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Global existence and blow-up of solutions to a nonlocal quasilinear degenerate parabolic system
Nonlinear Analysis 67 (2007) 1387–1402 www.elsevier.com/locate/na
Global existence and blow-up of solutions to a nonlocal quasilinear degenerate parabolic systemI Fucai Li ∗ Department of Mathematics, Nanjing University, Nanjing 210093, PR China Received 25 March 2005; accepted 19 July 2006
Abstract This paper investigates the properties of nonnegative solutions of a quasilinear degenerate parabolic system Z p−2 ∇u) = a v α (x, t)dx, u t − div(|∇u| Ω Z vt − div(|∇v|q−2 ∇v) = b u β (x, t)dx Ω
with zero Dirichlet boundary conditions in a smooth bounded domain Ω ⊂ R N (N ≥ 1), where p, q > 2, α, β ≥ 1, and a, b > 0 are positive constants. Under appropriate hypotheses, we first establish the local existence and uniqueness of solutions, then we show that whether or not the solution blows up in finite time depends on the initial data and the relations between αβ ( p −R 1)(q − 1). In the special case Rof α = q − β = p − 1, we conclude that the solution exists globally if R and R 1 and p−1 dx q−1 dx ≤ 1/(ab), while if p−1 dx q−1 dx > 1/(ab) then the solution blows up in finite time. Here φ ψ φ ψ Ω Ω Ω Ω φ(x) and ψ(x) denote the unique solution of the following elliptic problem −div(|∇φ| p−2 ∇φ) = 1 in Ω , φ(x)|∂ Ω = 0 and −div(|∇ψ|q−2 ∇ψ) = 1 in Ω , ψ(x)|∂ Ω = 0, respectively. c 2006 Elsevier Ltd. All rights reserved.
MSC: 35K65; 35K50; 35K57; 35A07 Keywords: Quasilinear degenerate parabolic system; Nonlinear source; Global existence; Blow-up
1. Introduction and main results This paper deals with the following nonlocal quasilinear degenerate parabolic system in a smooth bounded domain Ω ⊂ R N (N ≥ 1):
I This project was supported partially by NSFC (Grant 10501047), and partially by Nanjing University Talent Development Foundation. ∗ Fax: +86 2583597130.
E-mail address: fucai [email protected]. c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.07.024
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F. Li / Nonlinear Analysis 67 (2007) 1387–1402
u t − div(|∇u| p−2 ∇u) = a vt − div(|∇v|q−2 ∇v) = b
Z ZΩ Ω
v α (x, t)dx,
(x, t) ∈ Ω × (0, T ],
u β (x, t)dx,
(x, t) ∈ Ω × (0, T ],
v(x, t) = 0,
u(x, t) = 0, u(x, 0) = u 0 (x),
(1.1)
(x, t) ∈ ∂Ω × [0, T ],
v(x, 0) = v0 (x),
x ∈ Ω,
where p, q > 2, α, β ≥ 1, and a, b > 0 are positive constants. In the last three decades, many authors have studied the following degenerate parabolic problem: u t − div(|∇u| p−2 ∇u) = f (u),
(x, t) ∈ Ω × (0, T ],
u(x, t) = 0, (x, t) ∈ ∂Ω × [0, T ], u(x, 0) = u 0 (x), x ∈ Ω
(1.2)
under different conditions. In [1,6,10,11,18–22,24,26,31], the existence, uniqueness and regularity of solutions were obtained. When f (u) = −u q , q > 0 or f (u) ≡ 0 extinction phenomenon of the solution may appear, see [27,29]; however, if f (u) = u q , q > 1 the solution may blow up in finite time, see [9,13,26,30]. Roughly speaking, the results in [9,13,26,30] read: (1) the solution u exists globally if q < p − 1; (2) u blows up in finite time if q > p − 1 and u 0 (x) is sufficiently large. The (1) and (2) stated above are also true for the nonlocal degenerate parabolic equation: Z p−2 u t − div(|∇u| ∇u) = u q dx Ω
with null Dirichlet boundary condition, see [17] for details. The purpose of this paper is to investigate qualitative properties of the solutions to the complicated problem (1.1). The system in (1.1) is degenerate and appears, for example, in the theory of non-Newtonian filtration fluids [1, 12]. From a physical point of view, we need only to consider the nonnegative solutions. Moreover, if we assume that u 0 (x), v0 (x) ≥ 0 in Ω , by Lemma 2.2 (see Section 2 below), we can obtain that (u(x, t), v(x, t)) ≥ (0, 0) a.e. in (Ω × (0, T )) × (Ω × (0, T )). Thus we only consider the nonnegative solutions in later sections. Since the equations in (1.1) are degenerate for |∇u| = 0 or |∇v| = 0, one cannot expect the existence of a classical solution for (1.1). The most of studies on p-Laplacian equations concerned with weak solutions, see [1,4,6–8,10,11,18–22,24,26,28,31]. We start from the definition of solution. Let Q T = Ω × (0, T ], ST = ∂Ω × [0, T ], T > 0. Throughout this paper we assume that: 1, p 1,q (A) u 0 (x) ∈ C(Ω ) ∩ W0 (Ω ), v0 (x) ∈ C(Ω ) ∩ W0 (Ω ) and ∂u 0 (x)∂η < 0, ∂v0 (x)∂η < 0 on the boundary ∂Ω , where η denotes the unit outer normal vector on ∂Ω . Definition 1.1. A pair of functions (u, v) is called a solution of the problem (1.1) on Q T × Q T if and only if 1, p 1,q u ∈ C(0, T ; L ∞ (Ω )) ∩ L p (0, T ; W0 (Ω )), v ∈ C(0, T ; L ∞ (Ω )) ∩ L q (0, T ; W0 (Ω )), u t ∈ L 2 (0, T ; L 2 (Ω )), vt ∈ L 2 (0, T ; L 2 (Ω )), u(x, t)|t=0 = u 0 (x), v(x, t)|t=0 = v0 (x), and the equalities Z Z Z t2 Z u(x, t2 )ψ1 (x, t2 )dx − u(x, t1 )ψ1 (x, t1 )dx = uψ1t dxdt Ω
Ω
t2
Z
Z
|∇u| p−2 ∇u · ∇ψ1 dxdt + a
− t1
Z Ω
Ω
Z
t2
Z
t1
Z
Ω
Ω
Ω
ψ1 (x, t)
v(x, t1 )ψ2 (x, t1 )dx =
|∇v|q−2 ∇q · ∇ψ2 dxdt + b
−
Z
t1
v(x, t2 )ψ2 (x, t2 )dx −
Ω
t1 t2
Z
Z
t2
t2
t1
Z Ω
Ω
Z
t1
Z
Z
ψ2 (x, t)
Ω
u α (x, t)dx dxdt,
(1.3)
vψ2t dxdt
Z Ω
u β (x, t)dx dxdt
(1.4)
hold for all 0 < t1 < t2 < T , where ψ1 (x, t), ψ2 (x, t) ∈ C 1,1 (Q T ) such that ψ1 (x, T ) = ψ2 (x, T ) = 0 and ψ1 (x, t) = ψ2 (x, t) = 0 on ST .
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F. Li / Nonlinear Analysis 67 (2007) 1387–1402
Similarly, to define a subsolution (u(x, t), v(x, t)) we need only to require that ψ1 (x, t), ψ2 (x, t) ≥ 0, (u(x, 0), v(x, 0)) ≤ (u 0 (x), v0 (x)) on Ω × Ω , (u(x, t), u(x, t)) ≤ (0, 0) on ST × ST and the equalities in (1.3) and (1.4) are replaced by ≤. A supersolution can be defined similarly. Definition 1.2. We say the solution (u, v) of the problem (1.1) blows up in finite time if there exists a positive constant T ∗ , T ∗ < ∞ such that lim |u(·, t)| L ∞ (Ω ) + |v(·, t)| L ∞ (Ω ) = +∞. t→T ∗−
We say the solution (u, v) exists globally if sup t∈(0,+∞)
|u(·, t)| L ∞ (Ω ) + |v(·, t)| L ∞ (Ω ) < +∞.
Our first result is on the local existence and uniqueness of solutions. Theorem 1.3. Suppose that (u 0 , v0 ) ≥ (0, 0) and satisfies the condition (A), then there exists a constant T0 > 0 such 1, p that the problem (1.1) admits a unique solution (u, v) ∈ Q T0 × Q T0 , u ∈ C(0, T0 ; L ∞ (Ω )) ∩ L p (0, T0 ; W0 (Ω )), 1,q and v ∈ C(0, T0 ; L ∞ (Ω )) ∩ L q (0, T0 ; W0 (Ω )). The next two theorems show that the solution (u, v) may exist globally or blow up in finite time depending on p and q, the exponents α and β, the initial data u 0 (x) and v0 (x) and the Lebesgue’s measure of Ω . Theorem 1.4. Suppose that (u 0 , v0 ) satisfies the conditions in Theorem 1.3 and one of the following conditions holds: (1) αβ < ( p − 1)(q − 1); (2) αβ = ( p − 1)(q − 1) and |Ω | is sufficiently small; (3) αβ > ( p − 1)(q − 1) and u 0 (x), v0 (x) are sufficiently small. Then the solution of problem (1.1) exists globally. Theorem 1.5. Suppose that (u 0 , v0 ) satisfies the conditions in Theorem 1.3 and one of the following conditions holds: (1) αβ > ( p − 1)(q − 1) and u 0 (x), v0 (x) are sufficiently large, (2) αβ = ( p − 1)(q − 1) and Ω contains a sufficiently large ball. Then the solution of problem (1.1) blows up in finite time. Let φ(x) and ψ(x) be the unique solution of the following elliptical problem: −div(|∇φ| p−2 ∇φ) = 1
in Ω ,
φ(x) = 0
−div(|∇ψ|q−2 ∇ψ) = 1
in Ω ,
ψ(x) = 0
on ∂Ω
(1.5)
and on ∂Ω ,
(1.6)
respectively. In the special case of αβ = ( p − 1)(q − 1) with α = q − 1 and β = p − 1, we have the following optimal results: Theorem 1.6. Suppose that (u 0 , v0 ) satisfies the conditions in Theorem 1.3, and that α = q − 1 and β = p − 1. R R (1) If RΩ φ p−1 dx RΩ ψ q−1 dx ≤ 1/(ab), then the solution of problem (1.1) exists globally; (2) If Ω φ p−1 dx Ω ψ q−1 dx > 1/(ab), then the solution of problem (1.1) blows up in finite time. Remark 1.7. In the case of linear diffusions, namely, the problem (1.1) with p = q = 2, the solution exists globally if αβ equals to the diffusive power, see [15,16]. However, the above conclusion is not true for nonlinear diffusions, our results in Theorems 1.4–1.6 explain that how the nonlinear diffusions affect the behaviour of the solution. This paper is organized as follows. In Section 2, we shall establish the local existence theory of solution and prove Theorem 1.3. In Section 3, we shall discuss the global existence of solution and prove Theorem 1.4. In Section 4, we shall discuss the blow-up results of the solution and prove Theorem 1.5. The special case α = q − 1 and β = p − 1 will be discussed in the last section.
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2. Local solution We first give a maximum principle for the nondegenerate parabolic system, whose proof is similar to that given in [23], we omit it here. Lemma 2.1. Suppose that w(x, t), z(x, t) ∈ C 2,1 (Q T ) ∩ C(Q T ) and satisfy Z N N X X wt − ai j (x, t)wxi x j + ci (x, t)wxi ≥ e(x, t)z(x, t)dx, (x, t) ∈ Q T , i, j=1
zt −
N X
Ω
i=1
bi j (x, t)z xi x j +
i, j=1
N X
di (x, t)z xi ≥
Z
i=1
w(x, t) ≥ 0,
z(x, t) ≥ 0,
(x, t) ∈ ST ,
w(x, 0) ≥ 0,
z(x, 0) ≥ 0,
(x, t) ∈ Ω ,
g(x, t)w(x, t)dx, Ω
(x, t) ∈ Q T ,
where ai j (x, t), bi j (x, t), c(x, t), d(x, t), e(x, t) and g(x, t) are real bounded continuous functions, e(x, t), g(x, t) ≥ 0, and the matrixes A = (ai j ), B = (bi j ) are symmetric and positive definite. Then w(x, t) ≥ 0, z(x, t) ≥ 0, (x, t) ∈ Q T . The following comparison lemma of problem (1.1) plays a crucial role in the proof of our results. Lemma 2.2. Suppose that (u, v), (u, v) are a nonnegative subsolution and supersolution of the problem (1.1) on Q T × Q T , respectively. Then (u, v) ≤ (u, v) a.e. on Q T × Q T . Proof. For any small δ 0, ρδ (ξ ) = ξ/δ, 1,
> 0, set ξ ≤ 0, 0 < ξ < δ, ξ ≥ δ.
Then ρδ (ξ ) is a piecewise differentiable function. Let ψ1 (x, t) = ρδ ((u − u)(x, t)) and ψ2 (x, t) = ρδ ((v − v)(x, t)), it is easy to verify that ψ1 (x, t) and ψ2 (x, t) are admissible test functions in (1.3) and (1.4). Since (u, v) and (u, v) are subsolution and supersolution of (1.1), let t1 = τ , t2 = τ + h, τ , h > 0, τ + h < T and w = u − u, z = v − v, then we obtain Z Z w(x, τ + h)ψ1 (x, τ + h)dx − w(x, τ )ψ1 (x, τ )dx Ω
Ω
τ +h
Z
Z
≤ τ
Ω τ +h
Z
Ω
Z
τ
Ω
ψ1 (x, s)
Z
τ
Ω
Z
α
(|∇u| p−2 ∇u − |∇u| p−2 ∇u) · ∇ψ1 dxds α
(v − v )dx dxds, Z z(x, τ + h)ψ2 (x, τ + h)dx − z(x, τ )ψ2 (x, τ )dx +a
Z
wψ1s dxds −
τ +h
Z
(2.1)
Ω
Ω
τ +h
Z
Z zψ2s dxds −
≤ τ
Ω τ +h
Z +b
τ
τ +h
Z Z Ω
ψ2 (x, s)
τ
Z Ω
Z Ω
(|∇v|q−2 ∇v − |∇v|q−2 ∇v) · ∇ψ2 dxds
(u β − u β )dx dxds.
Dividing (2.1) and (2.2) by h and integrating τ over (0, t) gives Z t Z 1 (w(x, τ + h)ψ1 (x, τ + h) − w(x, τ )ψ1 (x, τ ))dxdτ 0 h Ω Z Z t Z τ +h Z Z t Z τ +h Z 1 1 α α ≤ wψ1s dxdsdτ + a ψ1 (x, s) (v − v )dx dxdsdτ 0 h τ Ω 0 h τ Ω Ω
(2.2)
F. Li / Nonlinear Analysis 67 (2007) 1387–1402
Z
t
Z
1 h
τ +h
1391
Z
(|∇u| p−2 ∇u − |∇u| p−2 ∇u) · ∇ψ1 dxdsdτ, − τ Ω 0 Z t Z 1 (z(x, τ + h)ψ2 (x, τ + h) − z(x, τ )ψ2 (x, τ ))dxdτ h 0 Ω Z Z t Z τ +h Z Z t Z τ +h Z 1 1 β β zψ2s dxdsdτ + b ψ2 (x, s) (u − u )dx dxdsdτ ≤ 0 h τ Ω Ω Ω 0 h τ Z t Z τ +h Z 1 (|∇v|q−2 ∇v − |∇v|q−2 ∇v) · ∇ψ2 dxdsdτ. − h τ Ω 0
(2.3)
(2.4)
By the properties of Steklov’s averages ([6], Lemma 1.3.2), we get t
Z 0
t
Z 0
t
Z 0
t
Z 0
1 h
Z
τ +h
Z
τ
Ω
wψ1s dxdsdτ →
Z tZ 0 t
Ω
wψ1s dxds
as h → 0+ ,
Z Z Z Z 1 τ +h zψ2s dxdsdτ → zψ2s dxds as h → 0+ , h τ Ω 0 Ω Z Z 1 τ +h (|∇u| p−2 ∇u − |∇u| p−2 ∇u) · ∇ψ1 dxdsdτ h τ Ω Z tZ → (|∇u| p−2 ∇u − |∇u| p−2 ∇u) · ∇ψ1 dxds as h → 0+ , 1 h
Z
0 Ω τ +h Z
τ
Ω
(2.5) (2.6)
(2.7)
(|∇v|q−2 ∇v − |∇v|q−2 ∇v) · ∇ψ2 dxdsdτ
Z tZ
(|∇v|q−2 ∇v − |∇v|q−2 ∇v) · ∇ψ2 dxds as h → 0+ , 0 Ω Z Z Z 1 τ +h α α ψ1 (x, s) (v − v )dx dxdsdτ h τ Ω Ω Z Z tZ → ψ1 (x, s) (v α − v α )dx dxds as h → 0+ , →
t
Z 0
t
Z 0
1 h
Z
0 Ω τ +h Z
τ
Ω
Z tZ → 0
Ω
ψ2 (x, s) ψ2 (x, s)
Z Ω
Z Ω
(u β − u β )dx dxdsdτ
(u β − u β )dx dxds
as h → 0+ .
Ω
0
t
(2.9)
Ω
Now we claim that Z t Z 1 (w(x, τ + h)ψ1 (x, τ + h) − w(x, τ )ψ1 (x, τ ))dxdτ 0 h Ω Z → (w(x, t)ψ1 (x, t) − w(x, 0)ψ1 (x, 0))dx as h → 0+ , Ω Z t Z 1 (z(x, τ + h)ψ2 (x, τ + h) − z(x, τ )ψ2 (x, τ ))dxdτ 0 h Ω Z → (z(x, t)ψ2 (x, t) − z(x, 0)ψ2 (x, 0))dx as h → 0+ . In fact, Z
(2.8)
Z 1 (w(x, τ + h)ψ1 (x, τ + h) − w(x, τ )ψ1 (x, τ ))dxdτ h Ω
(2.10)
(2.11)
(2.12)
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F. Li / Nonlinear Analysis 67 (2007) 1387–1402
Z Z 1 t 1 t w(x, τ + h)ψ1 (x, τ + h)dxdτ − w(x, τ )ψ1 (x, τ )dxdτ = Ω h 0 Ω h 0 Z Z Z Z 1 t 1 t+h = w(x, s)ψ1 (x, s)dxds − w(x, s)ψ1 (x, s)dxds Ω h 0 Ω h h Z Z Z Z 1 t+h 1 h = w(x, s)ψ1 (x, s)dxds − w(x, s)ψ1 (x, s)dxds Ω h t Ω h 0 Z Z w(x, 0)ψ1 (x, 0)dx as h → 0+ . w(x, t)ψ1 (x, t)dx − → Z
Z
Ω
Ω
The equality (2.12) can be obtained similarly. By (2.1)–(2.12) we obtain Z tZ Z Z wρδ0 (w)ws dxds w(x, 0)ρδ (w(x, 0))dx + w(x, t)ρδ (w(x, t))dx ≤ 0 Ω ΩZ Z Ω t (|∇u| p−2 ∇u − |∇u| p−2 ∇u) · ∇ρδ (u − u)dxds − 0Z Ω Z tZ ρδ (w(x, s)) (v α − v α )dx dxds, +a Z Ω
z(x, t)ρδ (z(x, t))dx ≤
(2.13)
Ω
Ω
0
Z tZ z(x, 0)ρδ (z(x, 0))dx + zρδ0 (z)z s dxds ΩZ Z 0 Ω t − (|∇v|q−2 ∇v − |∇v|q−2 ∇v) · ∇ρδ (v − v)dxds 0 Ω Z Z tZ +b ρδ (z(x, s)) (u β − u β )dx dxds. Z
Ω
0
(2.14)
Ω
Now we deal with the terms in (2.13) and (2.14). First, we have Z Z tZ Z tZ Z tZ α α ρδ (w(x, s)) (v − v )dx dxds ≤ α|Ω | η1 (v − v)+ dxds ≤ M1 (v − v)+ dxds, 0 Ω Ω 0 Ω 0 Ω Z Z tZ Z tZ Z tZ ρδ (z(x, s)) (u β − u β )dx dxds ≤ β|Ω | η2 (u − u)+ dxds ≤ M2 (u − u)+ dxds 0
Ω
Ω
0
Ω
0
Ω
for some positive constants M1 and M2 , and Z t Z Z tZ 0 w+ |ρδ0 (w)||ws |dxds ≤ (w)w dxds wρ s δ 0Z ΩZ 0 Ω 1 δ = w+ |ws |dxds → 0 as δ → 0+ , δ 0 Ω Z t Z Z tZ 0 zρδ (z)z s dxds ≤ z + |ρδ0 (z)||z s |dxds 0 Ω 0Z ΩZ 1 δ = z + |z s |dxds → 0 as δ → 0+ . δ 0 Ω Second, by Lemma 1.4.4 in [6], we get (|∇u| p−2 ∇u − |∇u| p−2 ∇u) · ∇ρδ (u − u) ≥ min{0, γ1 |∇(u − u)+ | p }, (|∇v|q−2 ∇v − |∇v|q−2 ∇v) · ∇ρδ (v − v) ≥ min{0, γ2 |∇(v − v)+ |q } R R for some γ1 , γ2 > 0. Finally, we have Ω w(x, 0)ρδ (w(x, 0))dx ≡ 0, Ω z(x, 0) × ρδ (z(x, 0))dx ≡ 0 and ρδ0 ≥ 0 a.e. in R, wρδ0 (w(x, t))ws , zρδ0 (z(x, t))z s increase and tend to w+ , z + as δ → 0+ , respectively. Hence we may let δ → 0+ in (2.13) and (2.14) to yield Z tZ Z w+ (x, t)dx ≤ (a M1 + 1) z + (x, s)dxds, Ω
0
Ω
F. Li / Nonlinear Analysis 67 (2007) 1387–1402
Z Ω
Hence, Z Ω
z + (x, t)dx ≤ (bM2 + 1)
Z tZ 0
Ω
1393
w+ (x, s)dxds.
(w+ (x, t) + z + (x, t))dx ≤ (a M1 + bM2 + 1)
Z tZ 0
Ω
(w+ (x, s) + z + (x, s))dxds.
R By the Gronwall’s inequality we obtain Ω (w+ (x, t) + z + (x, t))dx = 0, i.e. u ≤ u, v ≤ v a.e. on Q T . This completes the proof. Proof of Theorem 1.3. Consider the following approximate problems for the problem (1.1): Z p−2 u nt − div (|∇u n |2 + εn ) 2 ∇u n = a vnα (x, t)dx, (x, t) ∈ Q T , Ω Z q−2 2 2 u βn (x, t)dx, (x, t) ∈ Q T , vnt − div (|∇vn | + σn ) ∇vn = b Ω
u n (x, t) = 0, u n (x, 0) =
u n (x, t) = 0,
u ε0n (x),
(2.15)
(x, t) ∈ ST ,
vn (x, 0) = u σ0n (x),
x ∈ Ω.
Here {εn }, {σn } are strictly decreasing sequences, 0 < εn , σn < 1, and εn → 0+ , σn → 0+ as n → ∞. u ε0n ∈ C0∞ (Ω ) and v0σn ∈ C0∞ (Ω ) are approximation functions for the initial data u 0 (x) and v0 (x), respectively. |u ε0n | L ∞ (Ω ) ≤ 1, p |u 0 | L ∞ (Ω ) , |∇u ε0n | L ∞ (Ω ) ≤ |∇u 0 | L ∞ (Ω ) for all εn , and u ε0n → u 0 strongly in W0 (Ω ); |v0σn | L ∞ (Ω ) ≤ |v0 | L ∞ (Ω ) , 1,q |∇v0σn | L ∞ (Ω ) ≤ |∇v0 | L ∞ (Ω ) for all σn , and v0σn → v0 strongly in W0 (Ω ). (2.15) is a nondegenerate problem for each fixed εn and σn ; it is easy to prove that it admits a classic solution (u n , vn ) by using the Schauder’s fixed point theorem. The uniqueness of solutions is obtained by using Lemma 2.1. To find limit functions u(x, t) and v(x, t) of the sequence {(u n , vn )}, we divide our proof into four steps: Step 1: There exist a small constant T0 > 0 and a positive constant M1 , independent of n, such that: |u n | L ∞ (Q T0 ) , |vn | L ∞ (Q T0 ) ≤ M1 .
(2.16)
To this end, we consider the following Cauchy problems: dU1 dV1 β = a|Ω |V1α , = b|Ω |U1 , dt dt U1 (0) = |u 0 | L ∞ (Ω ) , V1 (0) = |v0 | L ∞ (Ω ) ,
(2.17)
dV2 dU2 β = −a|Ω |V2α , = −b|Ω |U2 , dt dt U2 (0) = −|u 0 | L ∞ (Ω ) , V2 (0) = −|v0 | L ∞ (Ω ) .
(2.18)
It is easy to verify that there exists a constant t0 ∈ (0, T ) such that (2.17) and (2.18) admits a solution (U1 (t), V1 (t)) and (U2 (t), V2 (t)) on [0, t0 ] respectively, moreover, the t0 depends only on |u 0 | L ∞ (Ω ) , |v0 | L ∞ (Ω ) . By Lemma 2.2 we get |u n (x, t)|, |vn (x, t)| ≤ max{U1 (t), V1 (t), −U2 (t), −V2 (t)}. Setting T0 = t0 /2 and M1 = max{U1 (t), V1 (t), −U2 (t), −V2 (t)}, we draw our conclusion. Step 2: There exists a constant M2 > 0, independent of n, such that: |∇u n | L p (Q T0 ) , |∇vn | L q (Q T0 ) ≤ M2 .
(2.19)
In fact, multiplying the first equation in (2.15) by u n , the second equation in (2.15) by vn and integrating the results over Q T0 we have: Z Z T0 Z p−2 1 2 u n (x, T0 )dx + (|∇u n |2 + εn ) 2 |∇u n |2 dxdt 2 Ω 0 Ω Z Z Z T0 Z 1 εn 2 α = (u (x)) dx + a u n (x, t)dx vn (x, t)dx dt, 2 Ω 0 0 Ω Ω
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F. Li / Nonlinear Analysis 67 (2007) 1387–1402
Z Z T0 Z q−2 1 vn2 (x, T0 )dx + (|∇vn |2 + σn ) 2 |∇vn |2 dxdt 2 Ω 0 Ω Z Z T0 Z Z 1 vn (x, t)dx u βn (x, t)dx dt. = (v0σn (x))2 dx + b 2 Ω 0 Ω Ω By |u ε0n | L ∞ (Ω ) ≤ |u 0 | L ∞ (Ω ) , |v0σn | L ∞ (Ω ) ≤ |v0 | L ∞ (Ω ) and (2.16) we get Z T0 Z 1 |∇u n | p dxdt ≤ |u 0 |2L ∞ (Ω ) + T0 |Ω |2 M1α+1 , 2 0 Ω Z T0 Z 1 β+1 |∇vn |q dxdt ≤ |v0 |2L ∞ (Ω ) + T0 |Ω |2 M1 . 2 0 Ω Set M2 = max
1 1 β+1 |u 0 |2L ∞ (Ω ) + aT0 |Ω |2 M1α+1 , |v0 |2L ∞ (Ω ) + bT0 |Ω |2 M1 , 2 2
we draw our conclusion. Step 3: There exists a constant M3 > 0, independent of n, such that: |u nt | L 2 (Q T ) , |vnt | L 2 (Q T ) ≤ M3 . 0
(2.20)
0
To do so, multiplying the first equation in (2.15) by u nt , the second equation in (2.15) by vnt and integrating the results over Q T0 we get: Z T0 Z Z T0 Z p−2 u 2nt (x, t)dxdt = − (|∇u n |2 + εn ) 2 ∇u n · ∇u nt dxdt 0 Ω 0Z Ω Z T0 Z α +a u nt (x, t) vn (x, t)dx dxdt. T0
Z 0
Z Ω
Z
0 T0
Ω
Z
Ω
q−2
(|∇vn |2 + σn ) 2 ∇vn · ∇vnt dxdt 0Z Ω Z T0 Z +b vnt (x, t) u βn (x, t)dx dxdt.
2 (x, t)dxdt = − vnt
0
|u ε0n | L ∞ (Ω )
Ω
Ω
|u 0 | L ∞ (Ω ) , |v0σn | L ∞ (Ω )
By H¨older’s inequality, ≤ ≤ |v0 | L ∞ (Ω ) , inequality (2.16) and the equalities Z T0 Z Z Z p−2 1 1 (|∇u n |2 + εn ) 2 ∇u n · ∇u nt dxdt = (|∇u n (x, T0 )|2 + εn ) p/2 dx − (|∇u ε0n |2 + εn ) p/2 dx, 2 Ω 2 Ω 0 Ω Z T0 Z Z Z q−2 1 1 (|∇vn |2 + σn ) 2 ∇vn · ∇vnt dxdt = (|∇vn (x, T0 )|2 + σn )q/2 dx − (|∇v0σn |2 + σn )q/2 dx, 2 Ω 2 Ω 0 Ω we obtain: Z T0 Z
Z Z 1 1 2 p/2 ≤ − (|∇u n (x, T0 )| + εn ) dx + (|∇u ε0n |2 + εn ) p/2 dx 2 Ω 2 Ω 0 Ω 2 Z T0 Z Z T0 Z 1 1 a 2 α 2 2 + ηa|Ω | u nt dxdt + |Ω | vn dx dt ≤ M 0 , η 0 Ω 0 Ω Z Z Z T0 Z 1 1 2 q/2 2 vnt dxdt ≤ − (|∇vn (x, T0 )| + σn ) dx + (|∇v0σn |2 + σn )q/2 dx 2 Ω 2 Ω 0 Ω 2 Z T0 Z Z T0 Z 1 1 b 2 β 2 2 + ηb|Ω | vnt dxdt + |Ω | u n dx dt ≤ M 0 η 0 Ω 0 Ω u 2nt dxdt
for some M 0 > 0 and some small η > 0.
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F. Li / Nonlinear Analysis 67 (2007) 1387–1402
Therefore, by virtue of (2.16), (2.19) and (2.20) and the Ascoli–Arzel´a theorem, we can choose a subsequence, still denoted by {(u n , vn )} for convenience, such that: u n → u, vn → v
a.e. for (x, t) ∈ Q T0 ,
(2.21)
∇u n → ∇u
weakly in L (0, T0 ; L (Ω )),
(2.22)
∇vn → ∇v
weakly in L q (0, T0 ; L q (Ω )),
(2.23)
p
u nt → u t , vnt → vt |∇u n |
p−2
(u n )xi → wi
|∇vn |q−2 (vn )xi → z i
p
weakly in L 2 (0, T0 ; L 2 (Ω )), p/( p−1)
weakly in L
(0, T0 ; L
(2.24) p/( p−1)
(Ω )),
(2.25)
weakly in L q/(q−1) (0, T0 ; L q/(q−1) (Ω )).
(2.26)
Step 4: we show that wi = xi and z i = xi . Multiplying the first equation in (2.15) by ψ1 (u n − u), the second equation in (2.15) by ψ2 (vn − v) and integrating the results over Q T0 we have: Z T0 Z Z T0 Z p−2 ψ1 (u n − u)u nt dxdt + ψ1 (|∇u n |2 + εn ) 2 ∇u n · ∇(u n − u)dxdt 0 Ω 0 Ω Z Z T0 Z Z T0 Z p−2 α 2 ψ1 (u n − u) vn (x, t)dx dxdt, + (u n − u)(|∇u n | + εn ) 2 ∇u n · ∇ψ1 dxdt = a |∇v|q−2 v
|∇u| p−2 u
Ω
0
T0
Z 0
Z
0
ψ2 (vn − v)vnt dxdt +
T0
Z
Z
ψ2 (|∇vn |2 + σn )
Ω
Ω
q−2 2
∇vn · ∇(vn − v)dxdt Z Z T0 Z Z T0 Z q−2 + (vn − v)(|∇vn |2 + σn ) 2 ∇vn · ∇ψ2 dxdt = b ψ2 (vn − v) u βn (x, t)dx dxdt. Ω
0
Ω
Ω
0
0
Ω
Ω
Using (2.16), (2.21) and (2.24) we get: Z T0 Z lim ψ1 |∇u n | p−2 ∇u n · ∇(u n − u)dxdt = 0, n→∞ 0
Z lim
n→∞ 0
Ω
T0
Z
ψ2 |∇vn |q−2 ∇vn · ∇(vn Ω C 1,1 (Q T0 ), ψ1 , ψ2 ≥ 0.
− v)dxdt = 0,
where ψ1 , ψ2 ∈ The left arguments are as same as those of Theorem 2.1 in [30], so we omit them. We complete the existence part by a standard limiting process. The uniqueness of the solution is obvious. In fact, assume that (u 1 , v1 ), (u 2 , v2 ) are two nonnegative solutions of (1.1), using Lemma 2.2 repeatedly, we can get u 1 = u 2 , v1 = v2 a.e. in Ω × [0, T0 ]. 3. Global existence of solution In this section we investigate the global existence property of solution to the problem (1.1). Our approach is a combination of comparison principle (see Lemma 2.2) and upper solution techniques, which is quite different from that used in [9,13,14,26,30] to study the problem (1.2). Proof of Theorem 1.4. Denote φ(x) and ψ(x) be the unique solution of the elliptical problems defined in (1.5) and (1.6), respectively. Then we have φ(x), ψ(x) > 0 in Ω , ∂φ(x)/∂η < 0 and ∂ψ(x)/∂η < 0 on the boundary ∂Ω , and there exist two positive constants M1 and M2 such that: sup φ(x) = M1 ,
sup ψ(x) = M2 ,
x∈Ω
x∈Ω
(3.1)
see [3,5]. Let W (x, t) = K 1 (φ(x) + 1) and Z (x, t) = K 2 (ψ(x) + 1), where K 1 and K 2 > 0 will be determined later. (1) In case of αβ < ( p − 1)(q − 1), we can choose sufficiently large constants K 1 and K 2 such that K 1 ≥ u 0 (x), K 2 ≥ v0 (x) and p−1
K1
≥ a|Ω |(M2 + 1)α K 2α ,
q−1
K2
β
≥ b|Ω |(M1 + 1)β K 1 .
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F. Li / Nonlinear Analysis 67 (2007) 1387–1402
In fact, we can take them freely provided that: q−1 ( p−1)(q−1)−αβ q+α−1 αβ α , K 1 > max max u 0 (x), ab q−1 |Ω | q−1 (M1 + 1) q−1 (M2 + 1)α x∈Ω p−1 ( p−1)(q−1)−αβ p+β−1 β αβ K 2 > max max v0 (x), a p−1 b|Ω | p−1 (M1 + 1)β (M2 + 1) p−1 . x∈Ω Thus, Z Z α dx, ≥ a|Ω |(M2 + 1)α K 2α ≥ a Ω Z β ≥ b|Ω |(M1 + 1)β K 1 ≥ b W β dx.
p−1
Wt − div(|∇W | p−2 ∇W ) = K 1
q−1
Z t − div(|∇ Z |q−2 ∇ Z ) = K 2
(3.2) (3.3)
Ω
Noticing W (x, t) = Z (x, t) = 0 on ∂Ω × [0, +∞), W (x, 0) ≥ u 0 (x) and Z (x, 0) ≥ v0 (x) in Ω , we obtain (u(x, t), v(x, t)) ≤ (W (x, t), Z (x, t)) in (Ω × [0, +∞)) × (Ω × [0, +∞)) by Lemma 2.2. Hence (u(x, t), v(x, t)) exists globally. (2) In this case, αβ = ( p − 1)(q − 1). We choose K 1 ≥ u 0 (x), K 2 ≥ v0 (x), then (3.2) and (3.3) hold provided that: − q−1 q+α−1 αβ α α q−1 q−1 |Ω | ≤ ab (M1 + 1) (M2 + 1) and − p−1 p+β−1 αβ β β |Ω | ≤ a p−1 b(M1 + 1) (M2 + 1) p−1 . Similarly to the case (1), we obtain that (u(x, t), v(x, t)) exists globally. (3) In this case, αβ > ( p − 1)(q − 1). To ensure (3.2) and (3.3) hold, we need only to choose sufficiently positive small K 1 , K 2 > 0 such that q−1 − αβ−( p−1)(q−1) q+α−1 αβ α α q−1 q−1 q−1 (M1 + 1) (M2 + 1) K 1 < ab |Ω | and p−1 − αβ−( p−1)(q−1) p+β−1 αβ β β p−1 p−1 p−1 K 2 < a b|Ω | (M1 + 1) (M2 + 1) .
Therefore, for the sufficiently small (u 0 (x), v0 (x)) satisfying u 0 (x) ≤ W (x, 0) and v0 (x) ≤ Z (x, 0), we obtain (u(x, t), v(x, t)) ≤ (W (x, t), Z (x, t)) on (Ω × [0, +∞)) × (Ω × [0, +∞)) by Lemma 2.2. Hence, (u(x, t), v(x, t)) must exist globally. 4. Blow-up of solution In this section, we shall prove Theorem 1.5. According to Lemma 2.2, we need only to construct a blowing up subsolution to the problem (1.1). Proof of Theorem 1.5. (1) In this case, we use some ideas developed by [25], which is a powerful tool to construct blowing up subsolutions for both nondegenerate and degenerate parabolic equations, see [2,25]. Let ϕ ∈ C 1 (Ω ), ϕ(x) ≥ 0, ϕ(x) 6≡ 0 and ϕ|∂ Ω = 0. By translation, we may assume without loss of generality that 0 ∈ Ω and ϕ(0) > 0. Set |x| 1 |x| 1 V V , z(x, t) = w(x, t) = (T − t)γ1 (T − t)σ1 (T − t)γ2 (T − t)σ2
F. Li / Nonlinear Analysis 67 (2007) 1387–1402
1397
with A y2 V (y) = 1 + − 2 2A
,
y ≥ 0,
+
where γi , σi > 0 (i = 1, 2), A > 1 and 0 < T < 1 are to be determined. Notice the fact that: supp w+ (·, t) = B(0, R(T − t)σ1 ) ⊂ B(0, RT σ1 ) ⊂ Ω , supp z + (·, t) = B(0, R(T
− t)σ2 )
⊂ B(0,
RT σ2 )
⊂Ω
for sufficiently small T > 0 with R = (A(2 + A))1/2 . Denote yi = |x|/(T − t)σi , i = 1, 2. Computing directly we obtain: γ1 V (y1 ) + σ1 y1 V 0 (y1 ) γ2 V (y2 ) + σ2 y2 V 0 (y2 ) , z (x, t) = ; t (T − t)γ1 +1 (T − t)γ2 +1 N /A N /A −∆w(x, t) = , −∆z(x, t) = ; (T − t)γ1 +2σ1 (T − t)γ2 +2σ2
wt (x, t) =
and Z |x| M1 α V dx (T − t)γ2 α B(0,R(T −t)σ2 ) (T − t)σ2 Ω = M1 /(T − t)γ2 α−N σ2 , Z Z |x| M2 β V dx w β (x, t)dx = (T − t)γ1 β B(0,R(T −t)σ1 ) (T − t)σ1 Ω = M2 /(T − t)γ1 β−N σ1 , R R where M1 = B(0,R) V α (|ξ |)dξ and M2 = B(0,R) V β (|ξ |)dξ . By Z
z α (x, t)dx =
div(|∇w| p−2 ∇w) = |∇w| p−2 ∆w + ( p − 2)|∇w| p−4 (∇w)T (Hx (w))∇w N X N X ∂w ∂ 2 w ∂w = |∇w| p−2 ∆w + ( p − 2)|∇w| p−4 , ∂ xi ∂ xi ∂ x j ∂ x j j=1 i=1 div(|∇z|q−2 ∇z) = |∇z|q−2 ∆z + (q − 2)|∇z|q−4 (∇z)T (Hx (z))∇z N X N X ∂z ∂ 2 z ∂z , = |∇z|q−2 ∆z + (q − 2)|∇z|q−4 ∂ xi ∂ xi ∂ x j ∂ x j j=1 i=1 where Hx (w), Hx (z) denotes the Hessian matrix of w(x, t), z(x, t) respect to x, respectively, we get: p−2 N /A diam(Ω ) |div(|∇w| p−2 ∇w)| ≤ (T − t)γ1 +2σ1 (T − t)γ1 +2σ1 p−4 2 diam(Ω ) diam(Ω ) N /A + ( p − 2) γ +2σ γ +2σ 1 1 1 1 (T − t) (T − t) (T − t)γ1 +2σ1 p−2 N ( p − 1)(diam(Ω )) = , A(T − t)(γ1 +2σ1 )( p−1) q−2 diam(Ω ) N /A q−2 |div(|∇z| ∇z)| ≤ (T − t)γ2 +2σ2 (T − t)γ2 +2σ2 q−4 2 diam(Ω ) diam(Ω ) N /A + (q − 2) (T − t)γ2 +2σ2 (T − t)γ2 +2σ2 (T − t)γ2 +2σ2 N (q − 1)(diam(Ω ))q−2 = . A(T − t)(γ2 +2σ2 )(q−1)
(4.1) (4.2)
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F. Li / Nonlinear Analysis 67 (2007) 1387–1402
If 0 ≤ yi ≤ A, we have 1 ≤ V (yi ) ≤ 1 + A/2 and V 0 (yi ) ≤ 0, i = 1, 2. Hence: Z N ( p − 1)(diam(Ω )) p−2 a M1 γ1 (1 + A/2) p−2 + − , w α dx ≤ ∇z) − a z t − div(|∇z| γ1 +1 (γ1 +2σ1 )( p−1) (T − t)γ2 α−N σ2 (T − t) A(T − t) Ω Z bM2 γ2 (1 + A/2) N ( p − 1)(diam(Ω ))q−2 − z β dx ≤ + z t − div(|∇z|q−2 ∇z) − b . γ +1 (γ2 +2σ2 )(q−1) 2 (T − t)γ1 β−N σ1 (T − t) A(T − t) Ω If yi ≥ A, we have V (yi ) ≤ 1 and V 0 (yi ) ≤ −1, i = 1, 2. Then: Z γ − σ1 A N ( p − 1)(diam(Ω )) p−2 p−2 , z α dx ≤ ∇w) − a + wt − div(|∇w| (T − t)γ1 +1 A(T − t)(γ1 +2σ1 )( p−1) Ω Z γ −σA N (q − 1)(diam(Ω ))q−2 w β dx ≤ + . z t − div(|∇z|q−2 ∇z) − b γ +1 (T − t) 2 A(T − t)(γ2 +2σ2 )(q−1) Ω
(4.3) (4.4)
(4.5) (4.6)
If 0 < y1 ≤ A and y2 > A, we have (4.3) and (4.6) hold. If y1 > A and 0 < y2 ≤ A, we have (4.4) and (4.5) hold. Since p, q > 2 and αβ > ( p − 1)(q − 1) > 1, we can choose two constants σ1 , σ2 > 0, which is sufficiently small, such that: 1 + α + σ2 N + σ1 N α 1 − 2σ1 ( p − 1) < , αβ − 1 p−2 1 + β + σ1 N + σ2 Nβ 1 − 2σ2 (q − 1) < . αβ − 1 q −2 Choose two constants γ1 , γ2 > 0 satisfying 1 + α + σ2 N + σ1 N α 1 − 2σ1 ( p − 1) < γ1 < , αβ − 1 p−2 1 + β + σ1 N + σ2 Nβ 1 − 2σ2 (q − 1) < γ2 < , αβ − 1 q −2 then we have (γ1 + 2σ1 )( p − 1) < γ1 + 1 < γ2 α − N σ2 , (γ2 + 2σ2 )(q − 1) < γ2 + 1 < γ1 β − N σ1 , where the condition αβ > ( p − 1)(q − 1) is used. Select A > max{1, γ1 /σ1 , γ2 /σ2 }, then, for sufficiently small T > 0, (4.3)–(4.6) imply that: Z wt − div(|∇w| p−2 ∇w) − a z α dx ≤ 0, (x, t) ∈ Q T , Z Ω q−2 z t − div(|∇z| ∇z) − b w β dx ≤ 0, (x, t) ∈ Q T . Ω
Since ϕ(0) > 0 and ϕ is a continuous function, there exist two positive constants ρ and such that ϕ(x) ≥ for all x ∈ B(0, ρ) ⊂ Ω . Taking T small enough to ensure that B(0, RT σ ) ⊂ B(0, ρ), then z ≤ 0 on ∂Ω × [0, T ]. From (4.1) and (4.2) it follows that (w(x, 0), z(x, 0)) ≤ (K 0 ϕ(x), K 0 ϕ(x)) for sufficiently large M. By Lemma 2.2 we have (w, z) ≤ (u, v) provided that (u 0 (x), v0 (x)) ≥ (K 0 ϕ(x), K 0 ϕ(x)) and (u, v) can exist no later than t = T . This shows that (u, v) blows up in finite time. (2) Since p > 2, q > 2 and αβ = ( p − 1)(q − 1), we can choose two constants l1 , l2 > 1 such that: l1 q −1 α = = . p−1 l2 β Then l2 β = ( p − 1)l1 > 1,
l1 β = (q − 1)l2 > 1.
F. Li / Nonlinear Analysis 67 (2007) 1387–1402
1399
Without loss of generality, we can assume that 0 ∈ Ω . Denote B R (0) a ball centering at 0 with radius R satisfying B R (0) ⊂⊂ Ω . By Lemma 2.2, we need only to show that the solution of problem (1.1) on (B R (0)×[0, T ])×(B R (0)× [0, T ]) blows up in finite time. Let φ(r ) and ψ(r ) (r = |x|) be the solution of the following elliptical problem: −(r N −1 |φ 0 | p−2 φ 0 )0 = r N −1 ,
r ∈ (0, R), φ 0 (0) = 0, φ(R) = 0,
−(r N −1 |ψ 0 |q−2 ψ 0 )0 = r N −1 ,
r ∈ (0, R), ψ 0 (0) = 0, ψ(R) = 0,
and
respectively. Then: φ(r ) = (( p − 1)/ p)(1/N )1/( p−1) (R p/( p−1) − r p/( p−1) ), ψ(r ) = ((q − 1)/q)(1/N )1/(q−1) (R q/(q−1) − r q/(q−1) ). By the assumptions on u 0 (x) and v0 (x), we can choose sufficiently small constant s0 such that: u 0 (r ) ≥ s0l1 φ(r ), v0 (r ) ≥ s0l2 ψ(r ), r ∈ [0, R). RR RR Denote c1 = 0 ψ α dr and c2 = 0 φ β dr , then: Z 1 Z R c1 = ψ α (r )dr = ((q − 1)/q)(1/N )1/(q−1) R 1+qα/( p−1) (1 − s q/(q−1) )qα/(q−1) ds > 1/a, 0
0 R
Z c2 =
β
φ (r )dr = (( p − 1)/ p)(1/N )
1/( p−1)
R
1+ pβ/( p−1)
1
Z
(1 − s p/( p−1) ) pβ/( p−1) ds > 1/b
0
0
for sufficiently large R. Now we consider the Cauchy problem: s 0 (t) = min{(ac1 − 1)/M1 , (bc2 − 1)/M2 }s δ (t),
s(0) = s0 ,
where M1 = (( p − 1)/ p)(1/N )1/( p−1) R p/( p−1) ,
M2 = ((q − 1)/q)(1/N )1/(q−1) R q/(q−1)
and δ = min{( p − 1)l1 − l1 + 1, (q − 1)l2 − l2 + 1}. It is easy to know that there exists a constant T 0 > 0 such that limt→T 0 s(t) = +∞. Define two functions w(r, t) = s l1 φ(r ) and z(r, t) = s l2 ψ(r ), we assert that (w, z) is a subsolution of the problem (1.1) on (B R (0) × [0, T ]) × (B R (0) × [0, T ]), T < T 0 . In fact, a series of computations shows: Z R 1−N N −1 p−2 wt − r (r |wr | wr )r − a z α dr = l1 φs l1 −1 (t)s 0 (t) + s ( p−1)l1 (t) − ac1 s l2 α (t) 0
z t − r 1−N (r N −1 |zr |q−2 zr )r − b
Z
R
= l1 φs l1 −1 (t)(s 0 (t) − (ac1 − 1)φ −1 s ( p−1)l1 −l1 +1 s(t)) ≤ l1 φs l1 −1 (t)(s 0 (t) − M1−1 (ac1 − 1)s ( p−1)l1 −l1 +1 s(t)) ≤ 0, (r, t) ∈ B R (0) × (0, T ),
w β dr = l2 ψs l2 −1 (t)s 0 (t) + s (q−1)l2 (t) − bc2 s l1 β (t)
0
r N −1 |wr | p−2 wr
r =0
= 0,
w(R, t) = s l1 φ(R) = 0, w(r, 0) =
s0l1 φ(r )
≤ u 0 (r ),
= l2 ψs l2 −1 (t)(s 0 (t) − (bc2 − 1)ψ −1 s (q−1)l2 −l2 +1 s(t)) ≤ l2 ψs l2 −1 (t)(s 0 (t) − M2−1 (bc2 − 1)s (q−1)l2 −l2 +1 s(t)) ≤ 0, (r, t) ∈ (x, t) ∈ B R (0) × (0, T ), r N −1 |zr | p−2 zr = 0, t ∈ [0, T ], r =0
z(R, t) = s l2 ψ(R) = 0, z(r, 0) =
s0l2 ψ(r )
t ∈ [0, T ],
≤ v0 (r ),
r ∈ [0, R].
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F. Li / Nonlinear Analysis 67 (2007) 1387–1402
Using Lemma 2.2 we get (w, z) ≤ (u, v) on Q T × Q T . Thus (u, v) must blow up in finite time since (w, z) does. 5. The special case α = q − 1 and β = p − 1 In this section, we investigate the problem (1.1) for the special case α =Rq − 1 and β = p − 1.R Let φ(x) and ψ(x) be two functions defined by (1.5) and (1.6) respectively, and denote ρ = Ω φ p−1 dx and θ = Ω ψ q−1 dx, we first give the following lemma: Lemma 5.1. Suppose that ρ θ ≥ 1/(ab) and (u 0 (x), v0 (x)) satisfies the conditions in Theorem 1.3, then there exist two positive constants k1 and k2 such that u(x, t) ≥ k1 φ(x),
v(x, t) ≥ k2 ψ(x),
(x, t) ∈ Q T .
Proof. By ρ θ ≥ 1/(ab) and the assumptions on u 0 (x) and v0 (x), we can take appropriate constants k1 and k2 such that u 0 (x) ≥ k1 φ(x), v0 (x) ≥ k2 ψ(x), x ∈ Ω and p−1
a θ ≥ k1
q−1
/k2
≥ (b ρ)−1 .
Let w1 (x, t) = k1 φ(x) and w2 (x, t) = k2 ψ(x), then we have: Z q−1 p−1 q−1 p−2 w1t − div(|∇w1 | ∇w1 ) − w2 dx = k1 − ak2 θ ≤ 0, ZΩ p−1 q−1 p−1 q−2 w2t − div(|∇w2 | ∇w2 ) − w1 dx = k2 − bk1 ρ ≤ 0, Ω
w1 (x, t) = 0,
w2 (x, t) = 0,
w1 (x, 0) ≤ u 0 (x),
(x, t) ∈ Q T , (x, t) ∈ Q T ,
(x, t) ∈ ST ,
w2 (x, 0) ≤ v0 (x),
x ∈ Ω.
Therefore, we get (w1 , w2 ) ≤ (u, v) on Q T × Q T by using Lemma 2.2.
Now we prove Theorem 1.6. Proof of Theorem 1.6. (1) Since ρ θ ≤ 1/(ab), then there exist two positive constants K 1 and K 2 such that p−1
a θ ≤ K1
q−1
/K 2
≤ (b ρ)−1
and u 0 (x) ≤ K 1 φ(x), v0 (x) ≤ K 2 ψ(x). Set W (x, t) = K 1 φ(x) and Z (x, t) = K 2 ψ(x), then we have Z p−1 q−1 p−2 Wt − div(|∇W | ∇W ) = K 1 ≥ a K 2 θ ≥ a Z q−1 dx. Ω Z q−1 p−1 Z t − div(|∇ Z |q−2 ∇ Z ) = K 2 ≥ bK 1 θ ≥ b W p−1 dx. Ω
Noticing W (x, t) = Z (x, t) = 0 on ∂Ω × [0, +∞), we obtain (u(x, t), v(x, t)) ≤ (W (x, t), Z (x, t)) on (Ω × [0, +∞)) × (Ω × [0, +∞)) by Lemma 2.2. Therefore, (u(x, t), v(x, t)) exists globally. (2) In view of ρ θ > 1/(ab), we can choose a smooth subdomain Ω1 ⊂⊂ Ω such that ρ˜ θ˜ > 1/(ab), where R R p−1 q−1 ρ˜ = Ω φ1 dx, θ˜ = Ω ψ1 dx, and φ1 , ψ1 are the unique solution of the elliptic problem −div(|∇φ1 | p−2 ∇φ1 ) = 1
in Ω1 ,
φ1 (x) = 0
−div(|∇ψ1 | p−2 ∇ψ1 ) = 1
in Ω1 ,
ψ1 (x) = 0
on ∂Ω1 ,
and on ∂Ω1 ,
respectively. Denote σ = min{infx∈Ω2 k1 φ1 (x), infx∈Ω2 k2 ψ1 (x)}, Ω2 ⊂⊂ Ω1 , then σ > 0, where k1 and k2 are determined by Lemma 5.1.
F. Li / Nonlinear Analysis 67 (2007) 1387–1402
1401
Let (u 1 , v1 ) be the unique solution of the following problem: Z p−2 v1α (x, t)dx, (x, t) ∈ Ω2 × (0, T ], u 1t − div(|∇u 1 | ∇u 1 ) = a Ω2 Z β q−2 u 1 (x, t)dx, (x, t) ∈ Ω2 × (0, T ], v1t − div(|∇v1 | ∇v1 ) = b
(5.1)
Ω2
u 1 (x, t) = σ,
v1 (x, t) = σ,
(x, t) ∈ ∂Ω2 × [0, T ],
u 1 (x, 0) = σ,
v1 (x, 0) = σ,
x ∈ Ω2 .
Using Lemmas 5.1 and 2.2, it is easy to verify that: (u, v) ≥ (u 1 , v1 ) on (Ω2 × [0, T ]) × (Ω2 × [0, T ]). Let K 0 = max{supx∈Ω2 φ1 (x), supx∈Ω2 ψ1 (x)}, consider the Cauchy problem: p−1
K 0 s10 (t) = −s1
q−1
K 0 s20 (t) = −s2 s1 (0) = σ/K 0 ,
˜ q−1 (t), (t) + a θs 2 p−1
(t) + bρs ˜ 1
(t),
(5.2)
s2 (0) = σ/K 0 .
Multiplying the first equation in (5.2) by (ρ˜ + 1), the second equation in (5.2) by (θ˜ + 1) and combining them together, we can get p−1
K 0 (ρ˜ + 1)s10 (t) + K 0 (θ˜ + 1)s20 (t) = (abρ˜ θ˜ − 1)(s1
q−1
(t) + s2
(t)).
Noticing p > 2, q > 2 and abρ˜ θ˜ > 1 we obtain that there exists a constant T 0 > 0 such that: lim (s1 (t) + s2 (t)) = +∞.
t→T 0
Set z 1 (x, t) = s1 (t)φ1 (x), z 2 (x, t) = s2 (t)ψ1 (x), now we assert that (z 1 , z 2 ) is a subsolution of problem (5.1). Calculating directly we can get: Z q−1 p−1 p−2 ˜ q−1 (t) z 1t − div(|∇z 1 | ∇z 1 ) − z 2 dx = φ1 (x)s10 (t) + s1 (t) − a θs 2 Ω2
p−1
≤ K 0 s10 (t) + s1 z 2t − div(|∇z 2 |q−2 ∇z 2 ) −
Z
p−1
Ω2
z1
q−1
˜ (t) − a θs 2
q−1
(t) = 0,
p−1
dx = ψ1 (x)s20 (t) + s2
(t) − bρs ˜ 1
q−1
p−1
≤ K 0 s20 (t) + s2
(t) − bρs ˜ 1
z 1 (x, t) = s1 (t)φ1 (x) = 0,
z 2 (x, t) = s2 (t)ψ1 (x) = 0,
z 1 (x, 0) = s1 (0)φ1 (x) ≤ σ,
z 2 (x, 0) = s2 (0)ψ1 (x) ≤ σ,
(x, t) ∈ Ω2 × (0, T ],
(t)
(t) = 0,
(x, t) ∈ Ω2 × (0, T ],
(x, t) ∈ ∂Ω2 × [0, T ], x ∈ Ω2 .
Using Lemma 2.2 it follows that (z 1 , z 2 ) ≤ (u 1 , v1 ) on Ω2 × [0, T ]. Therefore, (u 1 , v1 ) blows up in finite time. Our conclusion comes from the fact that the relation (u, v) ≥ (u 1 , v1 ) holds on (Ω2 × [0, T ]) × (Ω2 × [0, T ]). Acknowledgement The author thanks the anonymous referee for his careful reading and valuable suggestions. References [1] N.D. Alikakos, L.C. Evans, Continuing of the gradient for weak solutions of a degenerate parabolic equation, J. Math. Pures Appl. 62 (1983) 253–268. [2] F. Andreu, J.M. Maz´on, F. Simondon, J. Toledo, Blow-up for a class of nonlinear parabolic problems, Asymptot. Anal. 29 (2002) 143–155. [3] M.F. Bidanut-V´eron, M. Garc´ıa-Huidobro, Regular and singular solutions of a quasilinear equation with weights, Asymptot. Anal. 28 (2001) 115–150. [4] J.A.A. Crespo, I.P. Alonso, Global behavior of the Cauchy problem for some critical nonlinear parabolic equations, SIAM J. Math. Anal. 31 (2000) 1270–1294. [5] J.I. D´ıaz, Nonlinear Partial Differential Equations and Free Boundaries, in: Elliptic Equations, vol. 1, Pitman, London, 1985.
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F. Li / Nonlinear Analysis 67 (2007) 1387–1402
[6] E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, Berlin, New York, 1993. [7] E. DiBenedetto, On the local behavior of solutions of degenerate parabolic equations with measurable coefficients, Ann. Sc. Norm. Super. Pisa Cl. Sci. 13 (1986) 487–535. [8] M.A. Herrero, J.L. V´azquez, Asymptotic behavior of the solutions of a strongly nonlinear parabolic problem, Ann. Fac. Sci. Toulouse Math. 3 (1981) 113–127. [9] H. Ishii, Asymptotic stability and blowing up of solutions of some nonlinear equations, J. Differential Equations 26 (1977) 291–319. [10] A.S. Kalashnikov, Some problems of the qualitative theory of nonlinear degenerate second order parabolic equations, Russian Math. Surveys 42 (1987) 169–222. [11] T. Kilpel¨ainen, P. Lindqvist, On the Dirichlet boundary value problem for a degenerate parabolic equation, SIAM J. Math. Anal. 27 (1996) 661–683. [12] O.A. Ladyzenkaya, New equation for the description of incompressible fluids and solvability in the large boundary problem for them, Proc. Steklov Inst. Math. 102 (1967) 95–118. [13] H.A. Levine, L.E. Payne, Nonexistence of global weak solutions of classes of nonlinear wave and parabolic equations, J. Math. Anal. Appl. 55 (1976) 329–334. [14] H.A. Levine, L.E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions for the porous medium equation backward in time, J. Differential Equations 16 (1974) 319–334. [15] F.C. Li, Y.P. Chen, C.H. Xie, Asymptotic behavior of solution for nonlocal reaction–diffusion system, Acta Math. Sci. Ser. B Engl. Ed. 23 (2003) 261–273. [16] F.C. Li, S.X. Huang, C.H. Xie, Global existence and blow-up of solution to a nonlocal reaction–diffusion system, Discrete Contin. Dyn. Syst. 9 (2003) 1519–1532. [17] F.C. Li, C.H. Xie, Global and blow-up solutions to a p-Laplacian equation with nonlocal source, Comput. Math. Appl. 46 (2003) 1525–1533. [18] M. Nakao, L p -estimates of solutions of some nonlinear degenerate diffusion equations, J. Math. Soc. Japan 37 (1985) 41–63. [19] M. Nakao, Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations, Nonlinear Anal. 10 (1986) 299–314. [20] Y. Ohara, L ∞ -estimates of solutions of some nonlinear degenerate parabolic equations, Nonlinear Anal. 18 (1992) 413–426. [21] Y. Ohara, Gradient estimates for some quasilinear parabolic equations with nonmonotonic perturbations, Adv. Math. Sci. Appl. 6 (1996) 531–540. ˆ [22] M. Otani, Y. Sugiyama, Lipschitz continuous solutions of some doubly nonlinear parabolic equations, Discrete Contin. Dyn. Syst. 8 (2002) 647–670. [23] C.V. Pao, Blowing-up of solution for a non-local reaction–diffusion problem in combustion theory, J. Math. Anal. Appl. 166 (1992) 591–600. [24] M.M. Porzio, V. Vespri, H¨older estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations 103 (1993) 146–178. [25] P. Souplet, Blow-up in nonlocal reaction–diffusion equations, SIAM J. Math. Anal. 29 (1998) 1301–1334. [26] M. Tsutsumi, Existence and nonexistence of global solutions for nonlinear parabolic equations, Publ. Res. Inst. Math. Sci. 8 (1972) 221–229. [27] M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption, J. Math. Anal. Appl. 132 (1988) 187–212. [28] X. Xu, On the initial-boundary-value problem for u t − div(|∇u| p−2 ∇u) = 0, Arch. Ration. Mech. Anal. 127 (1994) 319–335. [29] H.J. Yuan, Extinction and positivity of the evolution p-Laplacian equation, J. Math. Anal. Appl. 196 (1995) 754–763. [30] J. Zhao, Existence and nonexistence of solutions for u t − div(|∇u| p−2 ∇u) = f (∇u, u, x, t), J. Math. Anal. Appl. 172 (1993) 130–146. [31] S.L. Zhou, A priori L ∞ -estimate and existence of solutions for some nonlinear parabolic equations, Nonlinear Anal. 42 (2000) 887–904.
Global existence and blow-up of solutions to a nonlocal quasilinear degenerate parabolic systemI Fucai Li ∗ Department of Mathematics, Nanjing University, Nanjing 210093, PR China Received 25 March 2005; accepted 19 July 2006
Abstract This paper investigates the properties of nonnegative solutions of a quasilinear degenerate parabolic system Z p−2 ∇u) = a v α (x, t)dx, u t − div(|∇u| Ω Z vt − div(|∇v|q−2 ∇v) = b u β (x, t)dx Ω
with zero Dirichlet boundary conditions in a smooth bounded domain Ω ⊂ R N (N ≥ 1), where p, q > 2, α, β ≥ 1, and a, b > 0 are positive constants. Under appropriate hypotheses, we first establish the local existence and uniqueness of solutions, then we show that whether or not the solution blows up in finite time depends on the initial data and the relations between αβ ( p −R 1)(q − 1). In the special case Rof α = q − β = p − 1, we conclude that the solution exists globally if R and R 1 and p−1 dx q−1 dx ≤ 1/(ab), while if p−1 dx q−1 dx > 1/(ab) then the solution blows up in finite time. Here φ ψ φ ψ Ω Ω Ω Ω φ(x) and ψ(x) denote the unique solution of the following elliptic problem −div(|∇φ| p−2 ∇φ) = 1 in Ω , φ(x)|∂ Ω = 0 and −div(|∇ψ|q−2 ∇ψ) = 1 in Ω , ψ(x)|∂ Ω = 0, respectively. c 2006 Elsevier Ltd. All rights reserved.
MSC: 35K65; 35K50; 35K57; 35A07 Keywords: Quasilinear degenerate parabolic system; Nonlinear source; Global existence; Blow-up
1. Introduction and main results This paper deals with the following nonlocal quasilinear degenerate parabolic system in a smooth bounded domain Ω ⊂ R N (N ≥ 1):
I This project was supported partially by NSFC (Grant 10501047), and partially by Nanjing University Talent Development Foundation. ∗ Fax: +86 2583597130.
E-mail address: fucai [email protected]. c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.07.024
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F. Li / Nonlinear Analysis 67 (2007) 1387–1402
u t − div(|∇u| p−2 ∇u) = a vt − div(|∇v|q−2 ∇v) = b
Z ZΩ Ω
v α (x, t)dx,
(x, t) ∈ Ω × (0, T ],
u β (x, t)dx,
(x, t) ∈ Ω × (0, T ],
v(x, t) = 0,
u(x, t) = 0, u(x, 0) = u 0 (x),
(1.1)
(x, t) ∈ ∂Ω × [0, T ],
v(x, 0) = v0 (x),
x ∈ Ω,
where p, q > 2, α, β ≥ 1, and a, b > 0 are positive constants. In the last three decades, many authors have studied the following degenerate parabolic problem: u t − div(|∇u| p−2 ∇u) = f (u),
(x, t) ∈ Ω × (0, T ],
u(x, t) = 0, (x, t) ∈ ∂Ω × [0, T ], u(x, 0) = u 0 (x), x ∈ Ω
(1.2)
under different conditions. In [1,6,10,11,18–22,24,26,31], the existence, uniqueness and regularity of solutions were obtained. When f (u) = −u q , q > 0 or f (u) ≡ 0 extinction phenomenon of the solution may appear, see [27,29]; however, if f (u) = u q , q > 1 the solution may blow up in finite time, see [9,13,26,30]. Roughly speaking, the results in [9,13,26,30] read: (1) the solution u exists globally if q < p − 1; (2) u blows up in finite time if q > p − 1 and u 0 (x) is sufficiently large. The (1) and (2) stated above are also true for the nonlocal degenerate parabolic equation: Z p−2 u t − div(|∇u| ∇u) = u q dx Ω
with null Dirichlet boundary condition, see [17] for details. The purpose of this paper is to investigate qualitative properties of the solutions to the complicated problem (1.1). The system in (1.1) is degenerate and appears, for example, in the theory of non-Newtonian filtration fluids [1, 12]. From a physical point of view, we need only to consider the nonnegative solutions. Moreover, if we assume that u 0 (x), v0 (x) ≥ 0 in Ω , by Lemma 2.2 (see Section 2 below), we can obtain that (u(x, t), v(x, t)) ≥ (0, 0) a.e. in (Ω × (0, T )) × (Ω × (0, T )). Thus we only consider the nonnegative solutions in later sections. Since the equations in (1.1) are degenerate for |∇u| = 0 or |∇v| = 0, one cannot expect the existence of a classical solution for (1.1). The most of studies on p-Laplacian equations concerned with weak solutions, see [1,4,6–8,10,11,18–22,24,26,28,31]. We start from the definition of solution. Let Q T = Ω × (0, T ], ST = ∂Ω × [0, T ], T > 0. Throughout this paper we assume that: 1, p 1,q (A) u 0 (x) ∈ C(Ω ) ∩ W0 (Ω ), v0 (x) ∈ C(Ω ) ∩ W0 (Ω ) and ∂u 0 (x)∂η < 0, ∂v0 (x)∂η < 0 on the boundary ∂Ω , where η denotes the unit outer normal vector on ∂Ω . Definition 1.1. A pair of functions (u, v) is called a solution of the problem (1.1) on Q T × Q T if and only if 1, p 1,q u ∈ C(0, T ; L ∞ (Ω )) ∩ L p (0, T ; W0 (Ω )), v ∈ C(0, T ; L ∞ (Ω )) ∩ L q (0, T ; W0 (Ω )), u t ∈ L 2 (0, T ; L 2 (Ω )), vt ∈ L 2 (0, T ; L 2 (Ω )), u(x, t)|t=0 = u 0 (x), v(x, t)|t=0 = v0 (x), and the equalities Z Z Z t2 Z u(x, t2 )ψ1 (x, t2 )dx − u(x, t1 )ψ1 (x, t1 )dx = uψ1t dxdt Ω
Ω
t2
Z
Z
|∇u| p−2 ∇u · ∇ψ1 dxdt + a
− t1
Z Ω
Ω
Z
t2
Z
t1
Z
Ω
Ω
Ω
ψ1 (x, t)
v(x, t1 )ψ2 (x, t1 )dx =
|∇v|q−2 ∇q · ∇ψ2 dxdt + b
−
Z
t1
v(x, t2 )ψ2 (x, t2 )dx −
Ω
t1 t2
Z
Z
t2
t2
t1
Z Ω
Ω
Z
t1
Z
Z
ψ2 (x, t)
Ω
u α (x, t)dx dxdt,
(1.3)
vψ2t dxdt
Z Ω
u β (x, t)dx dxdt
(1.4)
hold for all 0 < t1 < t2 < T , where ψ1 (x, t), ψ2 (x, t) ∈ C 1,1 (Q T ) such that ψ1 (x, T ) = ψ2 (x, T ) = 0 and ψ1 (x, t) = ψ2 (x, t) = 0 on ST .
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F. Li / Nonlinear Analysis 67 (2007) 1387–1402
Similarly, to define a subsolution (u(x, t), v(x, t)) we need only to require that ψ1 (x, t), ψ2 (x, t) ≥ 0, (u(x, 0), v(x, 0)) ≤ (u 0 (x), v0 (x)) on Ω × Ω , (u(x, t), u(x, t)) ≤ (0, 0) on ST × ST and the equalities in (1.3) and (1.4) are replaced by ≤. A supersolution can be defined similarly. Definition 1.2. We say the solution (u, v) of the problem (1.1) blows up in finite time if there exists a positive constant T ∗ , T ∗ < ∞ such that lim |u(·, t)| L ∞ (Ω ) + |v(·, t)| L ∞ (Ω ) = +∞. t→T ∗−
We say the solution (u, v) exists globally if sup t∈(0,+∞)
|u(·, t)| L ∞ (Ω ) + |v(·, t)| L ∞ (Ω ) < +∞.
Our first result is on the local existence and uniqueness of solutions. Theorem 1.3. Suppose that (u 0 , v0 ) ≥ (0, 0) and satisfies the condition (A), then there exists a constant T0 > 0 such 1, p that the problem (1.1) admits a unique solution (u, v) ∈ Q T0 × Q T0 , u ∈ C(0, T0 ; L ∞ (Ω )) ∩ L p (0, T0 ; W0 (Ω )), 1,q and v ∈ C(0, T0 ; L ∞ (Ω )) ∩ L q (0, T0 ; W0 (Ω )). The next two theorems show that the solution (u, v) may exist globally or blow up in finite time depending on p and q, the exponents α and β, the initial data u 0 (x) and v0 (x) and the Lebesgue’s measure of Ω . Theorem 1.4. Suppose that (u 0 , v0 ) satisfies the conditions in Theorem 1.3 and one of the following conditions holds: (1) αβ < ( p − 1)(q − 1); (2) αβ = ( p − 1)(q − 1) and |Ω | is sufficiently small; (3) αβ > ( p − 1)(q − 1) and u 0 (x), v0 (x) are sufficiently small. Then the solution of problem (1.1) exists globally. Theorem 1.5. Suppose that (u 0 , v0 ) satisfies the conditions in Theorem 1.3 and one of the following conditions holds: (1) αβ > ( p − 1)(q − 1) and u 0 (x), v0 (x) are sufficiently large, (2) αβ = ( p − 1)(q − 1) and Ω contains a sufficiently large ball. Then the solution of problem (1.1) blows up in finite time. Let φ(x) and ψ(x) be the unique solution of the following elliptical problem: −div(|∇φ| p−2 ∇φ) = 1
in Ω ,
φ(x) = 0
−div(|∇ψ|q−2 ∇ψ) = 1
in Ω ,
ψ(x) = 0
on ∂Ω
(1.5)
and on ∂Ω ,
(1.6)
respectively. In the special case of αβ = ( p − 1)(q − 1) with α = q − 1 and β = p − 1, we have the following optimal results: Theorem 1.6. Suppose that (u 0 , v0 ) satisfies the conditions in Theorem 1.3, and that α = q − 1 and β = p − 1. R R (1) If RΩ φ p−1 dx RΩ ψ q−1 dx ≤ 1/(ab), then the solution of problem (1.1) exists globally; (2) If Ω φ p−1 dx Ω ψ q−1 dx > 1/(ab), then the solution of problem (1.1) blows up in finite time. Remark 1.7. In the case of linear diffusions, namely, the problem (1.1) with p = q = 2, the solution exists globally if αβ equals to the diffusive power, see [15,16]. However, the above conclusion is not true for nonlinear diffusions, our results in Theorems 1.4–1.6 explain that how the nonlinear diffusions affect the behaviour of the solution. This paper is organized as follows. In Section 2, we shall establish the local existence theory of solution and prove Theorem 1.3. In Section 3, we shall discuss the global existence of solution and prove Theorem 1.4. In Section 4, we shall discuss the blow-up results of the solution and prove Theorem 1.5. The special case α = q − 1 and β = p − 1 will be discussed in the last section.
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F. Li / Nonlinear Analysis 67 (2007) 1387–1402
2. Local solution We first give a maximum principle for the nondegenerate parabolic system, whose proof is similar to that given in [23], we omit it here. Lemma 2.1. Suppose that w(x, t), z(x, t) ∈ C 2,1 (Q T ) ∩ C(Q T ) and satisfy Z N N X X wt − ai j (x, t)wxi x j + ci (x, t)wxi ≥ e(x, t)z(x, t)dx, (x, t) ∈ Q T , i, j=1
zt −
N X
Ω
i=1
bi j (x, t)z xi x j +
i, j=1
N X
di (x, t)z xi ≥
Z
i=1
w(x, t) ≥ 0,
z(x, t) ≥ 0,
(x, t) ∈ ST ,
w(x, 0) ≥ 0,
z(x, 0) ≥ 0,
(x, t) ∈ Ω ,
g(x, t)w(x, t)dx, Ω
(x, t) ∈ Q T ,
where ai j (x, t), bi j (x, t), c(x, t), d(x, t), e(x, t) and g(x, t) are real bounded continuous functions, e(x, t), g(x, t) ≥ 0, and the matrixes A = (ai j ), B = (bi j ) are symmetric and positive definite. Then w(x, t) ≥ 0, z(x, t) ≥ 0, (x, t) ∈ Q T . The following comparison lemma of problem (1.1) plays a crucial role in the proof of our results. Lemma 2.2. Suppose that (u, v), (u, v) are a nonnegative subsolution and supersolution of the problem (1.1) on Q T × Q T , respectively. Then (u, v) ≤ (u, v) a.e. on Q T × Q T . Proof. For any small δ 0, ρδ (ξ ) = ξ/δ, 1,
> 0, set ξ ≤ 0, 0 < ξ < δ, ξ ≥ δ.
Then ρδ (ξ ) is a piecewise differentiable function. Let ψ1 (x, t) = ρδ ((u − u)(x, t)) and ψ2 (x, t) = ρδ ((v − v)(x, t)), it is easy to verify that ψ1 (x, t) and ψ2 (x, t) are admissible test functions in (1.3) and (1.4). Since (u, v) and (u, v) are subsolution and supersolution of (1.1), let t1 = τ , t2 = τ + h, τ , h > 0, τ + h < T and w = u − u, z = v − v, then we obtain Z Z w(x, τ + h)ψ1 (x, τ + h)dx − w(x, τ )ψ1 (x, τ )dx Ω
Ω
τ +h
Z
Z
≤ τ
Ω τ +h
Z
Ω
Z
τ
Ω
ψ1 (x, s)
Z
τ
Ω
Z
α
(|∇u| p−2 ∇u − |∇u| p−2 ∇u) · ∇ψ1 dxds α
(v − v )dx dxds, Z z(x, τ + h)ψ2 (x, τ + h)dx − z(x, τ )ψ2 (x, τ )dx +a
Z
wψ1s dxds −
τ +h
Z
(2.1)
Ω
Ω
τ +h
Z
Z zψ2s dxds −
≤ τ
Ω τ +h
Z +b
τ
τ +h
Z Z Ω
ψ2 (x, s)
τ
Z Ω
Z Ω
(|∇v|q−2 ∇v − |∇v|q−2 ∇v) · ∇ψ2 dxds
(u β − u β )dx dxds.
Dividing (2.1) and (2.2) by h and integrating τ over (0, t) gives Z t Z 1 (w(x, τ + h)ψ1 (x, τ + h) − w(x, τ )ψ1 (x, τ ))dxdτ 0 h Ω Z Z t Z τ +h Z Z t Z τ +h Z 1 1 α α ≤ wψ1s dxdsdτ + a ψ1 (x, s) (v − v )dx dxdsdτ 0 h τ Ω 0 h τ Ω Ω
(2.2)
F. Li / Nonlinear Analysis 67 (2007) 1387–1402
Z
t
Z
1 h
τ +h
1391
Z
(|∇u| p−2 ∇u − |∇u| p−2 ∇u) · ∇ψ1 dxdsdτ, − τ Ω 0 Z t Z 1 (z(x, τ + h)ψ2 (x, τ + h) − z(x, τ )ψ2 (x, τ ))dxdτ h 0 Ω Z Z t Z τ +h Z Z t Z τ +h Z 1 1 β β zψ2s dxdsdτ + b ψ2 (x, s) (u − u )dx dxdsdτ ≤ 0 h τ Ω Ω Ω 0 h τ Z t Z τ +h Z 1 (|∇v|q−2 ∇v − |∇v|q−2 ∇v) · ∇ψ2 dxdsdτ. − h τ Ω 0
(2.3)
(2.4)
By the properties of Steklov’s averages ([6], Lemma 1.3.2), we get t
Z 0
t
Z 0
t
Z 0
t
Z 0
1 h
Z
τ +h
Z
τ
Ω
wψ1s dxdsdτ →
Z tZ 0 t
Ω
wψ1s dxds
as h → 0+ ,
Z Z Z Z 1 τ +h zψ2s dxdsdτ → zψ2s dxds as h → 0+ , h τ Ω 0 Ω Z Z 1 τ +h (|∇u| p−2 ∇u − |∇u| p−2 ∇u) · ∇ψ1 dxdsdτ h τ Ω Z tZ → (|∇u| p−2 ∇u − |∇u| p−2 ∇u) · ∇ψ1 dxds as h → 0+ , 1 h
Z
0 Ω τ +h Z
τ
Ω
(2.5) (2.6)
(2.7)
(|∇v|q−2 ∇v − |∇v|q−2 ∇v) · ∇ψ2 dxdsdτ
Z tZ
(|∇v|q−2 ∇v − |∇v|q−2 ∇v) · ∇ψ2 dxds as h → 0+ , 0 Ω Z Z Z 1 τ +h α α ψ1 (x, s) (v − v )dx dxdsdτ h τ Ω Ω Z Z tZ → ψ1 (x, s) (v α − v α )dx dxds as h → 0+ , →
t
Z 0
t
Z 0
1 h
Z
0 Ω τ +h Z
τ
Ω
Z tZ → 0
Ω
ψ2 (x, s) ψ2 (x, s)
Z Ω
Z Ω
(u β − u β )dx dxdsdτ
(u β − u β )dx dxds
as h → 0+ .
Ω
0
t
(2.9)
Ω
Now we claim that Z t Z 1 (w(x, τ + h)ψ1 (x, τ + h) − w(x, τ )ψ1 (x, τ ))dxdτ 0 h Ω Z → (w(x, t)ψ1 (x, t) − w(x, 0)ψ1 (x, 0))dx as h → 0+ , Ω Z t Z 1 (z(x, τ + h)ψ2 (x, τ + h) − z(x, τ )ψ2 (x, τ ))dxdτ 0 h Ω Z → (z(x, t)ψ2 (x, t) − z(x, 0)ψ2 (x, 0))dx as h → 0+ . In fact, Z
(2.8)
Z 1 (w(x, τ + h)ψ1 (x, τ + h) − w(x, τ )ψ1 (x, τ ))dxdτ h Ω
(2.10)
(2.11)
(2.12)
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F. Li / Nonlinear Analysis 67 (2007) 1387–1402
Z Z 1 t 1 t w(x, τ + h)ψ1 (x, τ + h)dxdτ − w(x, τ )ψ1 (x, τ )dxdτ = Ω h 0 Ω h 0 Z Z Z Z 1 t 1 t+h = w(x, s)ψ1 (x, s)dxds − w(x, s)ψ1 (x, s)dxds Ω h 0 Ω h h Z Z Z Z 1 t+h 1 h = w(x, s)ψ1 (x, s)dxds − w(x, s)ψ1 (x, s)dxds Ω h t Ω h 0 Z Z w(x, 0)ψ1 (x, 0)dx as h → 0+ . w(x, t)ψ1 (x, t)dx − → Z
Z
Ω
Ω
The equality (2.12) can be obtained similarly. By (2.1)–(2.12) we obtain Z tZ Z Z wρδ0 (w)ws dxds w(x, 0)ρδ (w(x, 0))dx + w(x, t)ρδ (w(x, t))dx ≤ 0 Ω ΩZ Z Ω t (|∇u| p−2 ∇u − |∇u| p−2 ∇u) · ∇ρδ (u − u)dxds − 0Z Ω Z tZ ρδ (w(x, s)) (v α − v α )dx dxds, +a Z Ω
z(x, t)ρδ (z(x, t))dx ≤
(2.13)
Ω
Ω
0
Z tZ z(x, 0)ρδ (z(x, 0))dx + zρδ0 (z)z s dxds ΩZ Z 0 Ω t − (|∇v|q−2 ∇v − |∇v|q−2 ∇v) · ∇ρδ (v − v)dxds 0 Ω Z Z tZ +b ρδ (z(x, s)) (u β − u β )dx dxds. Z
Ω
0
(2.14)
Ω
Now we deal with the terms in (2.13) and (2.14). First, we have Z Z tZ Z tZ Z tZ α α ρδ (w(x, s)) (v − v )dx dxds ≤ α|Ω | η1 (v − v)+ dxds ≤ M1 (v − v)+ dxds, 0 Ω Ω 0 Ω 0 Ω Z Z tZ Z tZ Z tZ ρδ (z(x, s)) (u β − u β )dx dxds ≤ β|Ω | η2 (u − u)+ dxds ≤ M2 (u − u)+ dxds 0
Ω
Ω
0
Ω
0
Ω
for some positive constants M1 and M2 , and Z t Z Z tZ 0 w+ |ρδ0 (w)||ws |dxds ≤ (w)w dxds wρ s δ 0Z ΩZ 0 Ω 1 δ = w+ |ws |dxds → 0 as δ → 0+ , δ 0 Ω Z t Z Z tZ 0 zρδ (z)z s dxds ≤ z + |ρδ0 (z)||z s |dxds 0 Ω 0Z ΩZ 1 δ = z + |z s |dxds → 0 as δ → 0+ . δ 0 Ω Second, by Lemma 1.4.4 in [6], we get (|∇u| p−2 ∇u − |∇u| p−2 ∇u) · ∇ρδ (u − u) ≥ min{0, γ1 |∇(u − u)+ | p }, (|∇v|q−2 ∇v − |∇v|q−2 ∇v) · ∇ρδ (v − v) ≥ min{0, γ2 |∇(v − v)+ |q } R R for some γ1 , γ2 > 0. Finally, we have Ω w(x, 0)ρδ (w(x, 0))dx ≡ 0, Ω z(x, 0) × ρδ (z(x, 0))dx ≡ 0 and ρδ0 ≥ 0 a.e. in R, wρδ0 (w(x, t))ws , zρδ0 (z(x, t))z s increase and tend to w+ , z + as δ → 0+ , respectively. Hence we may let δ → 0+ in (2.13) and (2.14) to yield Z tZ Z w+ (x, t)dx ≤ (a M1 + 1) z + (x, s)dxds, Ω
0
Ω
F. Li / Nonlinear Analysis 67 (2007) 1387–1402
Z Ω
Hence, Z Ω
z + (x, t)dx ≤ (bM2 + 1)
Z tZ 0
Ω
1393
w+ (x, s)dxds.
(w+ (x, t) + z + (x, t))dx ≤ (a M1 + bM2 + 1)
Z tZ 0
Ω
(w+ (x, s) + z + (x, s))dxds.
R By the Gronwall’s inequality we obtain Ω (w+ (x, t) + z + (x, t))dx = 0, i.e. u ≤ u, v ≤ v a.e. on Q T . This completes the proof. Proof of Theorem 1.3. Consider the following approximate problems for the problem (1.1): Z p−2 u nt − div (|∇u n |2 + εn ) 2 ∇u n = a vnα (x, t)dx, (x, t) ∈ Q T , Ω Z q−2 2 2 u βn (x, t)dx, (x, t) ∈ Q T , vnt − div (|∇vn | + σn ) ∇vn = b Ω
u n (x, t) = 0, u n (x, 0) =
u n (x, t) = 0,
u ε0n (x),
(2.15)
(x, t) ∈ ST ,
vn (x, 0) = u σ0n (x),
x ∈ Ω.
Here {εn }, {σn } are strictly decreasing sequences, 0 < εn , σn < 1, and εn → 0+ , σn → 0+ as n → ∞. u ε0n ∈ C0∞ (Ω ) and v0σn ∈ C0∞ (Ω ) are approximation functions for the initial data u 0 (x) and v0 (x), respectively. |u ε0n | L ∞ (Ω ) ≤ 1, p |u 0 | L ∞ (Ω ) , |∇u ε0n | L ∞ (Ω ) ≤ |∇u 0 | L ∞ (Ω ) for all εn , and u ε0n → u 0 strongly in W0 (Ω ); |v0σn | L ∞ (Ω ) ≤ |v0 | L ∞ (Ω ) , 1,q |∇v0σn | L ∞ (Ω ) ≤ |∇v0 | L ∞ (Ω ) for all σn , and v0σn → v0 strongly in W0 (Ω ). (2.15) is a nondegenerate problem for each fixed εn and σn ; it is easy to prove that it admits a classic solution (u n , vn ) by using the Schauder’s fixed point theorem. The uniqueness of solutions is obtained by using Lemma 2.1. To find limit functions u(x, t) and v(x, t) of the sequence {(u n , vn )}, we divide our proof into four steps: Step 1: There exist a small constant T0 > 0 and a positive constant M1 , independent of n, such that: |u n | L ∞ (Q T0 ) , |vn | L ∞ (Q T0 ) ≤ M1 .
(2.16)
To this end, we consider the following Cauchy problems: dU1 dV1 β = a|Ω |V1α , = b|Ω |U1 , dt dt U1 (0) = |u 0 | L ∞ (Ω ) , V1 (0) = |v0 | L ∞ (Ω ) ,
(2.17)
dV2 dU2 β = −a|Ω |V2α , = −b|Ω |U2 , dt dt U2 (0) = −|u 0 | L ∞ (Ω ) , V2 (0) = −|v0 | L ∞ (Ω ) .
(2.18)
It is easy to verify that there exists a constant t0 ∈ (0, T ) such that (2.17) and (2.18) admits a solution (U1 (t), V1 (t)) and (U2 (t), V2 (t)) on [0, t0 ] respectively, moreover, the t0 depends only on |u 0 | L ∞ (Ω ) , |v0 | L ∞ (Ω ) . By Lemma 2.2 we get |u n (x, t)|, |vn (x, t)| ≤ max{U1 (t), V1 (t), −U2 (t), −V2 (t)}. Setting T0 = t0 /2 and M1 = max{U1 (t), V1 (t), −U2 (t), −V2 (t)}, we draw our conclusion. Step 2: There exists a constant M2 > 0, independent of n, such that: |∇u n | L p (Q T0 ) , |∇vn | L q (Q T0 ) ≤ M2 .
(2.19)
In fact, multiplying the first equation in (2.15) by u n , the second equation in (2.15) by vn and integrating the results over Q T0 we have: Z Z T0 Z p−2 1 2 u n (x, T0 )dx + (|∇u n |2 + εn ) 2 |∇u n |2 dxdt 2 Ω 0 Ω Z Z Z T0 Z 1 εn 2 α = (u (x)) dx + a u n (x, t)dx vn (x, t)dx dt, 2 Ω 0 0 Ω Ω
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Z Z T0 Z q−2 1 vn2 (x, T0 )dx + (|∇vn |2 + σn ) 2 |∇vn |2 dxdt 2 Ω 0 Ω Z Z T0 Z Z 1 vn (x, t)dx u βn (x, t)dx dt. = (v0σn (x))2 dx + b 2 Ω 0 Ω Ω By |u ε0n | L ∞ (Ω ) ≤ |u 0 | L ∞ (Ω ) , |v0σn | L ∞ (Ω ) ≤ |v0 | L ∞ (Ω ) and (2.16) we get Z T0 Z 1 |∇u n | p dxdt ≤ |u 0 |2L ∞ (Ω ) + T0 |Ω |2 M1α+1 , 2 0 Ω Z T0 Z 1 β+1 |∇vn |q dxdt ≤ |v0 |2L ∞ (Ω ) + T0 |Ω |2 M1 . 2 0 Ω Set M2 = max
1 1 β+1 |u 0 |2L ∞ (Ω ) + aT0 |Ω |2 M1α+1 , |v0 |2L ∞ (Ω ) + bT0 |Ω |2 M1 , 2 2
we draw our conclusion. Step 3: There exists a constant M3 > 0, independent of n, such that: |u nt | L 2 (Q T ) , |vnt | L 2 (Q T ) ≤ M3 . 0
(2.20)
0
To do so, multiplying the first equation in (2.15) by u nt , the second equation in (2.15) by vnt and integrating the results over Q T0 we get: Z T0 Z Z T0 Z p−2 u 2nt (x, t)dxdt = − (|∇u n |2 + εn ) 2 ∇u n · ∇u nt dxdt 0 Ω 0Z Ω Z T0 Z α +a u nt (x, t) vn (x, t)dx dxdt. T0
Z 0
Z Ω
Z
0 T0
Ω
Z
Ω
q−2
(|∇vn |2 + σn ) 2 ∇vn · ∇vnt dxdt 0Z Ω Z T0 Z +b vnt (x, t) u βn (x, t)dx dxdt.
2 (x, t)dxdt = − vnt
0
|u ε0n | L ∞ (Ω )
Ω
Ω
|u 0 | L ∞ (Ω ) , |v0σn | L ∞ (Ω )
By H¨older’s inequality, ≤ ≤ |v0 | L ∞ (Ω ) , inequality (2.16) and the equalities Z T0 Z Z Z p−2 1 1 (|∇u n |2 + εn ) 2 ∇u n · ∇u nt dxdt = (|∇u n (x, T0 )|2 + εn ) p/2 dx − (|∇u ε0n |2 + εn ) p/2 dx, 2 Ω 2 Ω 0 Ω Z T0 Z Z Z q−2 1 1 (|∇vn |2 + σn ) 2 ∇vn · ∇vnt dxdt = (|∇vn (x, T0 )|2 + σn )q/2 dx − (|∇v0σn |2 + σn )q/2 dx, 2 Ω 2 Ω 0 Ω we obtain: Z T0 Z
Z Z 1 1 2 p/2 ≤ − (|∇u n (x, T0 )| + εn ) dx + (|∇u ε0n |2 + εn ) p/2 dx 2 Ω 2 Ω 0 Ω 2 Z T0 Z Z T0 Z 1 1 a 2 α 2 2 + ηa|Ω | u nt dxdt + |Ω | vn dx dt ≤ M 0 , η 0 Ω 0 Ω Z Z Z T0 Z 1 1 2 q/2 2 vnt dxdt ≤ − (|∇vn (x, T0 )| + σn ) dx + (|∇v0σn |2 + σn )q/2 dx 2 Ω 2 Ω 0 Ω 2 Z T0 Z Z T0 Z 1 1 b 2 β 2 2 + ηb|Ω | vnt dxdt + |Ω | u n dx dt ≤ M 0 η 0 Ω 0 Ω u 2nt dxdt
for some M 0 > 0 and some small η > 0.
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F. Li / Nonlinear Analysis 67 (2007) 1387–1402
Therefore, by virtue of (2.16), (2.19) and (2.20) and the Ascoli–Arzel´a theorem, we can choose a subsequence, still denoted by {(u n , vn )} for convenience, such that: u n → u, vn → v
a.e. for (x, t) ∈ Q T0 ,
(2.21)
∇u n → ∇u
weakly in L (0, T0 ; L (Ω )),
(2.22)
∇vn → ∇v
weakly in L q (0, T0 ; L q (Ω )),
(2.23)
p
u nt → u t , vnt → vt |∇u n |
p−2
(u n )xi → wi
|∇vn |q−2 (vn )xi → z i
p
weakly in L 2 (0, T0 ; L 2 (Ω )), p/( p−1)
weakly in L
(0, T0 ; L
(2.24) p/( p−1)
(Ω )),
(2.25)
weakly in L q/(q−1) (0, T0 ; L q/(q−1) (Ω )).
(2.26)
Step 4: we show that wi = xi and z i = xi . Multiplying the first equation in (2.15) by ψ1 (u n − u), the second equation in (2.15) by ψ2 (vn − v) and integrating the results over Q T0 we have: Z T0 Z Z T0 Z p−2 ψ1 (u n − u)u nt dxdt + ψ1 (|∇u n |2 + εn ) 2 ∇u n · ∇(u n − u)dxdt 0 Ω 0 Ω Z Z T0 Z Z T0 Z p−2 α 2 ψ1 (u n − u) vn (x, t)dx dxdt, + (u n − u)(|∇u n | + εn ) 2 ∇u n · ∇ψ1 dxdt = a |∇v|q−2 v
|∇u| p−2 u
Ω
0
T0
Z 0
Z
0
ψ2 (vn − v)vnt dxdt +
T0
Z
Z
ψ2 (|∇vn |2 + σn )
Ω
Ω
q−2 2
∇vn · ∇(vn − v)dxdt Z Z T0 Z Z T0 Z q−2 + (vn − v)(|∇vn |2 + σn ) 2 ∇vn · ∇ψ2 dxdt = b ψ2 (vn − v) u βn (x, t)dx dxdt. Ω
0
Ω
Ω
0
0
Ω
Ω
Using (2.16), (2.21) and (2.24) we get: Z T0 Z lim ψ1 |∇u n | p−2 ∇u n · ∇(u n − u)dxdt = 0, n→∞ 0
Z lim
n→∞ 0
Ω
T0
Z
ψ2 |∇vn |q−2 ∇vn · ∇(vn Ω C 1,1 (Q T0 ), ψ1 , ψ2 ≥ 0.
− v)dxdt = 0,
where ψ1 , ψ2 ∈ The left arguments are as same as those of Theorem 2.1 in [30], so we omit them. We complete the existence part by a standard limiting process. The uniqueness of the solution is obvious. In fact, assume that (u 1 , v1 ), (u 2 , v2 ) are two nonnegative solutions of (1.1), using Lemma 2.2 repeatedly, we can get u 1 = u 2 , v1 = v2 a.e. in Ω × [0, T0 ]. 3. Global existence of solution In this section we investigate the global existence property of solution to the problem (1.1). Our approach is a combination of comparison principle (see Lemma 2.2) and upper solution techniques, which is quite different from that used in [9,13,14,26,30] to study the problem (1.2). Proof of Theorem 1.4. Denote φ(x) and ψ(x) be the unique solution of the elliptical problems defined in (1.5) and (1.6), respectively. Then we have φ(x), ψ(x) > 0 in Ω , ∂φ(x)/∂η < 0 and ∂ψ(x)/∂η < 0 on the boundary ∂Ω , and there exist two positive constants M1 and M2 such that: sup φ(x) = M1 ,
sup ψ(x) = M2 ,
x∈Ω
x∈Ω
(3.1)
see [3,5]. Let W (x, t) = K 1 (φ(x) + 1) and Z (x, t) = K 2 (ψ(x) + 1), where K 1 and K 2 > 0 will be determined later. (1) In case of αβ < ( p − 1)(q − 1), we can choose sufficiently large constants K 1 and K 2 such that K 1 ≥ u 0 (x), K 2 ≥ v0 (x) and p−1
K1
≥ a|Ω |(M2 + 1)α K 2α ,
q−1
K2
β
≥ b|Ω |(M1 + 1)β K 1 .
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F. Li / Nonlinear Analysis 67 (2007) 1387–1402
In fact, we can take them freely provided that: q−1 ( p−1)(q−1)−αβ q+α−1 αβ α , K 1 > max max u 0 (x), ab q−1 |Ω | q−1 (M1 + 1) q−1 (M2 + 1)α x∈Ω p−1 ( p−1)(q−1)−αβ p+β−1 β αβ K 2 > max max v0 (x), a p−1 b|Ω | p−1 (M1 + 1)β (M2 + 1) p−1 . x∈Ω Thus, Z Z α dx, ≥ a|Ω |(M2 + 1)α K 2α ≥ a Ω Z β ≥ b|Ω |(M1 + 1)β K 1 ≥ b W β dx.
p−1
Wt − div(|∇W | p−2 ∇W ) = K 1
q−1
Z t − div(|∇ Z |q−2 ∇ Z ) = K 2
(3.2) (3.3)
Ω
Noticing W (x, t) = Z (x, t) = 0 on ∂Ω × [0, +∞), W (x, 0) ≥ u 0 (x) and Z (x, 0) ≥ v0 (x) in Ω , we obtain (u(x, t), v(x, t)) ≤ (W (x, t), Z (x, t)) in (Ω × [0, +∞)) × (Ω × [0, +∞)) by Lemma 2.2. Hence (u(x, t), v(x, t)) exists globally. (2) In this case, αβ = ( p − 1)(q − 1). We choose K 1 ≥ u 0 (x), K 2 ≥ v0 (x), then (3.2) and (3.3) hold provided that: − q−1 q+α−1 αβ α α q−1 q−1 |Ω | ≤ ab (M1 + 1) (M2 + 1) and − p−1 p+β−1 αβ β β |Ω | ≤ a p−1 b(M1 + 1) (M2 + 1) p−1 . Similarly to the case (1), we obtain that (u(x, t), v(x, t)) exists globally. (3) In this case, αβ > ( p − 1)(q − 1). To ensure (3.2) and (3.3) hold, we need only to choose sufficiently positive small K 1 , K 2 > 0 such that q−1 − αβ−( p−1)(q−1) q+α−1 αβ α α q−1 q−1 q−1 (M1 + 1) (M2 + 1) K 1 < ab |Ω | and p−1 − αβ−( p−1)(q−1) p+β−1 αβ β β p−1 p−1 p−1 K 2 < a b|Ω | (M1 + 1) (M2 + 1) .
Therefore, for the sufficiently small (u 0 (x), v0 (x)) satisfying u 0 (x) ≤ W (x, 0) and v0 (x) ≤ Z (x, 0), we obtain (u(x, t), v(x, t)) ≤ (W (x, t), Z (x, t)) on (Ω × [0, +∞)) × (Ω × [0, +∞)) by Lemma 2.2. Hence, (u(x, t), v(x, t)) must exist globally. 4. Blow-up of solution In this section, we shall prove Theorem 1.5. According to Lemma 2.2, we need only to construct a blowing up subsolution to the problem (1.1). Proof of Theorem 1.5. (1) In this case, we use some ideas developed by [25], which is a powerful tool to construct blowing up subsolutions for both nondegenerate and degenerate parabolic equations, see [2,25]. Let ϕ ∈ C 1 (Ω ), ϕ(x) ≥ 0, ϕ(x) 6≡ 0 and ϕ|∂ Ω = 0. By translation, we may assume without loss of generality that 0 ∈ Ω and ϕ(0) > 0. Set |x| 1 |x| 1 V V , z(x, t) = w(x, t) = (T − t)γ1 (T − t)σ1 (T − t)γ2 (T − t)σ2
F. Li / Nonlinear Analysis 67 (2007) 1387–1402
1397
with A y2 V (y) = 1 + − 2 2A
,
y ≥ 0,
+
where γi , σi > 0 (i = 1, 2), A > 1 and 0 < T < 1 are to be determined. Notice the fact that: supp w+ (·, t) = B(0, R(T − t)σ1 ) ⊂ B(0, RT σ1 ) ⊂ Ω , supp z + (·, t) = B(0, R(T
− t)σ2 )
⊂ B(0,
RT σ2 )
⊂Ω
for sufficiently small T > 0 with R = (A(2 + A))1/2 . Denote yi = |x|/(T − t)σi , i = 1, 2. Computing directly we obtain: γ1 V (y1 ) + σ1 y1 V 0 (y1 ) γ2 V (y2 ) + σ2 y2 V 0 (y2 ) , z (x, t) = ; t (T − t)γ1 +1 (T − t)γ2 +1 N /A N /A −∆w(x, t) = , −∆z(x, t) = ; (T − t)γ1 +2σ1 (T − t)γ2 +2σ2
wt (x, t) =
and Z |x| M1 α V dx (T − t)γ2 α B(0,R(T −t)σ2 ) (T − t)σ2 Ω = M1 /(T − t)γ2 α−N σ2 , Z Z |x| M2 β V dx w β (x, t)dx = (T − t)γ1 β B(0,R(T −t)σ1 ) (T − t)σ1 Ω = M2 /(T − t)γ1 β−N σ1 , R R where M1 = B(0,R) V α (|ξ |)dξ and M2 = B(0,R) V β (|ξ |)dξ . By Z
z α (x, t)dx =
div(|∇w| p−2 ∇w) = |∇w| p−2 ∆w + ( p − 2)|∇w| p−4 (∇w)T (Hx (w))∇w N X N X ∂w ∂ 2 w ∂w = |∇w| p−2 ∆w + ( p − 2)|∇w| p−4 , ∂ xi ∂ xi ∂ x j ∂ x j j=1 i=1 div(|∇z|q−2 ∇z) = |∇z|q−2 ∆z + (q − 2)|∇z|q−4 (∇z)T (Hx (z))∇z N X N X ∂z ∂ 2 z ∂z , = |∇z|q−2 ∆z + (q − 2)|∇z|q−4 ∂ xi ∂ xi ∂ x j ∂ x j j=1 i=1 where Hx (w), Hx (z) denotes the Hessian matrix of w(x, t), z(x, t) respect to x, respectively, we get: p−2 N /A diam(Ω ) |div(|∇w| p−2 ∇w)| ≤ (T − t)γ1 +2σ1 (T − t)γ1 +2σ1 p−4 2 diam(Ω ) diam(Ω ) N /A + ( p − 2) γ +2σ γ +2σ 1 1 1 1 (T − t) (T − t) (T − t)γ1 +2σ1 p−2 N ( p − 1)(diam(Ω )) = , A(T − t)(γ1 +2σ1 )( p−1) q−2 diam(Ω ) N /A q−2 |div(|∇z| ∇z)| ≤ (T − t)γ2 +2σ2 (T − t)γ2 +2σ2 q−4 2 diam(Ω ) diam(Ω ) N /A + (q − 2) (T − t)γ2 +2σ2 (T − t)γ2 +2σ2 (T − t)γ2 +2σ2 N (q − 1)(diam(Ω ))q−2 = . A(T − t)(γ2 +2σ2 )(q−1)
(4.1) (4.2)
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F. Li / Nonlinear Analysis 67 (2007) 1387–1402
If 0 ≤ yi ≤ A, we have 1 ≤ V (yi ) ≤ 1 + A/2 and V 0 (yi ) ≤ 0, i = 1, 2. Hence: Z N ( p − 1)(diam(Ω )) p−2 a M1 γ1 (1 + A/2) p−2 + − , w α dx ≤ ∇z) − a z t − div(|∇z| γ1 +1 (γ1 +2σ1 )( p−1) (T − t)γ2 α−N σ2 (T − t) A(T − t) Ω Z bM2 γ2 (1 + A/2) N ( p − 1)(diam(Ω ))q−2 − z β dx ≤ + z t − div(|∇z|q−2 ∇z) − b . γ +1 (γ2 +2σ2 )(q−1) 2 (T − t)γ1 β−N σ1 (T − t) A(T − t) Ω If yi ≥ A, we have V (yi ) ≤ 1 and V 0 (yi ) ≤ −1, i = 1, 2. Then: Z γ − σ1 A N ( p − 1)(diam(Ω )) p−2 p−2 , z α dx ≤ ∇w) − a + wt − div(|∇w| (T − t)γ1 +1 A(T − t)(γ1 +2σ1 )( p−1) Ω Z γ −σA N (q − 1)(diam(Ω ))q−2 w β dx ≤ + . z t − div(|∇z|q−2 ∇z) − b γ +1 (T − t) 2 A(T − t)(γ2 +2σ2 )(q−1) Ω
(4.3) (4.4)
(4.5) (4.6)
If 0 < y1 ≤ A and y2 > A, we have (4.3) and (4.6) hold. If y1 > A and 0 < y2 ≤ A, we have (4.4) and (4.5) hold. Since p, q > 2 and αβ > ( p − 1)(q − 1) > 1, we can choose two constants σ1 , σ2 > 0, which is sufficiently small, such that: 1 + α + σ2 N + σ1 N α 1 − 2σ1 ( p − 1) < , αβ − 1 p−2 1 + β + σ1 N + σ2 Nβ 1 − 2σ2 (q − 1) < . αβ − 1 q −2 Choose two constants γ1 , γ2 > 0 satisfying 1 + α + σ2 N + σ1 N α 1 − 2σ1 ( p − 1) < γ1 < , αβ − 1 p−2 1 + β + σ1 N + σ2 Nβ 1 − 2σ2 (q − 1) < γ2 < , αβ − 1 q −2 then we have (γ1 + 2σ1 )( p − 1) < γ1 + 1 < γ2 α − N σ2 , (γ2 + 2σ2 )(q − 1) < γ2 + 1 < γ1 β − N σ1 , where the condition αβ > ( p − 1)(q − 1) is used. Select A > max{1, γ1 /σ1 , γ2 /σ2 }, then, for sufficiently small T > 0, (4.3)–(4.6) imply that: Z wt − div(|∇w| p−2 ∇w) − a z α dx ≤ 0, (x, t) ∈ Q T , Z Ω q−2 z t − div(|∇z| ∇z) − b w β dx ≤ 0, (x, t) ∈ Q T . Ω
Since ϕ(0) > 0 and ϕ is a continuous function, there exist two positive constants ρ and such that ϕ(x) ≥ for all x ∈ B(0, ρ) ⊂ Ω . Taking T small enough to ensure that B(0, RT σ ) ⊂ B(0, ρ), then z ≤ 0 on ∂Ω × [0, T ]. From (4.1) and (4.2) it follows that (w(x, 0), z(x, 0)) ≤ (K 0 ϕ(x), K 0 ϕ(x)) for sufficiently large M. By Lemma 2.2 we have (w, z) ≤ (u, v) provided that (u 0 (x), v0 (x)) ≥ (K 0 ϕ(x), K 0 ϕ(x)) and (u, v) can exist no later than t = T . This shows that (u, v) blows up in finite time. (2) Since p > 2, q > 2 and αβ = ( p − 1)(q − 1), we can choose two constants l1 , l2 > 1 such that: l1 q −1 α = = . p−1 l2 β Then l2 β = ( p − 1)l1 > 1,
l1 β = (q − 1)l2 > 1.
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Without loss of generality, we can assume that 0 ∈ Ω . Denote B R (0) a ball centering at 0 with radius R satisfying B R (0) ⊂⊂ Ω . By Lemma 2.2, we need only to show that the solution of problem (1.1) on (B R (0)×[0, T ])×(B R (0)× [0, T ]) blows up in finite time. Let φ(r ) and ψ(r ) (r = |x|) be the solution of the following elliptical problem: −(r N −1 |φ 0 | p−2 φ 0 )0 = r N −1 ,
r ∈ (0, R), φ 0 (0) = 0, φ(R) = 0,
−(r N −1 |ψ 0 |q−2 ψ 0 )0 = r N −1 ,
r ∈ (0, R), ψ 0 (0) = 0, ψ(R) = 0,
and
respectively. Then: φ(r ) = (( p − 1)/ p)(1/N )1/( p−1) (R p/( p−1) − r p/( p−1) ), ψ(r ) = ((q − 1)/q)(1/N )1/(q−1) (R q/(q−1) − r q/(q−1) ). By the assumptions on u 0 (x) and v0 (x), we can choose sufficiently small constant s0 such that: u 0 (r ) ≥ s0l1 φ(r ), v0 (r ) ≥ s0l2 ψ(r ), r ∈ [0, R). RR RR Denote c1 = 0 ψ α dr and c2 = 0 φ β dr , then: Z 1 Z R c1 = ψ α (r )dr = ((q − 1)/q)(1/N )1/(q−1) R 1+qα/( p−1) (1 − s q/(q−1) )qα/(q−1) ds > 1/a, 0
0 R
Z c2 =
β
φ (r )dr = (( p − 1)/ p)(1/N )
1/( p−1)
R
1+ pβ/( p−1)
1
Z
(1 − s p/( p−1) ) pβ/( p−1) ds > 1/b
0
0
for sufficiently large R. Now we consider the Cauchy problem: s 0 (t) = min{(ac1 − 1)/M1 , (bc2 − 1)/M2 }s δ (t),
s(0) = s0 ,
where M1 = (( p − 1)/ p)(1/N )1/( p−1) R p/( p−1) ,
M2 = ((q − 1)/q)(1/N )1/(q−1) R q/(q−1)
and δ = min{( p − 1)l1 − l1 + 1, (q − 1)l2 − l2 + 1}. It is easy to know that there exists a constant T 0 > 0 such that limt→T 0 s(t) = +∞. Define two functions w(r, t) = s l1 φ(r ) and z(r, t) = s l2 ψ(r ), we assert that (w, z) is a subsolution of the problem (1.1) on (B R (0) × [0, T ]) × (B R (0) × [0, T ]), T < T 0 . In fact, a series of computations shows: Z R 1−N N −1 p−2 wt − r (r |wr | wr )r − a z α dr = l1 φs l1 −1 (t)s 0 (t) + s ( p−1)l1 (t) − ac1 s l2 α (t) 0
z t − r 1−N (r N −1 |zr |q−2 zr )r − b
Z
R
= l1 φs l1 −1 (t)(s 0 (t) − (ac1 − 1)φ −1 s ( p−1)l1 −l1 +1 s(t)) ≤ l1 φs l1 −1 (t)(s 0 (t) − M1−1 (ac1 − 1)s ( p−1)l1 −l1 +1 s(t)) ≤ 0, (r, t) ∈ B R (0) × (0, T ),
w β dr = l2 ψs l2 −1 (t)s 0 (t) + s (q−1)l2 (t) − bc2 s l1 β (t)
0
r N −1 |wr | p−2 wr
r =0
= 0,
w(R, t) = s l1 φ(R) = 0, w(r, 0) =
s0l1 φ(r )
≤ u 0 (r ),
= l2 ψs l2 −1 (t)(s 0 (t) − (bc2 − 1)ψ −1 s (q−1)l2 −l2 +1 s(t)) ≤ l2 ψs l2 −1 (t)(s 0 (t) − M2−1 (bc2 − 1)s (q−1)l2 −l2 +1 s(t)) ≤ 0, (r, t) ∈ (x, t) ∈ B R (0) × (0, T ), r N −1 |zr | p−2 zr = 0, t ∈ [0, T ], r =0
z(R, t) = s l2 ψ(R) = 0, z(r, 0) =
s0l2 ψ(r )
t ∈ [0, T ],
≤ v0 (r ),
r ∈ [0, R].
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Using Lemma 2.2 we get (w, z) ≤ (u, v) on Q T × Q T . Thus (u, v) must blow up in finite time since (w, z) does. 5. The special case α = q − 1 and β = p − 1 In this section, we investigate the problem (1.1) for the special case α =Rq − 1 and β = p − 1.R Let φ(x) and ψ(x) be two functions defined by (1.5) and (1.6) respectively, and denote ρ = Ω φ p−1 dx and θ = Ω ψ q−1 dx, we first give the following lemma: Lemma 5.1. Suppose that ρ θ ≥ 1/(ab) and (u 0 (x), v0 (x)) satisfies the conditions in Theorem 1.3, then there exist two positive constants k1 and k2 such that u(x, t) ≥ k1 φ(x),
v(x, t) ≥ k2 ψ(x),
(x, t) ∈ Q T .
Proof. By ρ θ ≥ 1/(ab) and the assumptions on u 0 (x) and v0 (x), we can take appropriate constants k1 and k2 such that u 0 (x) ≥ k1 φ(x), v0 (x) ≥ k2 ψ(x), x ∈ Ω and p−1
a θ ≥ k1
q−1
/k2
≥ (b ρ)−1 .
Let w1 (x, t) = k1 φ(x) and w2 (x, t) = k2 ψ(x), then we have: Z q−1 p−1 q−1 p−2 w1t − div(|∇w1 | ∇w1 ) − w2 dx = k1 − ak2 θ ≤ 0, ZΩ p−1 q−1 p−1 q−2 w2t − div(|∇w2 | ∇w2 ) − w1 dx = k2 − bk1 ρ ≤ 0, Ω
w1 (x, t) = 0,
w2 (x, t) = 0,
w1 (x, 0) ≤ u 0 (x),
(x, t) ∈ Q T , (x, t) ∈ Q T ,
(x, t) ∈ ST ,
w2 (x, 0) ≤ v0 (x),
x ∈ Ω.
Therefore, we get (w1 , w2 ) ≤ (u, v) on Q T × Q T by using Lemma 2.2.
Now we prove Theorem 1.6. Proof of Theorem 1.6. (1) Since ρ θ ≤ 1/(ab), then there exist two positive constants K 1 and K 2 such that p−1
a θ ≤ K1
q−1
/K 2
≤ (b ρ)−1
and u 0 (x) ≤ K 1 φ(x), v0 (x) ≤ K 2 ψ(x). Set W (x, t) = K 1 φ(x) and Z (x, t) = K 2 ψ(x), then we have Z p−1 q−1 p−2 Wt − div(|∇W | ∇W ) = K 1 ≥ a K 2 θ ≥ a Z q−1 dx. Ω Z q−1 p−1 Z t − div(|∇ Z |q−2 ∇ Z ) = K 2 ≥ bK 1 θ ≥ b W p−1 dx. Ω
Noticing W (x, t) = Z (x, t) = 0 on ∂Ω × [0, +∞), we obtain (u(x, t), v(x, t)) ≤ (W (x, t), Z (x, t)) on (Ω × [0, +∞)) × (Ω × [0, +∞)) by Lemma 2.2. Therefore, (u(x, t), v(x, t)) exists globally. (2) In view of ρ θ > 1/(ab), we can choose a smooth subdomain Ω1 ⊂⊂ Ω such that ρ˜ θ˜ > 1/(ab), where R R p−1 q−1 ρ˜ = Ω φ1 dx, θ˜ = Ω ψ1 dx, and φ1 , ψ1 are the unique solution of the elliptic problem −div(|∇φ1 | p−2 ∇φ1 ) = 1
in Ω1 ,
φ1 (x) = 0
−div(|∇ψ1 | p−2 ∇ψ1 ) = 1
in Ω1 ,
ψ1 (x) = 0
on ∂Ω1 ,
and on ∂Ω1 ,
respectively. Denote σ = min{infx∈Ω2 k1 φ1 (x), infx∈Ω2 k2 ψ1 (x)}, Ω2 ⊂⊂ Ω1 , then σ > 0, where k1 and k2 are determined by Lemma 5.1.
F. Li / Nonlinear Analysis 67 (2007) 1387–1402
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Let (u 1 , v1 ) be the unique solution of the following problem: Z p−2 v1α (x, t)dx, (x, t) ∈ Ω2 × (0, T ], u 1t − div(|∇u 1 | ∇u 1 ) = a Ω2 Z β q−2 u 1 (x, t)dx, (x, t) ∈ Ω2 × (0, T ], v1t − div(|∇v1 | ∇v1 ) = b
(5.1)
Ω2
u 1 (x, t) = σ,
v1 (x, t) = σ,
(x, t) ∈ ∂Ω2 × [0, T ],
u 1 (x, 0) = σ,
v1 (x, 0) = σ,
x ∈ Ω2 .
Using Lemmas 5.1 and 2.2, it is easy to verify that: (u, v) ≥ (u 1 , v1 ) on (Ω2 × [0, T ]) × (Ω2 × [0, T ]). Let K 0 = max{supx∈Ω2 φ1 (x), supx∈Ω2 ψ1 (x)}, consider the Cauchy problem: p−1
K 0 s10 (t) = −s1
q−1
K 0 s20 (t) = −s2 s1 (0) = σ/K 0 ,
˜ q−1 (t), (t) + a θs 2 p−1
(t) + bρs ˜ 1
(t),
(5.2)
s2 (0) = σ/K 0 .
Multiplying the first equation in (5.2) by (ρ˜ + 1), the second equation in (5.2) by (θ˜ + 1) and combining them together, we can get p−1
K 0 (ρ˜ + 1)s10 (t) + K 0 (θ˜ + 1)s20 (t) = (abρ˜ θ˜ − 1)(s1
q−1
(t) + s2
(t)).
Noticing p > 2, q > 2 and abρ˜ θ˜ > 1 we obtain that there exists a constant T 0 > 0 such that: lim (s1 (t) + s2 (t)) = +∞.
t→T 0
Set z 1 (x, t) = s1 (t)φ1 (x), z 2 (x, t) = s2 (t)ψ1 (x), now we assert that (z 1 , z 2 ) is a subsolution of problem (5.1). Calculating directly we can get: Z q−1 p−1 p−2 ˜ q−1 (t) z 1t − div(|∇z 1 | ∇z 1 ) − z 2 dx = φ1 (x)s10 (t) + s1 (t) − a θs 2 Ω2
p−1
≤ K 0 s10 (t) + s1 z 2t − div(|∇z 2 |q−2 ∇z 2 ) −
Z
p−1
Ω2
z1
q−1
˜ (t) − a θs 2
q−1
(t) = 0,
p−1
dx = ψ1 (x)s20 (t) + s2
(t) − bρs ˜ 1
q−1
p−1
≤ K 0 s20 (t) + s2
(t) − bρs ˜ 1
z 1 (x, t) = s1 (t)φ1 (x) = 0,
z 2 (x, t) = s2 (t)ψ1 (x) = 0,
z 1 (x, 0) = s1 (0)φ1 (x) ≤ σ,
z 2 (x, 0) = s2 (0)ψ1 (x) ≤ σ,
(x, t) ∈ Ω2 × (0, T ],
(t)
(t) = 0,
(x, t) ∈ Ω2 × (0, T ],
(x, t) ∈ ∂Ω2 × [0, T ], x ∈ Ω2 .
Using Lemma 2.2 it follows that (z 1 , z 2 ) ≤ (u 1 , v1 ) on Ω2 × [0, T ]. Therefore, (u 1 , v1 ) blows up in finite time. Our conclusion comes from the fact that the relation (u, v) ≥ (u 1 , v1 ) holds on (Ω2 × [0, T ]) × (Ω2 × [0, T ]). Acknowledgement The author thanks the anonymous referee for his careful reading and valuable suggestions. References [1] N.D. Alikakos, L.C. Evans, Continuing of the gradient for weak solutions of a degenerate parabolic equation, J. Math. Pures Appl. 62 (1983) 253–268. [2] F. Andreu, J.M. Maz´on, F. Simondon, J. Toledo, Blow-up for a class of nonlinear parabolic problems, Asymptot. Anal. 29 (2002) 143–155. [3] M.F. Bidanut-V´eron, M. Garc´ıa-Huidobro, Regular and singular solutions of a quasilinear equation with weights, Asymptot. Anal. 28 (2001) 115–150. [4] J.A.A. Crespo, I.P. Alonso, Global behavior of the Cauchy problem for some critical nonlinear parabolic equations, SIAM J. Math. Anal. 31 (2000) 1270–1294. [5] J.I. D´ıaz, Nonlinear Partial Differential Equations and Free Boundaries, in: Elliptic Equations, vol. 1, Pitman, London, 1985.
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F. Li / Nonlinear Analysis 67 (2007) 1387–1402
[6] E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, Berlin, New York, 1993. [7] E. DiBenedetto, On the local behavior of solutions of degenerate parabolic equations with measurable coefficients, Ann. Sc. Norm. Super. Pisa Cl. Sci. 13 (1986) 487–535. [8] M.A. Herrero, J.L. V´azquez, Asymptotic behavior of the solutions of a strongly nonlinear parabolic problem, Ann. Fac. Sci. Toulouse Math. 3 (1981) 113–127. [9] H. Ishii, Asymptotic stability and blowing up of solutions of some nonlinear equations, J. Differential Equations 26 (1977) 291–319. [10] A.S. Kalashnikov, Some problems of the qualitative theory of nonlinear degenerate second order parabolic equations, Russian Math. Surveys 42 (1987) 169–222. [11] T. Kilpel¨ainen, P. Lindqvist, On the Dirichlet boundary value problem for a degenerate parabolic equation, SIAM J. Math. Anal. 27 (1996) 661–683. [12] O.A. Ladyzenkaya, New equation for the description of incompressible fluids and solvability in the large boundary problem for them, Proc. Steklov Inst. Math. 102 (1967) 95–118. [13] H.A. Levine, L.E. Payne, Nonexistence of global weak solutions of classes of nonlinear wave and parabolic equations, J. Math. Anal. Appl. 55 (1976) 329–334. [14] H.A. Levine, L.E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions for the porous medium equation backward in time, J. Differential Equations 16 (1974) 319–334. [15] F.C. Li, Y.P. Chen, C.H. Xie, Asymptotic behavior of solution for nonlocal reaction–diffusion system, Acta Math. Sci. Ser. B Engl. Ed. 23 (2003) 261–273. [16] F.C. Li, S.X. Huang, C.H. Xie, Global existence and blow-up of solution to a nonlocal reaction–diffusion system, Discrete Contin. Dyn. Syst. 9 (2003) 1519–1532. [17] F.C. Li, C.H. Xie, Global and blow-up solutions to a p-Laplacian equation with nonlocal source, Comput. Math. Appl. 46 (2003) 1525–1533. [18] M. Nakao, L p -estimates of solutions of some nonlinear degenerate diffusion equations, J. Math. Soc. Japan 37 (1985) 41–63. [19] M. Nakao, Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations, Nonlinear Anal. 10 (1986) 299–314. [20] Y. Ohara, L ∞ -estimates of solutions of some nonlinear degenerate parabolic equations, Nonlinear Anal. 18 (1992) 413–426. [21] Y. Ohara, Gradient estimates for some quasilinear parabolic equations with nonmonotonic perturbations, Adv. Math. Sci. Appl. 6 (1996) 531–540. ˆ [22] M. Otani, Y. Sugiyama, Lipschitz continuous solutions of some doubly nonlinear parabolic equations, Discrete Contin. Dyn. Syst. 8 (2002) 647–670. [23] C.V. Pao, Blowing-up of solution for a non-local reaction–diffusion problem in combustion theory, J. Math. Anal. Appl. 166 (1992) 591–600. [24] M.M. Porzio, V. Vespri, H¨older estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations 103 (1993) 146–178. [25] P. Souplet, Blow-up in nonlocal reaction–diffusion equations, SIAM J. Math. Anal. 29 (1998) 1301–1334. [26] M. Tsutsumi, Existence and nonexistence of global solutions for nonlinear parabolic equations, Publ. Res. Inst. Math. Sci. 8 (1972) 221–229. [27] M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption, J. Math. Anal. Appl. 132 (1988) 187–212. [28] X. Xu, On the initial-boundary-value problem for u t − div(|∇u| p−2 ∇u) = 0, Arch. Ration. Mech. Anal. 127 (1994) 319–335. [29] H.J. Yuan, Extinction and positivity of the evolution p-Laplacian equation, J. Math. Anal. Appl. 196 (1995) 754–763. [30] J. Zhao, Existence and nonexistence of solutions for u t − div(|∇u| p−2 ∇u) = f (∇u, u, x, t), J. Math. Anal. Appl. 172 (1993) 130–146. [31] S.L. Zhou, A priori L ∞ -estimate and existence of solutions for some nonlinear parabolic equations, Nonlinear Anal. 42 (2000) 887–904.