有關反應擴散方程系統的爆破臨界指數

行政院國家科學委員會專題研究計畫 成果報告

有關反應擴散方程系統的爆破臨界指數

中 華 民 國 93 年 11 月 4 日

中文摘要 本計畫中，我們研究以下系統的爆破臨界指數 , 0 , , , 0 , ,               t x bv u v v t x au v u u s q n t r p m t 其中m,n1。我們找到一個爆破臨界指數，使得當pqmax{m,r}max{n,s}時， 所有的解均為全域解(global solution); 當pqmax{m,r}max{n,s}時，則存在爆破 解(blow-up solution)。當pqmax{m,r}max{n,s}時，解是否全域則視m,n,r,s而 定。

Abstract

In this project, we study the blow-up critical exponent for the following system.

, 0 , , , 0 , ,               t x bv u v v t x au v u u s q n t r p m t

where m,n1 . We obtain that all solutions exist globally if

} , max{ } , max{m r n s

pq ; while there exist blowing up solutions if

} , max{ } , max{m r n s

pq . For the critical case pqmax{m,r}max{n,s} , the existence or nonexistence of global solutions depends on the relation between the exponent of m,n,r,s.

Critical exponent for a system of slow diffusion equations

with both reaction and absorption terms

Sheng-Chen Fu

Department of Mathematical Sciences, National Chengchi University,

64, S-2 Zhi-nan Road, Taipei 116, Taiwan

1 Introduction

In this paper, we consider the following degenerate parabolic system

ut= 4um+ vp− aur, x ∈ Ω, t > 0, (1.1)

vt= 4vn+ uq− bvs, x ∈ Ω, t > 0, (1.2)

with boundary condition

u = v = 0, x ∈ ∂Ω, t > 0, (1.3) and initial condition

u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Ω, (1.4)

where Ω is a bounded domain in RN _{with a smooth boundary ∂Ω, m, n > 1, p, q, r, s, a, b}

are positive constants. It is well-known that when m, n > 1 the problem (1.1)-(1.4) admits solutions only in some weak senses. Since we are interested only in nonnegative solutions, we therefore assume that the initial functions u0 and v0 are nonnegative and u0, v0 ∈ L∞(Ω).

The system (1.1)-(1.4) can be used to model as the cooperative reaction of two species in an ecological system. The presence of the u-population species encourages the growth of the v-population species but reduces its own growth and vice versa. The choice of m, n > 1 describes the density dependent diffusion phenomenon.

We say that the solution (u, v) of the problem (1.1)-(1.4) blows up in finite time if there exists a finite time T > 0 such that the solution is defined in (0, T ), and

lim sup

t%T

{||u(·, t)||∞+ ||v(·, t)||∞} = +∞.

∗_{November 3, 2004}

The motivation of this study is from [1] in which Bedjaoui and Souplet studied (1.1)-(1.4) for the case m = n = 1. By comparing with some suitable supersolutions and subsolutions, they obtained optimal conditions on the exponents p, q, r, s for the existence of blowing up solutions and the global boundedness of all solutions, respectively.

The main purpose of this paper is to study the existence or nonexistence of global so-lutions for the problem (1.1)-(1.4) when m, n > 1. Indeed, we show that all soso-lutions are globally bounded if pq < max{m, r} max{n, s}; while there are finite time blowing up solu-tions if pq > max{m, r} max{n, s} and the initial data are sufficiently large. For the critical case pq = max{m, r} max{n, s}, the existence or nonexistence of global solutions depends on the relation between the exponents m, n, r, s, and also the range of the parameters a, b. We mention here that the problem (1.1)-(1.4) with a = b = 0 has been investigated by many authors, see for example and the reference cited therein. Also, the problem for a slow diffusion equation with absorption of the type ut = 4um− aur has been studied extensively

(see, for example, [2-8] and the reference cited therein).

This paper is organized as follows. We study the case pq 6= max{m, r} max{n, s} in §2 and the critical case pq = max{m, r} max{n, s} in §3.

2 The case pq 6= max{m, r} max{n, s}

For convenience, we denote Ω × (0, T ) and ∂Ω × (0, T ) by QT and ST, respectively. Based

on [9], we use the following definition of (weak) solution throughout this paper.

Definition 2.1 A pair of functions (u, v) is called a solution of (1.1)-(1.4) in QT, 0 < T <

∞, if

(i) u, v ∈ L∞_(Q T)

T

C([0, T ]; L2_{(Ω)) and u}m_{, v}n _{∈ L}2_{(0, T ; H}1_(Ω)),

(ii) u and v satisfy the identities

Z Ωϕ(x, T )u(x, T )dx + Z Z QT 5ϕ · 5um_dxdt = Z Z QT [ϕ(vp_{− au}r_{) + ϕ} tu]dxdt + Z Ωϕ(x, 0)u0(x)dx, Z Ωϕ(x, T )v(x, T )dx + Z Z QT 5ϕ · 5vn_dxdt = Z Z QT [ϕ(uq_{− bv}s_{) + ϕ} tv]dxdt + Z Ωϕ(x, 0)v0(x)dx, for any ϕ ∈ C1_(Q T) such that ϕ = 0 on ST. (iii) um _{= 0, v}n _{= 0 on S}

T in the trace sense.

Definition 2.2 A pair of functions (u, v) is called a (weak) supersolution of (1.1)-(1.3) in

QT, 0 < T < ∞, with initial data (u0, v0), if

(i) u, v ∈ L∞_(Q T)

T

C([0, T ]; L2_{(Ω)) and u}m_{, v}n _{∈ L}2_{(0, T ; H}1_(Ω)),

(ii) u and v satisfy the inequalities

Z Ωϕ(x, T )u(x, T )dx + Z Z QT 5ϕ · 5umdxdt ≥ Z Z QT [ϕ(vp_{− au}r_{) + ϕ} tu]dxdt + Z Ωϕ(x, 0)u0(x)dx, Z Ωϕ(x, T )v(x, T )dx + Z Z QT 5ϕ · 5vndxdt ≥ Z Z QT [ϕ(uq− bvs) + ϕtv]dxdt + Z Ωϕ(x, 0)v0(x)dx,

for any nonnegative function ϕ ∈ C1_(Q

T) such that ϕ = 0 on ST.

(iii) um _{≥ 0, v}n_{≥ 0 on S}

T in the trace sense.

A (weak) subsolution is defined by replacing ≥ in (ii) and (iii) by ≤.

We shall use the following comparison principle to prove the existence or nonexistence of global solutions. The proof can be found in [9] and we omit it.

Lemma 2.1 Let (u, v) and (u, v) be a supersolution and a subsolution of (1.1)-(1.3) in QT,

T > 0, with initial data satisfying u(x, 0) ≥ u(x, 0) and v(x, 0) ≥ v(x, 0). Then u ≥ u and v ≥ v in QT.

We remark here that the global existence of solutions for (1.1)-(1.4) when p < m and

q < n has been obtained in [9]. Using the super-sub-solution method, we can also recover

this known results. Indeed, we have the following theorem for the subcritical case.

Theorem 2.2 Suppose that pq < max{m, r} max{n, s}. Then all solutions of (1.1)-(1.4)

are globally bounded.

Proof. Take R > 0 such that Ω ⊂ B(0; R). Let (u, v) = (C1e−L1|x|

2

, C2e−L2|x|

2

), where L1,

L2, C1, C2 are positive constants satisfying

L1 ≤ N/(4mR2), L2 ≤ N/(4nR2), C1 ≥ eL1R 2 |u0|∞, C2 ≥ eL2R 2 |v0|∞, and C₂pqeq|rL1−pL2|R2 _{≤ a}q_Cqr 1 ≤ aqbre−r|qL1−sL2|R 2 C₂rs, if pq < rs, C₂pqeq|mL1−pL2|R2 _{≤ (NmL} 1)qC1mq ≤ (NmL1)q(NnL2)me−m|qL1−nL2|R 2 Cmn 2 , if pq < mn, C₂pqeq|rL1−pL2|R2 _{≤ a}q_Cqr 1 ≤ aq(NnL2)re−r|qL1−nL2|R 2 Cnr 2 , if pq < nr, C₂pqeq|mL1−pL2|R2 _{≤ (NmL} 1)qC1mq ≤ bm(NmL1)qe−m|qL1−sL2|R 2 C₂ms, if pq < ms.

It is easy to check that (u, v) is a supersolution of (1.1)-(1.3) with u(x, 0) ≥ u0(x) and

v(x, 0) ≥ v0(x). Thus, by Lemma 2.1, we obtain that u ≤ u and v ≤ v as long as the solution

(u, v) exists. Therefore, (u, v) is globally bounded. This completes the proof.

Borrowing an idea from [1] and [10], we will construct a self-similar subsolution to prove the existence of blowing up solutions.

Theorem 2.3 Suppose that pq > max{m, r} max{n, s}. Then the solution of the problem

(1.1)-(1.4) blows up in finite time for initial data large enough.

Proof. We first consider the case m ≤ r and n ≤ s. Without loss of generality, we may

assume that 0 ∈ Ω. Since pq > rs, we have either s/q < (p+1)/(q+1) or r/p < (q+1)/(p+1). Therefore, we may without loss of generality assume that s/q < (p + 1)/(q + 1) (otherwise, we exchange the roles of u and v).

We choose λ and β such that

s q < λ < min{ p + 1 q + 1, p r} 1 λq − 1 < β < 1 s − 1.

Set α = λβ. Then it is easy to check that

βp > α + 1, βp > αr ≥ αm, αq > β + 1 > βs ≥ βn. (2.1) Pick a positive number l such that

l < 1

2min{βp − αm, β + 1 − βn, αq − βn}. (2.2) We seek a subsolution of the following form

u(x, t) = (T − t)−α_U Ã |x| (T − t)l ! , v(x, t) = (T − t)−β_V Ã |x| (T − t)l ! ,

where U(y) = (A2_−y2₎1/m

+ , V (y) = (A2−K−2y2)1/n+ , K ∈ (1,

q

(βn + 2l)/(βn)) is a constant, and A, T > 0 are constants to be determined later. To show (u, v) is a subsolution of (1.1)-(1.3), it suffices to show that

(T − t)−(α+1)_{{αU(y) + lyU}0_{(y)} + a(T − t)}−αr_Ur_(y)

−(T − t)−(αm+2l)[(Um)00(y) + (N − 1)/y(Um)0(y)]

≤ (T − t)−βp_Vp_{(y) (2.3)}

and

(T − t)−(β+1){βV (y) + lyV0(y)} + b(T − t)−βsVs(y)

−(T − t)−(βn+2l)_[(Vn₎00_{(y) + (N − 1)/y(V}n₎0_(y)]

≤ (T − t)−αq_Uq_{(y) (2.4)}

hold pointwisely for y > 0, y 6= A, KA.

For y > A, it is clear that (2.3) holds. For 0 < y < A, (2.3) is equivalent to (T − t)−(α+1) ( α(A2− y2)1/m− 2l my 2_(A2_{− y}2₎1/m−1 ) +2N(T − t)−(αm+2l)_{+ a(T − t)}−αr_(A2_{− y}2₎r/m ≤ (T − t)−βp_(A2_{− y}2_/K2₎p/n_{. (2.5)}

Since A2 _{− y}2_/K2 _{≥ A}2_{(1 − 1/K}2_{) > 0, it follows from (2.1) and (2.2) that (2.5) holds}

provided that T is sufficiently small.

For y > KA, it is clear that (2.4) holds. Let θ ∈ (q(βn)/(βn + 2l)K, 1) be fixed. For

θA < y < KA, (2.4) holds provided that

( β(A2_{− y}2_/K2_{) −} 2l n(y 2_/K2₎ ) + 2N(T − t)β+1−βn−2l_(A2_{− y}2_/K2₎1−1/n +b(T − t)β+1−βs_(A2_{− y}2_/K2₎1−1/n+s/n _{≤ 0. (2.6)}

Since β(A2 _{− y}2_/K2_{) − 2l/n(y}2_/K2_{) ≤ A}2_{(β − θ}2_/K2_{(β + 2l/n)) < 0, it follows from (2.1)}

and (2.2) that (2.6) holds provided that T is sufficiently small. For 0 < y < θA, (2.4) is equivalent to (T − t)−(β+1) ( β(A2 _{− y}2_/K2₎1/n₋ 2l n(y 2_/K2_)(A2_{− y}2_/K2₎1/n−1 ) +2N(T − t)−βn−2l _{+ b(T − t)}−βs_(A2_{− y}2_/K2₎s/n ≤ (T − t)−αq(A2− y2)q/m. (2.7)

Since A2_{− y}2 _{≥ A}2_{(1 − θ}2_{) > 0, it follows from (2.1) and (2.2) that (2.7) holds provided that}

T is sufficiently small.

Now, we fix T so that (u, v) is a subsolution of (1.1)-(1.3). For any t ∈ [0, T ), supp u(·, t) ⊂ supp v(·, t) ⊂ B(0; KATl_{) ⊂ Ω if A > 0 is sufficiently small. Hence it follows from}

Lem-ma 2.1 that the solution (u, v) of (1.1)-(1.4) blows up in finite time if u0 ≥ u(x, 0) and

v0 ≥ v(x, 0).

For m > r or n > s, we shall only consider the case m > r and n ≤ s since the proof for the other two cases is similar. Since ηr_{≤ η}m_{+ 1 if η ≥ 0, the solution of (1.1)-(1.3) is a}

supersolution of

ut= 4um+ vp− a(um+ 1), x ∈ Ω, t > 0, (2.8)

vt = 4vn+ uq− bvs, x ∈ Ω, t > 0, (2.9)

u = v = 0, x ∈ ∂Ω, t > 0. (2.10)

Proceeding all steps in the previous case with a slight modification, we can show that (u, v) is a subsolution of (2.8)-(2.10) and that the solution (u, v) of (1.1)-(1.4) blows up in finite time if u0 ≥ u(x, 0) and v0 ≥ v(x, 0). Hence the proof is complete.

3 The case pq = max{m, r} max{n, s}

Following the proofs of Theorems 2.2 and 2.3 with a slight modification, we get the following results for the critical case.

Theorem 3.1 Let pq = max{m, r} max{n, s}.

(i) Suppose that r > m and s > n and suppose also that a and b are sufficiently small. Then the solution of the problem (1.1)-(1.4) blows up in finite time for initial data large enough.

(ii) Suppose that r ≥ m, s ≥ n, and suppose also that aq_br _{≥ 1. Then all solutions of}

(1.1)-(1.4) are globally bounded.

Proof. (i) Without loss of generality, we may assume that 0 ∈ Ω. Since pq = rs, we have

either s/q ≤ (p + 1)/(q + 1) or r/p ≤ (q + 1)/(p + 1). Therefore, we may without loss of generality assume that s/q ≤ (p + 1)/(q + 1) (otherwise, we exchange the roles of u and v).

Set α = s/[(s − 1)q] and β = 1/(s − 1). It is easy to check that

βp ≥ α + 1, βp = αr > αm, αq = β + 1 = βs > βn. (3.1) Pick a positive number l such that

l ≤ 1

2min{βp − αm, β + 1 − βn, αq − βn}. (3.2) We seek a subsolution of the following form

u(x, t) = C1(T − t)−αU Ã |x| (T − t)l ! , v(x, t) = C2(T − t)−βV Ã |x| (T − t)l ! , where U(y) = (A2 _{− y}2₎1/m

+ , V (y) = (A2 − K−2y2)1/n+ , K, C1, C2 are positve constants

satisfying 1 ≤ K ≤ s βn + 2l βn , αA2/m_C 1 < [A2(1 − 1 K2)] p/n_Cp 2 < (βA2/n)−p[A2(1 − 1 K2] p/n_[A2_{(1 − θ}2_)]pq/m_Cpq 1 (3.3)

and A, T > 0 are constants to be determined later. To show (u, v) is a subsolution of (1.1)-(1.3), it suffices to show that

C1(T − t)−(α+1){αU (y) + lyU0(y)} − C1m(T − t)−(αm+2l)4 (Um)(y)

+aCr

1(T − t)−αrUr(y)

≤ C₂p(T − t)−βp_Vp_{(y) (3.4)}

and

C2(T − t)−(β+1){βV (y) + lyV0(y)} − C2n(T − t)−(βn+2l)4 (Vn)(y)

+bCs

2(T − t)−βsVs(y)

≤ C₁q(T − t)−αq_Uq_{(y) (3.5)}

hold pointwisely for y > 0, y 6= A, KA, where 4 = d2_/dy2_{+ (N − 1)/yd/dy.}

For y > A, it is clear that (3.4) holds. For 0 < y < A, (3.4) is equivalent to

C1(T − t)−(α+1) ( α(A2_{− y}2₎1/m₋ 2l my 2_(A2_{− y}2₎1/m−1 ) +2NCm 1 (T − t)−(αm+2l)+ aC1r(T − t)−αr(A2− y2)r/m ≤ C₂p(T − t)−βp(A2 − y2/K2)p/n. (3.6) Since A2_{− y}2_/K2 _{≥ A}2_{(1 − 1/K}2_{) > 0, it follows from (3.1), (3.2), and (3.3) that (3.6) holds}

provided that a and T are sufficiently small.

For y > KA, it is clear that (3.5) holds. Let θ ∈ (q(βn)/(βn + 2l)K, 1) be fixed. For

θA < y < KA, (3.5) holds provided that C2 ( β(A2_{− y}2_/K2_{) −} 2l n(y 2_/K2₎ ) + 2NCn 2(T − t)β+1−βn−2l(A2− y2/K2)1−1/n +bCs 2(T − t)β+1−βs(A2− y2/K2)1−1/n+s/n ≤ 0. (3.7)

Since β(A2_{− y}2_/K2_{) − 2l/n(y}2_/K2_{) ≤ A}2_{(β − θ}2_/K2_{(β + 2l/n)) < 0, it follows from (3.1),}

(3.2), and (3.3) that (3.7) holds provided that b and T are sufficiently small. For 0 < y < θA, (3.5) is equivalent to

C2(T − t)−(β+1) ( β(A2_{− y}2_/K2₎1/n₋2l n(y 2_/K2_)(A2_{− y}2_/K2₎1/n−1 ) +2NCn 2(T − t)−βn−2l+ bC2s(T − t)−βs(A2− y2/K2)s/n ≤ C₁q(T − t)−αq(A2− y2)q/m. (3.8) Since A2 _{− y}2 _{≥ A}2_{(1 − θ}2_{) > 0, it follows from (3.1), (3.2), and (3.3) that (3.8) holds}

provided that b and T is sufficiently small.

Now, we fix T so that (u, v) is a subsolution of (1.1)-(1.3). For any t ∈ [0, T ), supp u(·, t) ⊂ supp v(·, t) ⊂ B(0; KATl_{) ⊂ Ω if A > 0 is sufficiently small. Hence it follows from}

Lem-ma 2.1 that the solution (u, v) of (1.1)-(1.4) blows up in finite time if u0 ≥ u(x, 0) and

v0 ≥ v(x, 0).

(ii) Let (u, v) = (C1, C2), where C1 and C2 are positive constants satisfying C1 >

|u0|∞, C2 > |v0|∞, and C2pq ≤ aqC1qr ≤ aqbrC2rs. It is easy to check that (u, v) is a

su-persolution of (1.1)-(1.3) with u(x, 0) ≥ u0(x) and v(x, 0) ≥ v0(x). Thus, by Lemma 2.1, we

obtain that u ≤ u and v ≤ v as long as the solution (u, v) exists. Therefore, (u, v) is globally bounded. This completes the proof.

References

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[2] N. D. Alikakos and R. Rostamian, Stabilization of solutions of the equation ∂u/∂t =

4ϕ(u) − β(u), Nonlinear Anal. 6 (1982), 637-647.

[3] M. Bertsch, T. Nanbu and L. A. Peletier, Decay of solutions of a degenerate nonlinear diffusion equation, Nonlinear Anal. 6 (1982), 539-554.

[4] M. A. Herrero and J. L. Vazquez, The one-dimensional nonlinear heat equation with absorption: regularity of solutions and interfaces, SIAM J. Math. Anal. 18 (1987), 149-167.

[5] A. S. Kalashnikov, The propagation of disturbances of nonlinear heat conduction with absorption, USSR Comp. Math. Math. Phys. 14 (1974), 70-85.

[6] S. Kamin and L. A. Peletier, Large time behaviour of solutions of the porous media equation with absorption, Israel. J. Math. 55 (1986), 129-146.

[7] , The behaviour of the solutions of degenerate quasi-linear parabolic equations as t → ∞,

Acta Math. Acad. Sci. Hungar. 34 (1979), 157-163.

[8] B. F. Knerr, The behaviour of the solutions of the equation of nonlinear heat conduction with absorption in one dimension, Trans. Amer. Math. Soc. 249 (1979), 409-424. [9] L. Maddalena, Existence of global solution for reaction-diffusion systems with density

dependent diffusion, Nonlinear Anal. 8 (1984), 1383-1394.

[10] P. Souplet and F. B. Weissler, Self-similar subsolutions and blowup for nonlinear parabolic equations, J. Math. Anal. Appl. 212 (1997), 60-74.

.

計畫成果自評 如同先前所預期的，我們可找到一個爆破臨界指數，使得當指數pq小於臨界指 數時，所有的解均為全域解(global solution); 當指數pq大於臨界指數時，則存在 爆破解(blow-up solution)。而當指數pq等於臨界指數時，目前得到在某些情況下， 解是全域的，在某些情況下，解會產生爆破現象。這部分的問題尚未完全解決， 本人在日後將繼續鑽研。