9

and letψ2xbe the positive solution of the linear elliptic problem:

−Δψ2x 0, x∈Ω, ψ2x

Ωg x, y

dy, x∈∂Ω, 3.10

whereois a positive constant such that 0≤ψix≤1i1,2. We remark that

Ωfx, ydy <

1 and

Ωgx, ydy <1 ensure the existence of such₀. Let

ux aψ₁x, vx bψ₂x, 3.11

wherea₀^p1/pq−1, b₀^q1/pq−1. We now show thatu, vis a supsolution of problem 1.1–1.3for small initial datau0, v₀. Indeed, it follows fromb₀ a^q, a₀ b^pthat, for x∈Ω,

u_t−Δua₀b^p≥v^p, v_t−Δvb₀a^q≥u^q. 3.12 Whenx∈∂Ω,

ux a

Ωf x, y

dy≥

Ωf x, y

aψ₁ y

dy

Ωf x, y

uxdy,

vx b

Ωg x, y

dy≥

Ωg x, y

bψ₂ y

dy

Ωg x, y

vxdy.

3.13

Here we usedψix≤1i1,2. The above inequalities show thatu, vis a supsolution of problem1.1–1.3wheneveru₀x< aψ₁x, v0x< bψ₂x. Therefore, system1.1–1.3 has global solutions ifpq >1 and

Ωfx, ydy <1,

Ωgx, ydy <1 for anyx∈∂Ω.

4. Blowup Rate Estimate

In this section, we derive the precise blowup rate estimate. To this end, we first establish a partial relationship between the solution componentsux, tandvx, t, which will be very useful in the subsequent analysis. For definiteness, we may assumep≥q≥1. Ifq > p, we can proceed in the same way by changing the role ofuandvand then obtain the corresponding conclusion.

Lemma 4.1. Ifp≥q,fx, y gx, yand

Ωfx, ydy≤1 for anyx∈∂Ω, there exists a positive constantC0 such that the solution u, v of problem 1.1–1.3 with positive initial datau0, v0 satisfies

ux, t≥C₀v^p1/q1x, t, x, t∈Ω×0, T. 4.1

Proof. LetJx, t ux, t−C₀v^p1/q1x, t, whereC₀ is a positive constant to be chosen.

Forx, t∈Ω×0, T, a series of calculations show that

J_t−ΔJu_t−C₀p1

q1v^p−q/q1v_t−ΔuC₀

p1 p−q

q1₂ |∇v|²C₀p1

q1v^p−q/q1Δv

≥v^p−C₀p1

q1v^p−q/q1u^q v^p−q/q1

v^qp1/q1−C0

p1 q1u^q

v^p−q/q1 1

C₀^qu−J^q−C0

p1 q1u^q

.

4.2

If we chooseC₀such that 1/C₀^q≥C₀p1/q1, we have

Jt−ΔJv^p−q/q1θu, vJ≥0, 4.3

whereθu, vis a function ofuand vand lies betweenC₀p1/q1u−JandC₀p 1/q1u.

Whenx, t∈∂Ω×0, T, on the other hand, we have

Jx, t

Ωf x, y

u y, t

dy−C0

Ωfx, yvy, tdy

_p1/q1

. 4.4

DenoteHx :

Ωfx, ydy ≥ 0,x ∈ ∂Ω. Sincefx,·/≡0 for anyx ∈ ∂Ω,Hx > 0. It follows from Jensen’s inequality,Hx≤1,andp1/q1≥1 that

Ωf x, y

v^p1/q1 y, t

dy−

Ωfx, yvy, tdy

_p1/q1

≥Hx

Ωfx, yvy, t dy Hx

p1/q1

−

Ωfx, yvy, tdy

_p1/q1

≥0,

4.5

which implies that

Jx, t≥

Ωf x, y

u y, t

dy−C₀

Ωf x, y

v^p1/q1 y, t

dy

Ωf x, y

J y, t

dy, x∈∂Ω.

4.6

For the initial condition, we have

Jx,0 u₀x−C₀v₀^p1/q1x≥0, x∈Ω, 4.7

provided thatC0≤inf_x∈Ω{u0xv^−p1/q1₀ x}.

Summarily, if we takeC0 min{inf_x∈Ωu0xv^−p1/q1₀ x,q1/p1^1/q1}, then it follows from Theorem 2.1 in13 thatJx, t≥0, that is,

ux, t≥C0v^p1/q1x, t, x, t∈Ω×0, T, 4.8

which is desired.

Using this lemma, we could establish our blowup rate estimate. To derive our conclusion, we shall use some ideas of3 .

Proof ofTheorem 1.4. For simplicity, we introduceα p1/pq−1, β q1/pq−1.

LetFx, t u_t−δv^pandGx, t v_t−δu^q. A direct computation yields

Ft−ΔF≥pv^p−1G, Gt−ΔG≥qu^q−1F, x∈Ω, 0< t < T. 4.9

Forx, t∈∂Ω×0, T, we have from the boundary conditions that Fx, t ut−δv^p

Ωf x, y

u_t y, t

dy−δ

Ωfx, yvy, tdy _p

Ωf x, y

Fδv^p y, t

dy−δ

Ωfx, yvy, tdy _p

Ωf x, y

F y, t

dyδ

Ωf x, y

v^p y, t

dy−

Ωfx, yvy, tdy _p

.

4.10

It follows from

Ωfx, ydy≤1 and Jensen’s inequality that the difference in the last brace is nonnegative and thus

Fx, t≥

Ωf x, y

F y, t

dy, x∈∂Ω. 4.11

By similar arguments, we have

Gx, t≥

Ωf x, y

G y, t

dy, x, t∈∂Ω×0, T. 4.12

On the other hand, the hypothesisHimplies that

Fx,0≥0, Gx,0≥0 x∈Ω. 4.13

Hence, from4.9–4.13and the comparison principleseeRemark 2.3, we get

Fx, t≥0, Gx, t≥0, x, t∈Ω×0, T. 4.14

That is,

ut≥δv^p, vt≥δu^q, x, t∈Ω×0, T. 4.15 LetUt max_x∈Ωux, t, Vt max_x∈Ωvx, t.ThenUtand Vtare Lipschitz continuous and thus are differential almost everywheresee e.g.,24 . Moreover, we have from equations1.1that

Ut≤V^pt, Vt≤U^qt, a.e. t∈0, T. 4.16 We claim that

Vt≥kV^qp1/q1t, a.e. t∈0, T 4.17

for some positive constant k. Indeed, if we letxt, tbe the points at whichv attains its maximum, then relation4.1means that

uxt, t≥C₀V^p1/q1t, t∈0, T. 4.18

At any pointt1of differentiability ofVt, ift2> t1, Vt2−Vt1

t2−t1 ≥ vxt1, t2−vxt1, t1

t2−t1 v_txt1, t1 o1, ast₂−→t₁. 4.19 From4.15,4.18, and4.19, we can confirm our claim4.17.

Integrating4.17ont, Tyields

VtT−t^β≤k, t∈0, T, 4.20

which gives the upper estimate forVt. Namely, there exists a constantc4>0 such that Vt≤c₄T−t^−β, t∈0, T. 4.21

Then by4.16and4.21, we get

Ut≤V^pt≤c^p₄T−t^−pβ, t∈0, T. 4.22

Integrating this equality from 0 tot, we obtain

Ut≤c2T−t^−α, t∈0, T 4.23

for some positive constantc2. Thus we have established the upper estimates forUt.

To obtain the lower estimate forUt, we notice that4.16and4.18lead to

Ut≤k₂U^pq1/p1t 4.24

for a constantk₂. Integrating above equality ont, T, we see there exists a positive constant c₁such that

Ut≥c1T−t^−α, t∈0, T. 4.25

Finally, we give the lower estimate for Vt. Indeed, using the relationship 4.16, 4.23 and 4.25, we could prove that VtT −t^β is bounded from below; that is, there exists a positive constantc₃such that

Vt≥c3T−t^−β. 4.26

To see this, our approach is based on the contradiction arguments. Assume that there would exist two sequences{tn} ⊂0, Twitht_n → T⁻and{dn}withd_n → 0 asn → ∞such that

Vtn≤dnT−tn^−β, n1,2,3, . . . . 4.27 Then we could choose a corresponding sequence{sn}such thatt_n−s_nkT−t_n, wherekis a positive constant to be determined later. AsUt≤V^pt, we have

Utn≤Usn

_tn

sn

V^pτdτ. 4.28

From4.23and4.27, we obtain

Utn≤c₂T−s_n^−αV^ptntn−s_n

≤c2T−sn^−αd^p_nT−tn^−βptn−sn

≤c₂k1^−αT−t_n^−αkd^p_nT−t_n^−α.

4.29

Choosingksuch thatc2k1^−α≤c1/2, one can get Utn≤ c₁

2T−t_n^−αkd_n^pT−t_n^−α c₁

2 kd^p_n

T−t_n^−α, 4.30

which would contradict to4.25asnis large enough sinced_n → 0 asn → ∞.

Acknowledgments

The authors are very grateful to the anonymous referees for their careful reading and useful suggestions, which greatly improved the presentation of the paper. This work is supported in part by Natural Science Foundation Project of CQ CSTC2007BB2450, China Postdoctoral Science Foundation, the Key Scientific Research Foundation of Xihua University, and Youth Foundation of Science and Technology of UESTC.

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