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The existence of global solution and the blowup problem for some p-Laplace heat equations*
Acta Mathematica Scientia 2007,27B(2):274–282 http://actams.wipm.ac.cn
THE EXISTENCE OF GLOBAL SOLUTION AND THE BLOWUP PROBLEM FOR SOME p-LAPLACE HEAT EQUATIONS∗ Wang Yang (
)
Department of Mathematics, East China Normal University, Shanghai 200062, China E-mail: [email protected]
Abstract
This article consider, for the following heat equation
|x|u − ∆ u = u , u(x, t) = 0, u(x, 0) = u (x), t s
q
p
0
(x, t) ∈ Ω × (0, T ), (x, t) ∈ ∂Ω × (0, T ), u0 (x) ≥ 0,
u0 (x) ≡ 0
the existence of global solution under some conditions and give two sufficient conditions for the blow up of local solution in finite time, where Ω is a smooth bounded domain in . RN (N > p), 0 ∈ Ω, ∆p u = div(|∇u|p−2 ∇u), 0 ≤ s ≤ 2, p ≥ 2, p − 1 < q < Np−N+p N−p Key words Hardy–Sobolev inequality, decay of exponent, blow up 2000 MR Subject Classification
1
35K20
Introduction and Statements of Main Results In this article, we shall consider for the following heat equation ⎧ u t ⎪ − ∆p u = uq , (x, t) ∈ Ω × (0, T ), ⎪ ⎪ ⎨ |x|s u(x, t) = 0, (x, t) ∈ ∂Ω × (0, T ), ⎪ ⎪ ⎪ ⎩ u(x, 0) = u (x), u (x) ≥ 0, u (x) ≡ 0 0
0
(1.1)
0
the existence of global solution under some conditions and give two sufficient conditions for the blow up of local solution in finite time, where Ω is a smooth bounded domain in RN (N > p), +p 0 ∈ Ω, ∆p u = div(|∇u|p−2 ∇u), 0 ≤ s ≤ 2, p ≥ 2, p − 1 < q < N p−N N −p . Fujita in [1] in 1966 first considered the following equation ⎧ ⎨ u − ∆u = up , x ∈ RN , t > 0, t ⎩ u(x, 0) = u0 (x), x ∈ RN , ∗ Received
December 30, 2004; revised September 24, 2005. The author is supported by PhD Program Scholarship Fund of ECNU 2006.
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Wang: EXISTENCE OF GLOBAL SOLUTION AND BLOWUP PROBLEM
where p > 1 , ∆ =
N i=1
∂2 . ∂x2i
He showed that for 1 < p <
n+2 n
275
, the solution u(x, t; u0 ) must
blow up in finite time; and for p > n+2 n , the solution u(x, t; u0 ) globally exists in time if u0 is sufficiently small, say, dominated by a small Gaussian (which exponentially decays at x = ∞). The borderline case p = n+2 n belongs to global nonexistence and was settled later by Hayakawa in [2] and Kobayashi, Sirao, and Tanaka in [3]. Over the course of the last thirty to thirty-five years there was an explosion of interest in the blowup theorems for solutions of the nonlinear evolution equations. Blow up theorems have wide applications in partial differential equations, diverse chanics, and fluid mechanics. In chemistry, Pao in [4] described the following chemical reaction diffusion equation ut − ∇(D∇u) = f (x, u, ∇u), where D is called the diffusion coefficient. In 2001, Zhong Tan in [5] considered the existence and asymptotic estimates of global solutions and finite time blowup of local solution of chemical reaction diffusion equation with special diffusion coefficient D = |x|2 , that is, the following form ⎧ u t ⎪ − ∆u = up , (x, t) ∈ Ω × (0, T ), ⎪ ⎪ |x| ⎨ 2 u(x, t) = 0, (x, t) ∈ ∂Ω × (0, T ), ⎪ ⎪ ⎪ ⎩ u(x, 0) = u (x), u (x) ≥ 0, u (x) ≡ 0, 0
0
0
N +2 where Ω is a bounded domain in RN (N ≥ 3) with smooth boundary ∂Ω, 0 ∈ Ω, 1 < p < N −2 . It is easy to see that this equation is the special case of equation (1.1). When this article is accepted, the author was told that Tan Zhong had extended his result to p-Laplacian case with s = 2 only (see [6]). In this article, we will extend this problem to more general condition. More precisely, we will prove the existence of global solution by Hardy–Sobolev inequality and find two sufficient conditions for blow up in finite time by variational methods and classical concave methods (see [7–10]). Before giving the main results, we first give the following definitions: Definition 1.1 We define u to be the solution of (1.1) on Ω × (0, T ) if
u ∈ L∞ (0, T ; W01,p(Ω)),
T 0
Ω
|ut |2 dxdt < ∞, |x|s
and u satisfies (1.1) in the distribution sense. Let u(x, t; u0 ) be the solution with initial value u0 (x). Definition 1.2 We say a solution u blow up at T if u L∞ (Ω) → +∞ as t → T− . 1 Let E(u) = 1p Ω |∇u|p dx − q+1 |u|q+1 dx be the energy functional of (1.1). Ω Now we define p H(u) = |∇u| dx − |u|q+1 dx, Ω
K = {u ∈
W01,p (Ω); H(u)
Ω
= 0, u = 0},
d = inf{sup E(λu), u ∈ W01,p (Ω), u = 0}, λ≥0
then 0 < d = inf E(u) (see [9] and [11]). u∈K
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Let Σ1 and Σ2 be as follows: Σ1 = {u ∈ W01,p (Ω); E(u) < d, H(u) > 0} ∪ {0}, Σ2 = {u ∈ W01,p (Ω); E(u) < d, H(u) < 0}. The main results of this article are: Theorem 1.1 If u0 (x) ∈ Σ1 , then (1.1) exists a global solution u(x, t; u0 ). Moreover, there exists α > 0 such that ∇u(t) pp = O(e−αt ),
t → ∞.
(1.2)
Theorem 1.2 Let u(x, t; u0 ) be the solution of (1.1) and E(u0 ) ≤ 0 , then u(x, t; u0 ) will blow up in finite time. Theorem 1.3 Let u(x, t; u0 ) be the solution of (1.1) and u0 ∈ Σ2 , then u(x, t; u0 ) will blow up in finite time.
2
The Existence and Exponential Decay of the Solution
In this section, we will prove Theorem 1.1. First, we introduce some lemmas: Lemma 2.1 (Hardy–Sobolev inequality) (see [12]) Let RN = Rk × RN −k , 2 ≤ k ≤ N and x = (y, z) ∈ RN = Rk × RN −k . For given n, β satisfying 1 < n < N, 0 ≤ β ≤ n and −β) β < k, m(β, N, n) = n(N N −n , there exists positive constant C = C(β, n, N, k) such that for any u ∈ W01,n (RN ), N −β N |u(x)|m −n n dx ≤ C |∇u| dx . β |y| N N R R Remark When m = n = β, this inequality is the classical Hardy inequality. +p Lemma 2.2 (Sobolev–Poincare) Let p − 1 < q ≤ N p−N N −p . Then there exists a best constant M such that for all u ∈ W01,p (Ω), we have u q+1 ≤ M ∇u p , where M depends only on Ω, N, p, q. − p(q+1) q+1−p . Lemma 2.3 d = q+1−p p(q+1) M Proof From the definition of E(u), we have λp λq+1 p E(λu) = |∇u| dx − |u|q+1 dx, p Ω q+1 Ω then
Let
∂E(λu) ∂λ
∂E(λu) = λp−1 |∇u|p dx − λq |u|q+1 dx. ∂λ Ω Ω Ê |∇u|p dx 1/(q+1−p) = 0, then λ = Ê Ω |u|q+1 dx . Combining it with E(λu), we have Ω
q + 1 − p ∇u p q+1−p . p(q + 1) u q+1 p(q+1)
sup E(λu) =
λ≥0
Recall the definition of d, we complete the proof. Next we prove Theorem 1.1. Step 1: Proof of Existence.
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277
Let wn (x) = min{|x|−s , n}. According to the similar results in [13], for any n > 0, the following equation has the only global classical solution un ∈ C(Q∞ ) ∩ C 2,1 (Q∞ ), where Q∞ = Ω × (0, ∞). ⎧ q ⎪ ⎪ ⎨ wn (x)ut − ∆p u = min{n, u }, (x, t) ∈ Q∞ ,
(x, t) ∈ ∂Ω × (0, ∞),
u(x, t) = 0,
⎪ ⎪ ⎩
(2.1)
u(x, 0) = un0 .
In the above equation, un0 ∈ C0∞ (Ω) strongly converges to u0 in W01,p (Ω) and for some ε0 > 0, p(q + 1) E(un0 ) ≤ E(u0 ) + ε0 < d, d. |un0 |q+1 dx ≤ |u0 |q+1 dx + ε0 < q +1−p Ω Ω Multiplying the equation (1.1) by unt and integrating over Ω × [0, T ], for fixed T > 0, we obtain T wn (x)u2nt dxdt + E(un ) ≤ E(un0 ) < d. (2.2) Ω
0
Claim ∀t ≥ 0, un (t) ∈ Σ1 . If not, let t∗ = min{t|un (t) ∈ Σ1 }, then un (t∗ ) ∈ ∂Σ1 , thus ∗ ∗ p E(un (t )) = d or |∇un (t )| dx = |un (t∗ )|q+1 dx. Ω
Ω
Obviously, the first equation contradicts (2.2); for the second, after simple calculation, we have E(un (t∗ )) = then
Ω
q+1−p p(q + 1)
|∇un (t∗ )|p dx,
Ω
p(q+1)
|∇un (t∗ )|p dx ≥ M − q+1−p by using Lemma 2.2 and Lemma 2.3. Thus, we have E(un (t∗ )) ≥
q + 1 − p − p(q+1) M q+1−p = d, p(q + 1)
again contradicts to (2.2). So for any t ≥ 0, un (t) ∈ Σ1 . Hence |∇un |p dx > |un |q dx, Ω
Ω
and E(un ) >
q+1−p p(q + 1)
|∇un |p dx.
Ω
According to (2.2), we have 0
so
T
Ω
wn (x)u2nt dxdt +
T 0
Ω
q+1−p p(q + 1)
|∇un |p dx ≤ E(un0 ) < d,
(2.3)
Ω
Ω
wn (x)u2nt dxdt < d,
(2.4)
p(q + 1) d. q+1−p
(2.5)
|∇un |p dx <
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Furthermore, Lemma 2.2 gives, for any n > 0 , (q+1)/p q+1 q+1 |un | dx ≤ M |∇un |p , Ω
and (2.3) implies that E(un0 ) ≥
Ω q+1−p p(q+1)
Ω
|∇un |p dx; therefore,
p(q + 1) q+1−p p E(un0 ) |∇un |p dx q+1−p Ω p(q + 1) q+1−p p (E(u0 ) + ε0 ) ≤ M q+1 |∇un |p dx. q+1−p Ω
|un |q+1 dx ≤ M q+1
Ω
q+1−p p For simplicity, denote M q+1 p(q+1) (E(u )+ε ) by δ (due to Lemma 2.3 and E(u0 )+ 0 0 q+1−p ε0 < d, we have 0 < δ < 1). Let γ = 1 − δ, then ∀n and t > 0 q+1 |un (t)| dx ≤ (1 − γ) |∇un (t)|p dx. (2.6) Ω
Ω
On the other hand, multiplying (2.1) by un and by integrating over Ω × [0, T ], inequality (2.6) implies 1 2 ≤ ≤
1 2 1 2
2
T
|∇un |p dxdt
wn (x)|un (x, T )| dx + Ω
wn (x)|u0 (x)|2 dx +
Ω
0
T
Ω
0
|un |q+1 dxdt
Ω
wn (x)|u0 (x)|2 dx + (1 − γ) Ω
0
T
|∇un |p dxdt.
Ω
So for some γ > 0 , 1 2
wn (x)|un (x, T )|2 dx + γ Ω
0
T
|∇un |p dxdt ≤ Ω
1 2
wn (x)|u0 (x)|2 dx, Ω
T
|∇un |p dxdt ≤ C, where C is a constant that is independent of n and T . By (2.4), (2.5), and (2.6), there exists subsequence (still denoted by un ) and function u such that for all T ≥ 0, thus
0
Ω
un → u ∇un ∇u
wn (x)unt un u
a.e. on QT , ut |x|s/2
in Lp (QT ), in L2 (QT ), in L∞ (0, T ; W01,p (Ω)) weakly ∗,
where “” denotes weak convergence, “→” denotes strong convergence. Thus, we get a global solution through limiting (2.1). In fact, it can be obtained from Boccardo and Murat([7]) and the limit u satisfies for any T > 0 p(q + 1) d, (2.7) |∇u|p dx < q +1−p Ω
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Wang: EXISTENCE OF GLOBAL SOLUTION AND BLOWUP PROBLEM
T
0
Ω
|ut |2 dxdt < d, |x|s
(2.8)
which completes the proof of Step 1. Step 2: Proof of (1.2). By Step 1, H(u(t)) > 0, ∀t ≥ 0. Thus q+1−p 1 q+1−p p H(u) > |∇u| dx + |∇u|p dx E(u) = p(q + 1) Ω q+1 p(q + 1) Ω p and E(u0 ) > q+1−p p(q+1) Ω |∇u| dx. Lemma 2.2 implies p(q + 1) q+1−p p E(u0 ) |u|q+1 dx ≤ M q+1 |∇u|p dx. q+1−p Ω Ω Denote M q+1
q+1−p p
p(q+1) q+1−p E(u0 )
279
(2.9)
by δ (0 < δ < 1). Let γ = 1 − δ, then
|u|q+1 dx ≤ (1 − γ)
Ω
|∇u|p dx.
(2.10)
Ω
|u(t)|2 d Let T > t0 be a fixed figure, we have 12 dt dx = −H(u(t)), the Hardy–Sobolev Ω |x|s inequality and the H¨ older inequality imply T |u(t)|2 |u(T )|2 |u(t)|2 1 1 1 H(u(τ ))dτ = dx − dx ≤ dx s s 2 Ω |x| 2 Ω |x| 2 Ω |x|s t N −s+2 p2 2N N ≤ C(N, s) |∇u| N −s+2 dx ≤ C(Ω, N, s) |∇u|p dx s. (2.11) Ω
Ω
On the other hand, on [t0 , ∞) we have 1 1 p E(u(t)) = |∇u| dx − |u|q+1 dx p Ω q+1 Ω 1 1 H(u(t)) − = |∇u|p dx + |∇u|p dx p Ω q+1 Ω q+1−p 1 q+1−p p H(u(t)) ≥ |∇u| dx + |∇u|p dx, = p(q + 1) Ω q+1 p(q + 1) Ω thus, by (2.11) and (2.12), we have on [t0 , T ] T p2 p(q + 1) 2 E(u(t)) H(u(τ ))dτ ≤ C(Ω, N, s) = C(Ω)(E(u(t))) p , q+1−p t
(2.12)
(2.13)
2
p where C(Ω) = C(Ω, N, s)( p(q+1) q+1−p ) . Furthermore, (2.10) implies on [t0 , ∞), γ |∇u(t)|p dx ≤ H(u(t)).
(2.14)
Ω
By (2.12) and (2.14), we have E(u(t)) ≤
q +1−p γp(q + 1)
+
1 )H(u(t) . q+1
(2.15)
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T 2 By (2.13) and (2.15), on [t0 , T ], we obtain C1 t E(u(τ ))dτ ≤ E(u(t)) p , where C1 = −1 q+1−p 1 C(Ω)( γp(q+1) + q+1 ) . Then, by the arbitrariness of T > t0 , it follows that ∞ 2 C1 E(u(τ ))dτ ≤ E(u(t)) p . t
Let T > t0 be sufficiently large such that C1−1 ≤ T , it follows that ∞ 2 E(u(τ ))dτ ≤ T E(u(t)) p , on [t0 , ∞). Setting y(t) = and E(u(t)) that
∞ t
(2.16)
t
t
E(u(τ ))dτ , it follows from (2.16) and the monotonicity of y(t)e T 2
t
2
t
t
y(t) ≤ y(T )e1− T ≤ T E(u(T )) p e1− T ≤ T E(u(t0 )) p e1− T , On the other hand, we get ∞ E(u(τ ))dτ ≥ t
t
T +t
∀t > T.
(2.17)
E(u(τ ))dτ ≥ T E(u(T + t)). 2
t
This inequality together with inequality (2.17) leads to E(u(T + t)) ≤ E(u(t0 )) p e1− T for all t > T. 2 t p 1− T , which implies the exponential From (2.12), we obtain ∇u pp ≤ p(q+1) q+1−p E(u(t0 )) e t
decay of solutions ∇u p ≤ Ce− pT with some constant C > 0 for large t > T , and this completes the proof of (1.2).
3
Proofs of Theorem 1.2 and Theorem 1.3
In this section, we consider the finite time blowup of Equation (1.1). We first prove Theorem 1.2. Proof of Theorem 1.2 In fact, we can prove a more general result: If there exists some t0 such that E(u(t0 )) ≤ 0, then u(x, t; u0 ) will blow up in finite time. We shall employ the classical concavity method. Suppose that tmax = ∞ and denote t 2 f (t) = 12 t0 Ω |u| |x|s dxdτ. We perform standard manipulations: t t0
Ω
|uτ |2 1 dxdτ + |x|s p
|∇u|p dx −
Ω
1 q+1
|u|q+1 dx = E(u(t0 )),
|u|2 1 f (t) = dx, 2 Ω |x|s uut p dx = − |∇u| dx + |u|q+1 dx. f (t) = s Ω |x| Ω Ω
Then we have q+1−p f (t) = p
p
|∇u| dx − (q + 1)E(u(t0 )) + (q + 1) Ω
As E(u(t0 )) ≤ 0, we have q+1−p p
Ω
(3.1)
Ω
(3.2) (3.3)
t t0
|∇u|p dx − (q + 1)E(u(t0 )) > 0,
Ω
|uτ |2 dxdτ. |x|s
(3.4)
(3.5)
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281
for all t ≥ t0 . If we have tmax = ∞, then there exists a constant c > 0 such that ∀t > t1 > t0 there holds t1 |uτ |2 dxdτ ≥ c > 0, f (t) > (q + 1) s t0 Ω |x| this inequality would yield lim f (t) = lim f (t) = ∞. t→∞
t→∞
On the other hand, equation (3.4) and (3.5) imply t
|uτ |2 dxdτ, s t0 Ω |x| t |u (τ )|2 q + 1 t |u(τ )|2 τ dxdτ dxdτ f (t)f (t) ≥ s 2 |x|s t0 Ω |x| t0 Ω 2 q + 1 t uuτ ≥ dxdτ s 2 t0 Ω |x| t 2 q+1 q+1 (f (t) − f (t0 ))2 , = f (τ )dτ = 2 2 t0
f (t) ≥ (q + 1)
and as t → ∞, we have for some α > 0 , f (t)f (t) ≥ (1 + α)(f (t))2 . Hence (f (t))−α is concave on [t0 , ∞), (f (t))−α > 0 and lim f −α (t) = 0. This contradiction proves that tmax < ∞. t→∞ Proof of Theorem 1.3 Obviously, if u0 ∈ Σ2 , then by Lemma 2.1, we have p(q + 1) d< |∇u0 |p dx < |u0 |q+1 dx. q+1−p Ω Ω If u(x, t) is a global solution, then u(x, t) does not strongly converge to 0 in W01,p (Ω). Other wise, there exists t∗ , 0 < t∗ < ∞ such that E(u(t∗ )) < d. But Ω |u(t∗ )|q+1 dx = Ω |∇u(t∗ )|p dx. Using the same method as Step 1 in the proof of Theorem 1.1, we can obtain a contradiction. Thus u(x, t) can not strongly converge to 0 in W01,p (Ω). Furthermore, we have the following claim: Claim If u0 ∈ Σ2 , then there exists a constant η > 0 which is sufficently small and independent of t and rely on u0 such that q+1 |u(x, t)| dx ≥ (1 + η) |∇u(x, t)|p dx, (3.6) Ω
Ω
for any t ∈ [0, Tmax ). First, we have p(q + 1) d< q+1−p
|∇u|p dx <
Ω
|u|q+1 dx,
Ω
for all t ∈ [0, Tmax). Indeed, if there exists a t∗ ∈ [0, Tmax) such that ∗ q+1 |u(x, t )| dx = |∇u(x, t∗ )|p dx, Ω
then we have
|∇u(x, t∗ )|p dx =
Ω
But d > E(u(x, t∗ )) =
Ω
q+1−p p(q+1)
Ω
Ω
|u(x, t∗ )|q+1 dx ≥
p(q + 1) d. q+1−p
|∇u(x, t∗ )|p dx, which is a contradiction.
(3.7)
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For the proof of (3.6), we can apply the method of Ishii [8] and it is omitted here. Now, we can complete the proof of Theorem 1.3. We shall employ the same argument as t 2 the proof of Theorem 1.2. Suppose that tmax = ∞ and denote f (x) = 12 0 Ω |u| |x|s dxdτ . From (3.1)–(3.3) and (3.6), we obtain f (t) ≥ − |∇u|p dx + (1 + η) |∇u(x, t)|p dx = η |∇u(x, t)|p dx. Ω
Ω
Ω
If we have tmax = ∞, then this inequality would yield lim f (t) = lim f (t) = ∞. t→∞
By (3.1), (3.2), and (3.3) we have f (t) =
q+1−p p
|∇u|p dx − (q + 1)E(u0 ) + (q + 1)
Ω
t 0
which implies
f (t) ≥ (q + 1)
t 0
and
t→∞
Ω
Ω
|uτ |2 dxdτ, |x|s
|uτ |2 dxdτ, |x|s
q+1 (f (t) − f (0))2 . 2 By the argument as the proof of Theorem 1.2, we obtain a contradiction. f (t)f (t) ≥
Acknowledgments The author would like to thank Professor Feng Zhou for his constant encouragement and valuable suggestion. References 1 Fujita H. On the blowing up of solutions of the Cauchy problem for ut = ∆u + u1+α . J Fac Sci Univ Tokyo Sect I, 1966, 13: 109–124 2 Hayakawa K. On nonexistence of global solutions of some semilinear parabolic equations. Proc Japan Acad, 1973, 49: 503–505 3 Kobayashi K, Sirao T, Tanaka H. On the growing up problem for semilinear heat equations. J Math Soc Japan, 1977, 29: 407–424 4 Pao C V. Nonlinear Parabolic and Elliptic Equations. New York, London: Plenum Press, 1992 5 Tan Z. The reaction diffusion equations with special diffusion processes. Chinese Journal of Contemporary Mathematics, 2001, 22(4) 6 Tan Z. Non-Newton filtration equation with special medium void. Acta Mathematica Scientia, 2004, 24B(1): 118–128 7 Fila M. Boundedness of global solutions of nonlinear diffusion equations. J of Diff Equs, 1992, 98: 226–240 8 Ishii H. Asymptotic stability and blowing up of solutions of some nonlinear equations. J of Diff Equs, 1977, 26: 291–319 9 Payne L E, Sattinger D H. Saddle points and instability of non-linear hyperbolic equations. Israel J of Math, 1975, 22(3/4): 273–303 10 Ye Q X, Li Z Y. An Introduction to Reaction Diffusion Equations. Beijing: Science Press (in Chinese), 1985 11 Tsutsumi M. Existence and nonexistence of global solutions for nonlinear parabolic equations. Publ Res Inst Math Sci, 1972, 8: 211–229 12 Badiale M, Tarantello G. A Sobolev–Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch Rational Mech Anal, 2002, 163: 259–293 13 Ladyzenskaja O A, Solonnikov V A, Ural’ceva N N. Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs 23. Providence R L: Amer Math Soc, 1968
THE EXISTENCE OF GLOBAL SOLUTION AND THE BLOWUP PROBLEM FOR SOME p-LAPLACE HEAT EQUATIONS∗ Wang Yang (
)
Department of Mathematics, East China Normal University, Shanghai 200062, China E-mail: [email protected]
Abstract
This article consider, for the following heat equation
|x|u − ∆ u = u , u(x, t) = 0, u(x, 0) = u (x), t s
q
p
0
(x, t) ∈ Ω × (0, T ), (x, t) ∈ ∂Ω × (0, T ), u0 (x) ≥ 0,
u0 (x) ≡ 0
the existence of global solution under some conditions and give two sufficient conditions for the blow up of local solution in finite time, where Ω is a smooth bounded domain in . RN (N > p), 0 ∈ Ω, ∆p u = div(|∇u|p−2 ∇u), 0 ≤ s ≤ 2, p ≥ 2, p − 1 < q < Np−N+p N−p Key words Hardy–Sobolev inequality, decay of exponent, blow up 2000 MR Subject Classification
1
35K20
Introduction and Statements of Main Results In this article, we shall consider for the following heat equation ⎧ u t ⎪ − ∆p u = uq , (x, t) ∈ Ω × (0, T ), ⎪ ⎪ ⎨ |x|s u(x, t) = 0, (x, t) ∈ ∂Ω × (0, T ), ⎪ ⎪ ⎪ ⎩ u(x, 0) = u (x), u (x) ≥ 0, u (x) ≡ 0 0
0
(1.1)
0
the existence of global solution under some conditions and give two sufficient conditions for the blow up of local solution in finite time, where Ω is a smooth bounded domain in RN (N > p), +p 0 ∈ Ω, ∆p u = div(|∇u|p−2 ∇u), 0 ≤ s ≤ 2, p ≥ 2, p − 1 < q < N p−N N −p . Fujita in [1] in 1966 first considered the following equation ⎧ ⎨ u − ∆u = up , x ∈ RN , t > 0, t ⎩ u(x, 0) = u0 (x), x ∈ RN , ∗ Received
December 30, 2004; revised September 24, 2005. The author is supported by PhD Program Scholarship Fund of ECNU 2006.
No.2
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where p > 1 , ∆ =
N i=1
∂2 . ∂x2i
He showed that for 1 < p <
n+2 n
275
, the solution u(x, t; u0 ) must
blow up in finite time; and for p > n+2 n , the solution u(x, t; u0 ) globally exists in time if u0 is sufficiently small, say, dominated by a small Gaussian (which exponentially decays at x = ∞). The borderline case p = n+2 n belongs to global nonexistence and was settled later by Hayakawa in [2] and Kobayashi, Sirao, and Tanaka in [3]. Over the course of the last thirty to thirty-five years there was an explosion of interest in the blowup theorems for solutions of the nonlinear evolution equations. Blow up theorems have wide applications in partial differential equations, diverse chanics, and fluid mechanics. In chemistry, Pao in [4] described the following chemical reaction diffusion equation ut − ∇(D∇u) = f (x, u, ∇u), where D is called the diffusion coefficient. In 2001, Zhong Tan in [5] considered the existence and asymptotic estimates of global solutions and finite time blowup of local solution of chemical reaction diffusion equation with special diffusion coefficient D = |x|2 , that is, the following form ⎧ u t ⎪ − ∆u = up , (x, t) ∈ Ω × (0, T ), ⎪ ⎪ |x| ⎨ 2 u(x, t) = 0, (x, t) ∈ ∂Ω × (0, T ), ⎪ ⎪ ⎪ ⎩ u(x, 0) = u (x), u (x) ≥ 0, u (x) ≡ 0, 0
0
0
N +2 where Ω is a bounded domain in RN (N ≥ 3) with smooth boundary ∂Ω, 0 ∈ Ω, 1 < p < N −2 . It is easy to see that this equation is the special case of equation (1.1). When this article is accepted, the author was told that Tan Zhong had extended his result to p-Laplacian case with s = 2 only (see [6]). In this article, we will extend this problem to more general condition. More precisely, we will prove the existence of global solution by Hardy–Sobolev inequality and find two sufficient conditions for blow up in finite time by variational methods and classical concave methods (see [7–10]). Before giving the main results, we first give the following definitions: Definition 1.1 We define u to be the solution of (1.1) on Ω × (0, T ) if
u ∈ L∞ (0, T ; W01,p(Ω)),
T 0
Ω
|ut |2 dxdt < ∞, |x|s
and u satisfies (1.1) in the distribution sense. Let u(x, t; u0 ) be the solution with initial value u0 (x). Definition 1.2 We say a solution u blow up at T if u L∞ (Ω) → +∞ as t → T− . 1 Let E(u) = 1p Ω |∇u|p dx − q+1 |u|q+1 dx be the energy functional of (1.1). Ω Now we define p H(u) = |∇u| dx − |u|q+1 dx, Ω
K = {u ∈
W01,p (Ω); H(u)
Ω
= 0, u = 0},
d = inf{sup E(λu), u ∈ W01,p (Ω), u = 0}, λ≥0
then 0 < d = inf E(u) (see [9] and [11]). u∈K
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Let Σ1 and Σ2 be as follows: Σ1 = {u ∈ W01,p (Ω); E(u) < d, H(u) > 0} ∪ {0}, Σ2 = {u ∈ W01,p (Ω); E(u) < d, H(u) < 0}. The main results of this article are: Theorem 1.1 If u0 (x) ∈ Σ1 , then (1.1) exists a global solution u(x, t; u0 ). Moreover, there exists α > 0 such that ∇u(t) pp = O(e−αt ),
t → ∞.
(1.2)
Theorem 1.2 Let u(x, t; u0 ) be the solution of (1.1) and E(u0 ) ≤ 0 , then u(x, t; u0 ) will blow up in finite time. Theorem 1.3 Let u(x, t; u0 ) be the solution of (1.1) and u0 ∈ Σ2 , then u(x, t; u0 ) will blow up in finite time.
2
The Existence and Exponential Decay of the Solution
In this section, we will prove Theorem 1.1. First, we introduce some lemmas: Lemma 2.1 (Hardy–Sobolev inequality) (see [12]) Let RN = Rk × RN −k , 2 ≤ k ≤ N and x = (y, z) ∈ RN = Rk × RN −k . For given n, β satisfying 1 < n < N, 0 ≤ β ≤ n and −β) β < k, m(β, N, n) = n(N N −n , there exists positive constant C = C(β, n, N, k) such that for any u ∈ W01,n (RN ), N −β N |u(x)|m −n n dx ≤ C |∇u| dx . β |y| N N R R Remark When m = n = β, this inequality is the classical Hardy inequality. +p Lemma 2.2 (Sobolev–Poincare) Let p − 1 < q ≤ N p−N N −p . Then there exists a best constant M such that for all u ∈ W01,p (Ω), we have u q+1 ≤ M ∇u p , where M depends only on Ω, N, p, q. − p(q+1) q+1−p . Lemma 2.3 d = q+1−p p(q+1) M Proof From the definition of E(u), we have λp λq+1 p E(λu) = |∇u| dx − |u|q+1 dx, p Ω q+1 Ω then
Let
∂E(λu) ∂λ
∂E(λu) = λp−1 |∇u|p dx − λq |u|q+1 dx. ∂λ Ω Ω Ê |∇u|p dx 1/(q+1−p) = 0, then λ = Ê Ω |u|q+1 dx . Combining it with E(λu), we have Ω
q + 1 − p ∇u p q+1−p . p(q + 1) u q+1 p(q+1)
sup E(λu) =
λ≥0
Recall the definition of d, we complete the proof. Next we prove Theorem 1.1. Step 1: Proof of Existence.
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Let wn (x) = min{|x|−s , n}. According to the similar results in [13], for any n > 0, the following equation has the only global classical solution un ∈ C(Q∞ ) ∩ C 2,1 (Q∞ ), where Q∞ = Ω × (0, ∞). ⎧ q ⎪ ⎪ ⎨ wn (x)ut − ∆p u = min{n, u }, (x, t) ∈ Q∞ ,
(x, t) ∈ ∂Ω × (0, ∞),
u(x, t) = 0,
⎪ ⎪ ⎩
(2.1)
u(x, 0) = un0 .
In the above equation, un0 ∈ C0∞ (Ω) strongly converges to u0 in W01,p (Ω) and for some ε0 > 0, p(q + 1) E(un0 ) ≤ E(u0 ) + ε0 < d, d. |un0 |q+1 dx ≤ |u0 |q+1 dx + ε0 < q +1−p Ω Ω Multiplying the equation (1.1) by unt and integrating over Ω × [0, T ], for fixed T > 0, we obtain T wn (x)u2nt dxdt + E(un ) ≤ E(un0 ) < d. (2.2) Ω
0
Claim ∀t ≥ 0, un (t) ∈ Σ1 . If not, let t∗ = min{t|un (t) ∈ Σ1 }, then un (t∗ ) ∈ ∂Σ1 , thus ∗ ∗ p E(un (t )) = d or |∇un (t )| dx = |un (t∗ )|q+1 dx. Ω
Ω
Obviously, the first equation contradicts (2.2); for the second, after simple calculation, we have E(un (t∗ )) = then
Ω
q+1−p p(q + 1)
|∇un (t∗ )|p dx,
Ω
p(q+1)
|∇un (t∗ )|p dx ≥ M − q+1−p by using Lemma 2.2 and Lemma 2.3. Thus, we have E(un (t∗ )) ≥
q + 1 − p − p(q+1) M q+1−p = d, p(q + 1)
again contradicts to (2.2). So for any t ≥ 0, un (t) ∈ Σ1 . Hence |∇un |p dx > |un |q dx, Ω
Ω
and E(un ) >
q+1−p p(q + 1)
|∇un |p dx.
Ω
According to (2.2), we have 0
so
T
Ω
wn (x)u2nt dxdt +
T 0
Ω
q+1−p p(q + 1)
|∇un |p dx ≤ E(un0 ) < d,
(2.3)
Ω
Ω
wn (x)u2nt dxdt < d,
(2.4)
p(q + 1) d. q+1−p
(2.5)
|∇un |p dx <
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Furthermore, Lemma 2.2 gives, for any n > 0 , (q+1)/p q+1 q+1 |un | dx ≤ M |∇un |p , Ω
and (2.3) implies that E(un0 ) ≥
Ω q+1−p p(q+1)
Ω
|∇un |p dx; therefore,
p(q + 1) q+1−p p E(un0 ) |∇un |p dx q+1−p Ω p(q + 1) q+1−p p (E(u0 ) + ε0 ) ≤ M q+1 |∇un |p dx. q+1−p Ω
|un |q+1 dx ≤ M q+1
Ω
q+1−p p For simplicity, denote M q+1 p(q+1) (E(u )+ε ) by δ (due to Lemma 2.3 and E(u0 )+ 0 0 q+1−p ε0 < d, we have 0 < δ < 1). Let γ = 1 − δ, then ∀n and t > 0 q+1 |un (t)| dx ≤ (1 − γ) |∇un (t)|p dx. (2.6) Ω
Ω
On the other hand, multiplying (2.1) by un and by integrating over Ω × [0, T ], inequality (2.6) implies 1 2 ≤ ≤
1 2 1 2
2
T
|∇un |p dxdt
wn (x)|un (x, T )| dx + Ω
wn (x)|u0 (x)|2 dx +
Ω
0
T
Ω
0
|un |q+1 dxdt
Ω
wn (x)|u0 (x)|2 dx + (1 − γ) Ω
0
T
|∇un |p dxdt.
Ω
So for some γ > 0 , 1 2
wn (x)|un (x, T )|2 dx + γ Ω
0
T
|∇un |p dxdt ≤ Ω
1 2
wn (x)|u0 (x)|2 dx, Ω
T
|∇un |p dxdt ≤ C, where C is a constant that is independent of n and T . By (2.4), (2.5), and (2.6), there exists subsequence (still denoted by un ) and function u such that for all T ≥ 0, thus
0
Ω
un → u ∇un ∇u
wn (x)unt un u
a.e. on QT , ut |x|s/2
in Lp (QT ), in L2 (QT ), in L∞ (0, T ; W01,p (Ω)) weakly ∗,
where “” denotes weak convergence, “→” denotes strong convergence. Thus, we get a global solution through limiting (2.1). In fact, it can be obtained from Boccardo and Murat([7]) and the limit u satisfies for any T > 0 p(q + 1) d, (2.7) |∇u|p dx < q +1−p Ω
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Wang: EXISTENCE OF GLOBAL SOLUTION AND BLOWUP PROBLEM
T
0
Ω
|ut |2 dxdt < d, |x|s
(2.8)
which completes the proof of Step 1. Step 2: Proof of (1.2). By Step 1, H(u(t)) > 0, ∀t ≥ 0. Thus q+1−p 1 q+1−p p H(u) > |∇u| dx + |∇u|p dx E(u) = p(q + 1) Ω q+1 p(q + 1) Ω p and E(u0 ) > q+1−p p(q+1) Ω |∇u| dx. Lemma 2.2 implies p(q + 1) q+1−p p E(u0 ) |u|q+1 dx ≤ M q+1 |∇u|p dx. q+1−p Ω Ω Denote M q+1
q+1−p p
p(q+1) q+1−p E(u0 )
279
(2.9)
by δ (0 < δ < 1). Let γ = 1 − δ, then
|u|q+1 dx ≤ (1 − γ)
Ω
|∇u|p dx.
(2.10)
Ω
|u(t)|2 d Let T > t0 be a fixed figure, we have 12 dt dx = −H(u(t)), the Hardy–Sobolev Ω |x|s inequality and the H¨ older inequality imply T |u(t)|2 |u(T )|2 |u(t)|2 1 1 1 H(u(τ ))dτ = dx − dx ≤ dx s s 2 Ω |x| 2 Ω |x| 2 Ω |x|s t N −s+2 p2 2N N ≤ C(N, s) |∇u| N −s+2 dx ≤ C(Ω, N, s) |∇u|p dx s. (2.11) Ω
Ω
On the other hand, on [t0 , ∞) we have 1 1 p E(u(t)) = |∇u| dx − |u|q+1 dx p Ω q+1 Ω 1 1 H(u(t)) − = |∇u|p dx + |∇u|p dx p Ω q+1 Ω q+1−p 1 q+1−p p H(u(t)) ≥ |∇u| dx + |∇u|p dx, = p(q + 1) Ω q+1 p(q + 1) Ω thus, by (2.11) and (2.12), we have on [t0 , T ] T p2 p(q + 1) 2 E(u(t)) H(u(τ ))dτ ≤ C(Ω, N, s) = C(Ω)(E(u(t))) p , q+1−p t
(2.12)
(2.13)
2
p where C(Ω) = C(Ω, N, s)( p(q+1) q+1−p ) . Furthermore, (2.10) implies on [t0 , ∞), γ |∇u(t)|p dx ≤ H(u(t)).
(2.14)
Ω
By (2.12) and (2.14), we have E(u(t)) ≤
q +1−p γp(q + 1)
+
1 )H(u(t) . q+1
(2.15)
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T 2 By (2.13) and (2.15), on [t0 , T ], we obtain C1 t E(u(τ ))dτ ≤ E(u(t)) p , where C1 = −1 q+1−p 1 C(Ω)( γp(q+1) + q+1 ) . Then, by the arbitrariness of T > t0 , it follows that ∞ 2 C1 E(u(τ ))dτ ≤ E(u(t)) p . t
Let T > t0 be sufficiently large such that C1−1 ≤ T , it follows that ∞ 2 E(u(τ ))dτ ≤ T E(u(t)) p , on [t0 , ∞). Setting y(t) = and E(u(t)) that
∞ t
(2.16)
t
t
E(u(τ ))dτ , it follows from (2.16) and the monotonicity of y(t)e T 2
t
2
t
t
y(t) ≤ y(T )e1− T ≤ T E(u(T )) p e1− T ≤ T E(u(t0 )) p e1− T , On the other hand, we get ∞ E(u(τ ))dτ ≥ t
t
T +t
∀t > T.
(2.17)
E(u(τ ))dτ ≥ T E(u(T + t)). 2
t
This inequality together with inequality (2.17) leads to E(u(T + t)) ≤ E(u(t0 )) p e1− T for all t > T. 2 t p 1− T , which implies the exponential From (2.12), we obtain ∇u pp ≤ p(q+1) q+1−p E(u(t0 )) e t
decay of solutions ∇u p ≤ Ce− pT with some constant C > 0 for large t > T , and this completes the proof of (1.2).
3
Proofs of Theorem 1.2 and Theorem 1.3
In this section, we consider the finite time blowup of Equation (1.1). We first prove Theorem 1.2. Proof of Theorem 1.2 In fact, we can prove a more general result: If there exists some t0 such that E(u(t0 )) ≤ 0, then u(x, t; u0 ) will blow up in finite time. We shall employ the classical concavity method. Suppose that tmax = ∞ and denote t 2 f (t) = 12 t0 Ω |u| |x|s dxdτ. We perform standard manipulations: t t0
Ω
|uτ |2 1 dxdτ + |x|s p
|∇u|p dx −
Ω
1 q+1
|u|q+1 dx = E(u(t0 )),
|u|2 1 f (t) = dx, 2 Ω |x|s uut p dx = − |∇u| dx + |u|q+1 dx. f (t) = s Ω |x| Ω Ω
Then we have q+1−p f (t) = p
p
|∇u| dx − (q + 1)E(u(t0 )) + (q + 1) Ω
As E(u(t0 )) ≤ 0, we have q+1−p p
Ω
(3.1)
Ω
(3.2) (3.3)
t t0
|∇u|p dx − (q + 1)E(u(t0 )) > 0,
Ω
|uτ |2 dxdτ. |x|s
(3.4)
(3.5)
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for all t ≥ t0 . If we have tmax = ∞, then there exists a constant c > 0 such that ∀t > t1 > t0 there holds t1 |uτ |2 dxdτ ≥ c > 0, f (t) > (q + 1) s t0 Ω |x| this inequality would yield lim f (t) = lim f (t) = ∞. t→∞
t→∞
On the other hand, equation (3.4) and (3.5) imply t
|uτ |2 dxdτ, s t0 Ω |x| t |u (τ )|2 q + 1 t |u(τ )|2 τ dxdτ dxdτ f (t)f (t) ≥ s 2 |x|s t0 Ω |x| t0 Ω 2 q + 1 t uuτ ≥ dxdτ s 2 t0 Ω |x| t 2 q+1 q+1 (f (t) − f (t0 ))2 , = f (τ )dτ = 2 2 t0
f (t) ≥ (q + 1)
and as t → ∞, we have for some α > 0 , f (t)f (t) ≥ (1 + α)(f (t))2 . Hence (f (t))−α is concave on [t0 , ∞), (f (t))−α > 0 and lim f −α (t) = 0. This contradiction proves that tmax < ∞. t→∞ Proof of Theorem 1.3 Obviously, if u0 ∈ Σ2 , then by Lemma 2.1, we have p(q + 1) d< |∇u0 |p dx < |u0 |q+1 dx. q+1−p Ω Ω If u(x, t) is a global solution, then u(x, t) does not strongly converge to 0 in W01,p (Ω). Other wise, there exists t∗ , 0 < t∗ < ∞ such that E(u(t∗ )) < d. But Ω |u(t∗ )|q+1 dx = Ω |∇u(t∗ )|p dx. Using the same method as Step 1 in the proof of Theorem 1.1, we can obtain a contradiction. Thus u(x, t) can not strongly converge to 0 in W01,p (Ω). Furthermore, we have the following claim: Claim If u0 ∈ Σ2 , then there exists a constant η > 0 which is sufficently small and independent of t and rely on u0 such that q+1 |u(x, t)| dx ≥ (1 + η) |∇u(x, t)|p dx, (3.6) Ω
Ω
for any t ∈ [0, Tmax ). First, we have p(q + 1) d< q+1−p
|∇u|p dx <
Ω
|u|q+1 dx,
Ω
for all t ∈ [0, Tmax). Indeed, if there exists a t∗ ∈ [0, Tmax) such that ∗ q+1 |u(x, t )| dx = |∇u(x, t∗ )|p dx, Ω
then we have
|∇u(x, t∗ )|p dx =
Ω
But d > E(u(x, t∗ )) =
Ω
q+1−p p(q+1)
Ω
Ω
|u(x, t∗ )|q+1 dx ≥
p(q + 1) d. q+1−p
|∇u(x, t∗ )|p dx, which is a contradiction.
(3.7)
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For the proof of (3.6), we can apply the method of Ishii [8] and it is omitted here. Now, we can complete the proof of Theorem 1.3. We shall employ the same argument as t 2 the proof of Theorem 1.2. Suppose that tmax = ∞ and denote f (x) = 12 0 Ω |u| |x|s dxdτ . From (3.1)–(3.3) and (3.6), we obtain f (t) ≥ − |∇u|p dx + (1 + η) |∇u(x, t)|p dx = η |∇u(x, t)|p dx. Ω
Ω
Ω
If we have tmax = ∞, then this inequality would yield lim f (t) = lim f (t) = ∞. t→∞
By (3.1), (3.2), and (3.3) we have f (t) =
q+1−p p
|∇u|p dx − (q + 1)E(u0 ) + (q + 1)
Ω
t 0
which implies
f (t) ≥ (q + 1)
t 0
and
t→∞
Ω
Ω
|uτ |2 dxdτ, |x|s
|uτ |2 dxdτ, |x|s
q+1 (f (t) − f (0))2 . 2 By the argument as the proof of Theorem 1.2, we obtain a contradiction. f (t)f (t) ≥
Acknowledgments The author would like to thank Professor Feng Zhou for his constant encouragement and valuable suggestion. References 1 Fujita H. On the blowing up of solutions of the Cauchy problem for ut = ∆u + u1+α . J Fac Sci Univ Tokyo Sect I, 1966, 13: 109–124 2 Hayakawa K. On nonexistence of global solutions of some semilinear parabolic equations. Proc Japan Acad, 1973, 49: 503–505 3 Kobayashi K, Sirao T, Tanaka H. On the growing up problem for semilinear heat equations. J Math Soc Japan, 1977, 29: 407–424 4 Pao C V. Nonlinear Parabolic and Elliptic Equations. New York, London: Plenum Press, 1992 5 Tan Z. The reaction diffusion equations with special diffusion processes. Chinese Journal of Contemporary Mathematics, 2001, 22(4) 6 Tan Z. Non-Newton filtration equation with special medium void. Acta Mathematica Scientia, 2004, 24B(1): 118–128 7 Fila M. Boundedness of global solutions of nonlinear diffusion equations. J of Diff Equs, 1992, 98: 226–240 8 Ishii H. Asymptotic stability and blowing up of solutions of some nonlinear equations. J of Diff Equs, 1977, 26: 291–319 9 Payne L E, Sattinger D H. Saddle points and instability of non-linear hyperbolic equations. Israel J of Math, 1975, 22(3/4): 273–303 10 Ye Q X, Li Z Y. An Introduction to Reaction Diffusion Equations. Beijing: Science Press (in Chinese), 1985 11 Tsutsumi M. Existence and nonexistence of global solutions for nonlinear parabolic equations. Publ Res Inst Math Sci, 1972, 8: 211–229 12 Badiale M, Tarantello G. A Sobolev–Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch Rational Mech Anal, 2002, 163: 259–293 13 Ladyzenskaja O A, Solonnikov V A, Ural’ceva N N. Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs 23. Providence R L: Amer Math Soc, 1968