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Qualitative properties for solutions of semilinear heat equations with strong absorption
J. Math. Anal. Appl. 281 (2003) 382–394 www.elsevier.com/locate/jmaa
Qualitative properties for solutions of semilinear heat equations with strong absorption ✩ Jun-Jie Li Department of Mathematics, Yuquan Campus, Zhejiang University, Hangzhou 310027, People’s Republic of China Received 13 June 2001 Submitted by J. McKenna
Abstract We are concerned with nonnegative solutions to the Cauchy problem ut − ∆u + b(x, t)|u|p−1 u = 0 in RN × (0, ∞), u(x, 0) = u0 (x)
in RN .
Here 0 < p < 1, 0 < u0 (x) f0 exp{A|x|α } (f0 > 0, α 2, A > 0). We investigate the dependence of such phenomena as comparison principle, instantaneous shrinking of the support and existence on the behaviour of b(x, t). 2003 Elsevier Science (USA). All rights reserved. Keywords: Instantaneous shrinking of the support; Existence; Comparison principle
1. Introduction In this paper we continue the study begun in [13], and investigate qualitative properties of the nonnegative solutions to the Cauchy problem Lu := ut − ∆u + b(x, t)|u|p−1 u = 0 in RN × (0, ∞), (1.1) in RN , u(x, 0) = u0 (x) where ✩
This research was supported by the National Science Foundation of China and the Scientific Research Foundation for Returned Scholars of the Ministry of Education of China. E-mail address: [email protected]. 0022-247X/03/$ – see front matter 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0022-247X(03)00109-4
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0 < p < 1,
383
(1.2)
b(x, t) 0, and u0 (x) satisfies hypotheses (H1 ) u0 (x) ∈ C(RN ), 0 < u0 (x) f (x) := f0 exp{A|x|α } in RN (α 2, A > 0, f0 > 0). We are mainly interested in the following: (a) Comparison principles for subsolutions and supersolutions of (1.1); (b) instantaneous shrinking of the support for the solution u(x, t) of (1.1). By definition, the support of u(x, t) is bounded for t > 0 although the initial value u0 (x) is positive everywhere; and (c) global existence for solutions of problem (1.1). There were a lot of results on phenomenon (b). We mention here only one of the pioneering paper [3] and some recent results [6,8,10,11,13,14], where the reader can find further references. Most attention has been concentrated on problems with initial data vanishing at infinity. However, several authors have considered more general classes of initial data. For instance, [7] deals with a continuous u0 (x) of power-like growth at infinity, [11] assumed that u0 ∈ L2 (RN ), while in [13] u0 (x) is allowed to be of O(exp(|x|2h(|x|))) growth at infinity, where h(r) tends to zero as r → ∞. In Section 2 we study phenomenon (b) for solutions of (1.1) when u0 (x) is subject to (H1 ). When p = 1, and a subsolution u− (x, t) and a supersolution u+ (x, t) are subject to the condition u± (x, t) = O exp k|x|2 for some positive constant k, there were well-known results for property (a); see, for instance, [4,5] or [12]. But in our case solutions of (1.1) are in general not subject to the above condition. To our knowledge, there are few results on this aspect. As for property (c), it is well known that in general problem (1.1) does not possess a global solution. In Section 3 we focus our attention on the comparison principle, global existence and nonglobal existence for solutions of problem (1.1).
2. Instantaneous shrinking of the support In the sequel we use C to denote different constants, their values may change from line to the next. The statement that a constant depends only on the data means that this constant can be determined in terms of N , p, A, α and f0 . We also use Br (x0 ) to denote the ball in RN of radius r and centered at x0 . Definition 2.1. We say a function u(x, t) ∈ C(RN × [0, ∞)) ∩ C 2,1 (RN × (0, ∞)) is a solution of problem (1.1) if it is nonnegative and satisfies (1.1) in a classical sense. Remark 2.1. For α 2, when u0 (x) is subject to (H1 ), in general we do not know at the moment the precise and proper conditions on b(x, t) under which problem (1.1) is
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global solvable in the sense of Definition 2.1. However, in Section 3 we will get a global existence result to problem (1.1) when b(x, t) is under some condition, and this condition is in a certain sense sharp for α = 2. Definition 2.2. For M0 1, Eα,M0 is the set of all nonnegative function u(x, t) in C(RN × [0, ∞)) ∩ C 2,1 (RN × (0, ∞)) satisfying (I) if α = 2, for any T > 0, there are constants M(T ) and C(T ) such that in RN × [0, T ]; u(x, t) M(T ) exp C(T )|x|2 (II) if α > 2,
u(x, t) M0 f0 exp A|x|α
in RN × [0, ∞).
For an arbitrary nonnegative continuous function v(x, t) defined in RN × [0, T0 ) (0 < T0 ∞), we set ξ(t; v) = sup |x|; v(x, t > 0) , t ∈ [0, T0 ), (2.1) and S(t; v) = x ∈ RN ; v(x, t) > 0 ,
t ∈ [0, T0 ).
(2.2)
Definition 2.3. Assume that v(x, t) ∈ C(RN × [0, T0 )), v 0 and ξ(0; v) = ∞. We say that instantaneous shrinking of the support (briefly, ISS) occurs for v if there exists τ > 0 such that ξ(t; v) < ∞
for all t ∈ (0, τ ].
It is well known that one of the most normal way in proving an instantaneous shrinking of the support assertion for solutions of problem (1.1) is by using a comparison principle; see, for instance, [7,10,13]. But for α > 2, and b(x, t) is subject to the condition given in Theorem 2.1, a comparison principle here is inadequate. It is quite natural to think that, in order to guarantee the instantaneous shrinking of the support, the coefficient of the absorption term should be “strong” enough. In this section we first exploit the idea of [3,14] to prove Theorem 2.1. We will establish in Section 3 a comparison principle for problem (1.1) when b(x, t) satisfies a stronger condition than in that given in Theorem 2.1. Theorem 2.1. Assume (1.2) and (H1 ) hold, and let u(x, t) ∈ Eα,M0 be a solution of (1.1). If (a) for α = 2, b(x, t) b0 (2A)2 |x|2f 1−p (x),
(2.3)
(b) for α > 2, 1−p
b(x, t) b0 (Aα)2 M0
|x|2α−2f 1−p (x),
(2.4)
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where b0 > 1, then ISS occurs for u. Further, for α = 2, γ ξ(t; u) √ , ∀t ∈ (0, τ ], t
(2.5)
where γ depends only on the data and b0 , τ is small enough; for α > 2, and for any β > 1/(2(α − 1)), Cβ + l0 , ∀t ∈ (0, ∞), (2.6) tβ where Cβ depends only on the data, M0 , b0 and β; l0 depends only on the data, M0 and b0 . ξ(t; u)
Proof. Case 1: α = 2. Denote ω = 2/(1 − p). We first so choose ε0 > 0 that b0 (1 + ε0 )2−p ,
(2.7)
then take β0 = ε0 (2A)2 /(4ω), and then introduce the function ω U (x, t) = (1 + ε0 )f (x) 1 − tβ0 |x|2 + := (1 + ε0 )f (x)Z ω (x, t), where we have used the notation a+ = max{a, 0}. We wish to prove for small τ > 0, u(x, t) U (x, t) in RN × [0, τ ];
(2.8)
then assertion (2.5) easily follows from (2.8). Set Q+ = (x, t) ∈ RN × (0, ∞); Z(x, t) > 0 . It is obvious that U (x, t) ∈ C 2,1 (RN × [0, ∞)), 0 < Z(x, t) 1 in Q+ and LU = 0 in RN × (0, ∞) \ Q+ . A direct calculation yields b(x, t) LU = (1 + ε0 )Z ωp f p − β0 ω|x|2Z ω(1−p)−1 f − Z ω(1−p)∆f (1 + ε0 )1−p − ωZ ω(1−p)−1f ∆Z − 2ωZ ω(1−p)−1∇f · ∇Z − ω(ω − 1)|∇Z|2 f . Let l > 1 be fixed. We compute in Q+ ∩ {|x| l, t > 0}
∆Z(x, t) 0, ∆f (x) = (2A)2 |x|2 + 2NA f (x), 4 ∇f · ∇Z 0, |∇Z|2 2 . l By (2.3), (2.7) and using the above estimates, we get 4ω(ω − 1) ωp 2 2 2 >0 LU (1 + ε0 )Z f ε0 (2A) |x| − β0 ω|x| − l2
(2.9)
in Q+ ∩ {|x| l, t > 0}, provided that l is fixed large enough. From U (x, 0) > u(x, 0) = u0 (x) in RN and the continuity of U and u, there exists τ = τ (l) > 0 such that u(x, t) U (x, t),
(x, t) ∈ Bl (0) × [0, τ ].
(2.10)
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From (2.9), (2.10) and LU = 0 in RN × (0, ∞) \ Q+ , an application of Theorem 3.3 of Section 3 to u and U in {|x| l, 0 t τ } leads to u(x, t) U (x, t) in |x| l, 0 t τ . (2.11) (2.8) easily follows from (2.10) and (2.11). Case 2: α > 2. Fix some t 0 and choose x 0 ∈ S(t 0 ; u). Let a > 1, with a/l0 small, be fixed, and introduce the function 1 1 U (x, t) = (t 0 − t) 1−p |x 0 | (1−p)β exp A|x 0 |α 0 α−1 ω |x | + M0 |x − x 0 |ω f0 exp A|x|α . a A direct calculation yields p 1 1 LU = b(x, t)U p (x, t) − (t 0 − t) 1−p |x 0 | (1−p)β exp A|x 0|α 1−p
2 2α−2 + Aα(N + α − 2)|x|α−2 − (Aα) |x| 0 α−1 ω |x | × M0 f0 |x − x 0 |ω exp A|x|α a 0 α−1 ω−2 |x | − M0 f0 |x − x 0 |ω−2 exp A|x|α a ω(N + ω − 2) 0 2α−2 2Aαω α−2 0 2α−2 0 . (2.12) x + |x| |x | x − x |x | × i i i a2 a2 One can easily see that in Ba/|x 0 |α−1 (x 0 ), when |x 0 | l0 > 2a,
|x| 2a 0 |x | 1 + |x|. 1 + |xa0 |α |x 0 |α
(2.13)
By (2.13) we compute in Ba/|x 0 |α−1 (x 0 ) 2Aαω α−2 0 2α−2 ω(ω + N − 2) 0 2α−2 C 2α−2 0 |x| |x | x − x |x | |x| , x i i i + a2 a a2 p |x 0 | (1−p)β |x|2α−2 exp Ap|x 0 |α exp A(1 − p)|x|α p exp(−Ca) 0 2α−2+ (1−p)β |x | exp A|x 0 |α , C where C depends only on the data. By the following elementary inequality: for a 0, b 0 and ε > 0, (a + b)p εa p + (1 − ε)bp , we have U p (x, t)
p p b0 − 1 0 (t − t) 1−p |x0 | (1−p)β exp pA|x 0 |α 2b0 b0 + 1 p p |x 0 |α−1 |x − x 0 | pω + M0 f0 exp pA|x|α . 2b0 a
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From (2.4), (2.12) and the above estimates, we get p 1 1 b0 − 1 0 2α−2−1/β LU − + |x | (t 0 − t) 1−p |x 0 | (1−p)β exp A|x 0|α 1 − p C exp(Ca) 2 2α−2 α−2 + Aα(Aα + N − 2)|x| − (Aα) |x| |x 0 |α−1 |x − x 0 | pω × M0 f0 exp A|x|α a CM0 2α−2 |x 0 |α−1 |x − x 0 | pω |x| exp A|x|α − a a 0 α−1 |x | |x − x 0 | pω 1 + b0 M0 f0 (Aα)2 |x|2α−2 exp A|x|α . + 2 a
(2.14)
From (2.14), first choose a large, then fix l0 > 2a large enough to obtain LU > 0 in Ba/|x 0 |α−1 (x 0 ) × (0, t 0 ].
(2.15)
By standard comparison theorems, u − U must attain maximum on the parabolic boundary of the cylinder Ba/|x 0 |α−1 (x 0 ) × [0, t 0 ]. Since on ∂Ba/|x 0|α−1 (x 0 ) × [0, t 0 ], u U , but 0 < u(x 0 , t 0 ) = u(x 0 , t 0 ) − U (x 0 , t 0 ), there must exist some point x¯ ∈ Ba/|x 0 |α−1 (x 0 ) for which 1 1 (t 0 ) 1−p |x 0 | (1−p)β exp A|x 0 |α U (x, ¯ 0) < u(x, ¯ 0) α ¯ C exp A|x 0|α , M0 exp A|x|
(2.16)
where C depends only on the data. Hence from (2.16), the definition of S(t; u) and the arbitrarity of x 0 ∈ S(t 0 ; u) it easily follows assertion (2.6). ✷ Now we pass to an extinction phenomenon Theorem 2.2. Assume p and u0 (x) as in Theorem 2.1, and let u ∈ Eα,M0 be a solution of (1.1). If (a) for α = 2, b(x, t) b0 (2A)2 |x|2f 1−p (x) + b1 ,
(2.17)
(b) for α > 2, 1−p
b(x, t) b0 (Aα)2 M0
|x|2α−2f 1−p (x) + b1 ,
where b0 > 1 and b1 > 0, then u vanishes identically after some time T0 > 0. Proof. For α = 2, by (2.8) we have ω u(x, t) (1 + ε0 )f (x) 1 − β0 t|x|2 + ,
(x, t) ∈ RN × [0, τ ],
(2.18)
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0 in RN for some constant M 0 . For β1 > 0 to be fixed, set and hence u(x, τ ) M 1 1−p 1−p U (x, t) = M − β (t − τ ) 1 + . 0
From (2.17), one has LU > 0 in RN × (τ, ∞), provided that β1 is chosen small enough. Then by comparison (see Theorem 3.2 of Section 3) 1 1−p 1−p u(x, t) U (x, t) = M − β (t − τ ) 1 + 0
in RN × [τ, ∞),
1−p /β1 . which implies u(x, t) ≡ 0, t τ + M 0 For α > 2, by Theorem 2.1 and the definition of Eα,M0 , there exist M1 and l1 > l0 such that u(x, t) M1
in RN × [1, ∞),
u(x, t) = 0 for |x| l1 , t 1.
(2.19)
For β2 > 0 to be fixed, set 1−p 1 V (x, t) = M1 − β2 (t − 1) +1−p . From (2.18), we have LU > 0 in RN × (1, ∞), provided that β2 is chosen small enough. By (2.19), similarly as in above we can get 1−p
u(x, t) ≡ 0,
t 1+
M1 . β2
✷
The following theorem shows that in a certain sense condition (2.3) is sharp. Theorem 2.3. Assume α = 2 and 0 < p < 1, and let u ∈ C(RN × [0, T0 )) ∩ C 2,1 (RN × (0, T0 )) (0 < T0 ∞) be a solution of (1.1) with u0 (x) = f (x) = f0 exp(A|x|2). If
0 b(x, t) (2A)2 |x|2 + 2NA f 1−p (x), then ISS does not occur for u. Proof. Set V (x, t) = f (x). A straightforward calculation leads to
LV = − (2A)2 |x|2 + 2NA V + b(x, t)V p 0. Then by comparison (see Theorem 3.2 of Section 3), we have u(x, t) V (x, t) = f (x) in RN × [0, T0 ), whence the result.
✷
Remark 2.2. For the case α > 2, since the comparison principle is inadequate when 0 1−p b(x, t) (Aα)2 |x|2α−2M0 f 1−p (x), the counter example is left open.
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3. Existence and comparison principle We begin with a global existence result. Theorem 3.1. If 0 < p 1, and b(x, t) ∈ C α0 ,α0 /2 (RN × [0, ∞)) (α0 ∈ (0, 1)) satisfies
b(x, t) (Aα)2 |x|2α−2 + Aα(N + α − 2)|x|α−2 f 1−p (x), (3.1) then there exists a solution u(x, t) of (1.1) in the sense of Definition 2.1 with 0 u(x, t) f (x) in RN × [0, ∞).
(3.2)
Proof. For any large n, let un (x, t) be the unique solution of the approximated problem p−1 u = 0 in B (0) × (0, n2 ], (3.3) n n (un )t − ∆un + b(x, t)|un | on ∂Bn (0) × (0, n2 ], un (x, t) = 0 (3.4) un (x, 0) = u0 (x)ψn (x) in Bn (0), (3.5) where ψn (x) is a smooth cutoff function in Bn (0) satisfying 0 ψn 1, ψn (x) = 0 on ∂Bn (0) and ψn (x) = 1 in Bn−1 (0). Clearly, we have un (x, t) 0 in Bn (0) × [0, n2 ]. To estimate the upper bound of un (x, t) consider the function V (x, t) = f (x). From the hypotheses one has
LV = − (Aα)2 |x|2α−2 + Aα(N + α − 2)|x|α−2 f + b(x, t)f p 0, and hence, by comparison, un (x, t) V (x, t) = f (x)
in Bn (0) × [0, n2 ].
(3.6)
From (3.6), according to the well-known interior estimates of solutions and their continuity module (see, for instance, [1,2,4,9,12]), it follows that for any l 1 and n l + 4, we have un C 2+γ0 ,1+γ0 /2 (Bl (0)×[1/ l 2,l 2 ] Cl
(3.7)
un (x, t) − un (x , t ) ωl |x − x | + |t − t |1/2
(3.8)
and
for all (x, t), (x , t ) ∈ Bl (0) × [0, l 2 ], where the positive constant γ0 is independent of n and l; Cl and ωl (r) are independent of n, and ωl (r) tends to zero as r ↓ 0. The above estimates together with Arzela’s lemma and a diagonal argument imply that there exists u(x, t) ∈ C(RN × [0, ∞)) ∩ C 2,1 (RN × (0, ∞)) such that, after extracting a subsequence if necessary, lim un (x, t) = u(x, t) f (x)
n→∞
in RN × [0, ∞),
and u(x, t) solves (1.1) in the classical sense.
✷
Next, when b(x, t) is subject to some conditions, we establish a comparison theorem as follows:
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Theorem 3.2 (Comparison principle). Assume 0 < p 1. Let u− ∈ Eα,M0 , 0 u+ ∈ C(RN × [0, ∞)) ∩ C 2,1 (RN × (0, ∞)) satisfy − Lu 0 in RN × (0, ∞), Lu+ 0 in RN × (0, ∞), and u− (x, 0) u+ (x, 0) in RN . (a) If α = 2 and b(x, t) 0, then u− (x, t) u+ (x, t)
in RN × [0, ∞).
(b) If α > 2 and b(x, t) satisfies, in addition to b(x, t) 0, the following
1−p 1 b(x, t) (Aα)2 |x|2α−2 + Aα(N + α − 2 + θ0 )|x|α−2 M0 f (x) p in {|x| l1 } × [0, ∞), where θ0 > 0, l1 1, then u− (x, t) u+ (x, t)
(3.9)
in RN × [0, ∞).
Proof. The assertion for α = 2 was established in [13, Theorem 2.2]. Here we only give the proof for α > 2. Suppose the contrary, then there exists a point (x 0 , t 0 ) ∈ RN × (0, ∞) such that u− (x 0 , t 0 ) − u+ (x 0 , t 0 ) = 2a > 0.
(3.10)
Set
w(x, t) = u− (x, t) − u+ (x, t) + a ,
c(x, t) = (Aα)2 |x|2α−2 + Aα(N + α − 2 + θ0 )|x|α−2 χ{|x|l1 } (x),
where χ{·} (x) is the characteristic function of {·}. From the hypotheses we see that
(3.11) wt − ∆w + b(x, t) (u− )p − (u+ + a)p 0. By (3.9), (3.11), in view of the elementary inequality p(A − B) for A B > 0, (3.12) A1−p and proceeding similarly as in the proof of Theorem 2.2 in [13], we deduce that in the weak sense Ap − B p
(w+ )t − ∆w+ + c(x, t)w+ 0.
(3.13)
Now choose a cutoff function ω(x) ∈ satisfying 0 ω 1, ω(x) = 1 in 0 0 B1/2 (x ) and ω(x) = 0 on ∂B1 (x ). For any integer n > l1 such that B2 (x 0 ) ⊂ Bn (0), let ϕ(x, t) be the solution of the problem := ϕt + ∆ϕ − c(x, t)ϕ = 0 in Bn (0) × [0, t 0 ), (3.14) Lϕ ϕ=0 on ∂Bn (0) × [0, t 0 ), (3.15) ϕ(x, t 0 ) = ω(x) in Bn (0). (3.16) C02 (B1 (x 0 ))
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It is obvious that 0 ϕ(x, t) 1. Moreover, we have Claim. There exist constants M1 and M2 depending only on the data, θ0 , l1 , x 0 and t 0 such that exp{−A|x|α } ϕ(x, t) M1 , (x, t) ∈ Bn (0) × [0, t 0 ], (3.17) (1 + |x|2 )k1 and
α ∂ϕ M2 exp(−An ) ∂ν nk2
on ∂Bn (0) × [0, t 0 ],
(3.18)
provided n is large enough. Here α + N − 2 θ0 θ0 + , k2 = N − 1 + ; 2 4 2 ν is the outward unit normal to ∂Bn (0). k1 =
Suppose for the moment that this claim has been proved. From w(x, 0) < 0 and the continuity of w, there is τn ∈ (0, 1/n] such that w+ (x, t) = 0 in Bn (0) × [0, τn ]. We take ϕ(x, t) as the test function in the weak inequality (3.13), taking into account (3.14)–(3.16) and (3.18), to obtain w+ (x, t 0 ) w+ (x, t 0 )ϕ(x, t 0 ) B 1 (x 0 )
Bn (0)
2
t 0
t 0
w+ (ϕt + ∆ϕ − cϕ) − τn Bn (0)
t 0
∂ϕ ∂ν
τn ∂Bn (0)
=−
w+
w+
∂ϕ Cn−θ0 /2 , ∂ν
τn ∂Bn (0)
provided n is large enough; here C depends only on the data, M0 , θ0 , l1 , x 0 and t 0 . Letting n → ∞, we get w+ (x, t 0 ) 0, B 1 (x 0 ) 2
whence w+ (x, t 0 ) ≡ 0 in B1/2 (x 0 ), which contradicts (3.10). ✷ Proof of the claim. For λ0 > 1 to be fixed, set ψ(x, t) = M
exp(λ0 (t 0 − t)) exp(−A|x|α ) , (1 + |x|2 )k1
where M = (1 + (|x 0 | + 2)2 )k1 exp{A(|x 0 | + 2)α }. A quick calculation gives
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− λ0 + Aα(N + α − 2)|x|α−2 + c(x, t) ψ Lψ
4k1(1 + k1 ) + (Aα)2 |x|2α−2 + 4k1 Aα|x|α−2 + ψ. 1 + |x|2 From the definition of c(x, t), it is easily seen that Lψ 0, provided λ0 is large enough. Hence, by comparison, ϕ(x, t) ψ(x, t) M1
exp(−A|x|α ) (1 + |x|2)k1
in (x, t) ∈ Bn (0) × [0, t 0 ],
for some constant M1 depending only on the data, θ0 , l1 , x 0 and t 0 . = n1−α , in (Bn (0) \ B (0)) × [0, t 0 ] We turn now to the proof of (3.18). Denote R n−R consider the function α (n − |x|) exp(−An ) , V (x, t) = ϕ(x, t) − M k n2 where M is a positive constant to be fixed. One easily see that LV 0 in (Bn (0) \ Bn−R(0)) × (0, t 0 ), which implies that V attains or |x| = n. By (3.17), the definition of R its maximum only on t = t 0 , or |x| = n − R, large enough, we have V (x, t 0 ) 0, x ∈ Bn (0) \ B (0), V (x, t) 0 on and fixing M n−R ∂Bn−R(0) × [0, t 0 ] and V (x, t) = 0 on ∂Bn (0) × [0, t 0 ], and hence sup (Bn (0)\Bn−R(0))×[0,t 0]
V (x, t) = 0.
Therefore, ∂V 0 on ∂Bn (0) × [0, t 0 ], ∂ν which yields α ∂ϕ exp(−An ) on ∂Bn (0) × [0, t 0 ]. −M ∂ν nk2 On the other hand, it is clear that
∂ϕ 0 on ∂Bn (0) × [0, t 0 ]. ∂ν Thus inequality (3.18) immediately follows from (3.19) and (3.20). ✷
(3.19)
(3.20)
Along the lines of the proof of Theorem 3.2, one can easily see the validity of the following: Theorem 3.3. For 0 < T0 ∞, 0 < R0 , assume p and b(x, t) as in Theorem 3.2. Let R0 (0)) × (0, T0 ]) be nonnegative and u± ∈ C((RN \ BR0 (0)) × [0, T0]) ∩ C 2,1 ((RN \ B satisfy − N (0)) × (0, T ), R0 0 Lu 0 in (R \ B + R0 (0)) × (0, T0 ), Lu 0 in (RN \ B − u (x, 0) u+ (x, 0), x ∈ RN \ BR0 (0),
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and u− (x, t) u+ (x, t),
|x| = R0 , t ∈ [0, T0 ].
u−
∈ Eα,M0 . Then u (x, t) u (x, t) in RN \ BR0 (0) × [0, T0 ].
Assume further that −
+
From Theorems 3.1 and 3.2, we have Corollary 3.1. If 0 < p < 1 and b(x, t) ∈ C α0 ,α0 /2 (RN × [0, ∞)) (α0 ∈ (0, 1)) satisfies (a) for α = 2,
b(x, t) (2A)2 |x|2 + 2NA f 1−p (x),
(b) for α > 2, b(x, t)
1 (Aα)2 |x|2α−2 + Aα(N + α − 2 + θ0 )|x|α−2 f 1−p (x), p
then problem (1.1) is uniquely solvable in Eα,1 . The following result shows the sharpness of condition (3.1) for the case α = 2. Theorem 3.4. If 0 p 1, α = 2 and
0 b(x, t) (1 − ε1 )(2A)2|x|2 + 2NA f 1−p (x),
(3.21)
where ε1 ∈ (0, 1), then there is no global solution in E2,M0 with u0 (x) = f (x). Proof. Suppose the contrary, then there exists a global solution u(x, t) ∈ E2,M0 to problem (1.1). For any β > 0, we set A
V (β) (x, t) = f0 e 1−βt |x| , 2
(x, t) ∈ RN × [0, 1/β).
By (3.21) we compute
2 p βA 2A 2NA 2 (β) 2 V (β) + b(x, t) V (β) |x| V − |x| + LV = 1 − βt 1 − βt (1 − βt)2
2 2A βA 2 2 V (β) 0, |x| − ε |x| 1 1 − βt (1 − βt)2 provided β 4Aε1 . Then by comparison, (β)
A
u(x, t) V (β) (x, t) = f0 e 1−βt
|x|2
in RN × [0, 1/β),
which implies limt →1/β u(x, t) = ∞, x = 0, a contradiction. ✷ Remark 3.1. The counter example for the case α > 2, when b(x, t) satisfies
0 b(x, t) (1 − ε1 ) (Aα)2 |x|2α−2 + Aα(N + α − 2)|x|α−2 f 1−p (x), is left open since the lack of a comparison theorem.
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Acknowledgments The author wishes to express his thanks to Professor A.S. Kalashnikov, who brought the author’s attention to the present subject and encouraged him through fruitful discussions and valuable suggestions. This work was carried out during the author’s visit to Moscow State University, to which he is deeply grateful for its hospitality. The author is also greatly indebted to China Scholarship Council for providing this visit for him. The author would like to thank the referees for their very helpful comments and suggestions.
References [1] M. Bertsch, A class of degenerate diffusion equations with a singular nonlinear term, Nonlinear Anal. 7 (1983) 117–127. [2] E. DiBeneditto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients, Ann. Scuola Norm. Sup. 13 (1986) 485–535. [3] L.C. Evans, B.F. Knerr, Instantaneous shrinking of the support of non-negative solutions to certain nonlinear parabolic equations and variational inequalities, Illinois J. Math. 23 (1979) 153–166. [4] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice–Hall, 1964. [5] A.M. Il’in, A.S. Kalashnikov, O.A. Oleinik, Second order linear equations of parabolic type, Russian Math. Surveys 17 (1962) 1–143. [6] A.S. Kalashnikov, Some problems of the qualitative theory of nonlinear second-order parabolic equations, Russian Math. Surveys 42 (1987) 169–222. [7] A.S. Kalashnikov, On quasilinear degenerating parabolic equation with singular lower-order terms and growing initial values, Differentsial’nye Uravneniya 29 (1993) 999–1009. [8] A.S. Kalashnikov, Instantaneous shrinking of the support for solutions to certain parabolic equations and systems, Rend. Mat. Accad. Lincei Ser. 9 8 (1997) 263–272. [9] R. Kersner, Degenerate parabolic equations with general nonlinearities, Nonlinear Anal. 4 (1980) 1043– 1062. [10] R. Kersner, F. Nicolosi, The nonlinear heat equation with absorption: effects of variable coefficients, J. Math. Anal. Appl. 170 (1992) 551–566. [11] R. Kersner, A. Shishkov, Instantaneous shrinking of the support of energy solutions, J. Math. Anal. Appl. 198 (1996) 729–750. [12] O.A. Ladyzenskaja, N.N. Ural’cevar, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968. [13] J. Li, Instantaneous shrinking of the support of solutions to certain parabolic equations with unbounded initial data, Nonlinear Anal. 48 (2002) 1–12. [14] M. Ughi, On the porous media equation with either source or absorption, Cuad. Inst. Mat. Beppo Levi 22 (1991) 1–72.
Qualitative properties for solutions of semilinear heat equations with strong absorption ✩ Jun-Jie Li Department of Mathematics, Yuquan Campus, Zhejiang University, Hangzhou 310027, People’s Republic of China Received 13 June 2001 Submitted by J. McKenna
Abstract We are concerned with nonnegative solutions to the Cauchy problem ut − ∆u + b(x, t)|u|p−1 u = 0 in RN × (0, ∞), u(x, 0) = u0 (x)
in RN .
Here 0 < p < 1, 0 < u0 (x) f0 exp{A|x|α } (f0 > 0, α 2, A > 0). We investigate the dependence of such phenomena as comparison principle, instantaneous shrinking of the support and existence on the behaviour of b(x, t). 2003 Elsevier Science (USA). All rights reserved. Keywords: Instantaneous shrinking of the support; Existence; Comparison principle
1. Introduction In this paper we continue the study begun in [13], and investigate qualitative properties of the nonnegative solutions to the Cauchy problem Lu := ut − ∆u + b(x, t)|u|p−1 u = 0 in RN × (0, ∞), (1.1) in RN , u(x, 0) = u0 (x) where ✩
This research was supported by the National Science Foundation of China and the Scientific Research Foundation for Returned Scholars of the Ministry of Education of China. E-mail address: [email protected]. 0022-247X/03/$ – see front matter 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0022-247X(03)00109-4
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0 < p < 1,
383
(1.2)
b(x, t) 0, and u0 (x) satisfies hypotheses (H1 ) u0 (x) ∈ C(RN ), 0 < u0 (x) f (x) := f0 exp{A|x|α } in RN (α 2, A > 0, f0 > 0). We are mainly interested in the following: (a) Comparison principles for subsolutions and supersolutions of (1.1); (b) instantaneous shrinking of the support for the solution u(x, t) of (1.1). By definition, the support of u(x, t) is bounded for t > 0 although the initial value u0 (x) is positive everywhere; and (c) global existence for solutions of problem (1.1). There were a lot of results on phenomenon (b). We mention here only one of the pioneering paper [3] and some recent results [6,8,10,11,13,14], where the reader can find further references. Most attention has been concentrated on problems with initial data vanishing at infinity. However, several authors have considered more general classes of initial data. For instance, [7] deals with a continuous u0 (x) of power-like growth at infinity, [11] assumed that u0 ∈ L2 (RN ), while in [13] u0 (x) is allowed to be of O(exp(|x|2h(|x|))) growth at infinity, where h(r) tends to zero as r → ∞. In Section 2 we study phenomenon (b) for solutions of (1.1) when u0 (x) is subject to (H1 ). When p = 1, and a subsolution u− (x, t) and a supersolution u+ (x, t) are subject to the condition u± (x, t) = O exp k|x|2 for some positive constant k, there were well-known results for property (a); see, for instance, [4,5] or [12]. But in our case solutions of (1.1) are in general not subject to the above condition. To our knowledge, there are few results on this aspect. As for property (c), it is well known that in general problem (1.1) does not possess a global solution. In Section 3 we focus our attention on the comparison principle, global existence and nonglobal existence for solutions of problem (1.1).
2. Instantaneous shrinking of the support In the sequel we use C to denote different constants, their values may change from line to the next. The statement that a constant depends only on the data means that this constant can be determined in terms of N , p, A, α and f0 . We also use Br (x0 ) to denote the ball in RN of radius r and centered at x0 . Definition 2.1. We say a function u(x, t) ∈ C(RN × [0, ∞)) ∩ C 2,1 (RN × (0, ∞)) is a solution of problem (1.1) if it is nonnegative and satisfies (1.1) in a classical sense. Remark 2.1. For α 2, when u0 (x) is subject to (H1 ), in general we do not know at the moment the precise and proper conditions on b(x, t) under which problem (1.1) is
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global solvable in the sense of Definition 2.1. However, in Section 3 we will get a global existence result to problem (1.1) when b(x, t) is under some condition, and this condition is in a certain sense sharp for α = 2. Definition 2.2. For M0 1, Eα,M0 is the set of all nonnegative function u(x, t) in C(RN × [0, ∞)) ∩ C 2,1 (RN × (0, ∞)) satisfying (I) if α = 2, for any T > 0, there are constants M(T ) and C(T ) such that in RN × [0, T ]; u(x, t) M(T ) exp C(T )|x|2 (II) if α > 2,
u(x, t) M0 f0 exp A|x|α
in RN × [0, ∞).
For an arbitrary nonnegative continuous function v(x, t) defined in RN × [0, T0 ) (0 < T0 ∞), we set ξ(t; v) = sup |x|; v(x, t > 0) , t ∈ [0, T0 ), (2.1) and S(t; v) = x ∈ RN ; v(x, t) > 0 ,
t ∈ [0, T0 ).
(2.2)
Definition 2.3. Assume that v(x, t) ∈ C(RN × [0, T0 )), v 0 and ξ(0; v) = ∞. We say that instantaneous shrinking of the support (briefly, ISS) occurs for v if there exists τ > 0 such that ξ(t; v) < ∞
for all t ∈ (0, τ ].
It is well known that one of the most normal way in proving an instantaneous shrinking of the support assertion for solutions of problem (1.1) is by using a comparison principle; see, for instance, [7,10,13]. But for α > 2, and b(x, t) is subject to the condition given in Theorem 2.1, a comparison principle here is inadequate. It is quite natural to think that, in order to guarantee the instantaneous shrinking of the support, the coefficient of the absorption term should be “strong” enough. In this section we first exploit the idea of [3,14] to prove Theorem 2.1. We will establish in Section 3 a comparison principle for problem (1.1) when b(x, t) satisfies a stronger condition than in that given in Theorem 2.1. Theorem 2.1. Assume (1.2) and (H1 ) hold, and let u(x, t) ∈ Eα,M0 be a solution of (1.1). If (a) for α = 2, b(x, t) b0 (2A)2 |x|2f 1−p (x),
(2.3)
(b) for α > 2, 1−p
b(x, t) b0 (Aα)2 M0
|x|2α−2f 1−p (x),
(2.4)
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where b0 > 1, then ISS occurs for u. Further, for α = 2, γ ξ(t; u) √ , ∀t ∈ (0, τ ], t
(2.5)
where γ depends only on the data and b0 , τ is small enough; for α > 2, and for any β > 1/(2(α − 1)), Cβ + l0 , ∀t ∈ (0, ∞), (2.6) tβ where Cβ depends only on the data, M0 , b0 and β; l0 depends only on the data, M0 and b0 . ξ(t; u)
Proof. Case 1: α = 2. Denote ω = 2/(1 − p). We first so choose ε0 > 0 that b0 (1 + ε0 )2−p ,
(2.7)
then take β0 = ε0 (2A)2 /(4ω), and then introduce the function ω U (x, t) = (1 + ε0 )f (x) 1 − tβ0 |x|2 + := (1 + ε0 )f (x)Z ω (x, t), where we have used the notation a+ = max{a, 0}. We wish to prove for small τ > 0, u(x, t) U (x, t) in RN × [0, τ ];
(2.8)
then assertion (2.5) easily follows from (2.8). Set Q+ = (x, t) ∈ RN × (0, ∞); Z(x, t) > 0 . It is obvious that U (x, t) ∈ C 2,1 (RN × [0, ∞)), 0 < Z(x, t) 1 in Q+ and LU = 0 in RN × (0, ∞) \ Q+ . A direct calculation yields b(x, t) LU = (1 + ε0 )Z ωp f p − β0 ω|x|2Z ω(1−p)−1 f − Z ω(1−p)∆f (1 + ε0 )1−p − ωZ ω(1−p)−1f ∆Z − 2ωZ ω(1−p)−1∇f · ∇Z − ω(ω − 1)|∇Z|2 f . Let l > 1 be fixed. We compute in Q+ ∩ {|x| l, t > 0}
∆Z(x, t) 0, ∆f (x) = (2A)2 |x|2 + 2NA f (x), 4 ∇f · ∇Z 0, |∇Z|2 2 . l By (2.3), (2.7) and using the above estimates, we get 4ω(ω − 1) ωp 2 2 2 >0 LU (1 + ε0 )Z f ε0 (2A) |x| − β0 ω|x| − l2
(2.9)
in Q+ ∩ {|x| l, t > 0}, provided that l is fixed large enough. From U (x, 0) > u(x, 0) = u0 (x) in RN and the continuity of U and u, there exists τ = τ (l) > 0 such that u(x, t) U (x, t),
(x, t) ∈ Bl (0) × [0, τ ].
(2.10)
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From (2.9), (2.10) and LU = 0 in RN × (0, ∞) \ Q+ , an application of Theorem 3.3 of Section 3 to u and U in {|x| l, 0 t τ } leads to u(x, t) U (x, t) in |x| l, 0 t τ . (2.11) (2.8) easily follows from (2.10) and (2.11). Case 2: α > 2. Fix some t 0 and choose x 0 ∈ S(t 0 ; u). Let a > 1, with a/l0 small, be fixed, and introduce the function 1 1 U (x, t) = (t 0 − t) 1−p |x 0 | (1−p)β exp A|x 0 |α 0 α−1 ω |x | + M0 |x − x 0 |ω f0 exp A|x|α . a A direct calculation yields p 1 1 LU = b(x, t)U p (x, t) − (t 0 − t) 1−p |x 0 | (1−p)β exp A|x 0|α 1−p
2 2α−2 + Aα(N + α − 2)|x|α−2 − (Aα) |x| 0 α−1 ω |x | × M0 f0 |x − x 0 |ω exp A|x|α a 0 α−1 ω−2 |x | − M0 f0 |x − x 0 |ω−2 exp A|x|α a ω(N + ω − 2) 0 2α−2 2Aαω α−2 0 2α−2 0 . (2.12) x + |x| |x | x − x |x | × i i i a2 a2 One can easily see that in Ba/|x 0 |α−1 (x 0 ), when |x 0 | l0 > 2a,
|x| 2a 0 |x | 1 + |x|. 1 + |xa0 |α |x 0 |α
(2.13)
By (2.13) we compute in Ba/|x 0 |α−1 (x 0 ) 2Aαω α−2 0 2α−2 ω(ω + N − 2) 0 2α−2 C 2α−2 0 |x| |x | x − x |x | |x| , x i i i + a2 a a2 p |x 0 | (1−p)β |x|2α−2 exp Ap|x 0 |α exp A(1 − p)|x|α p exp(−Ca) 0 2α−2+ (1−p)β |x | exp A|x 0 |α , C where C depends only on the data. By the following elementary inequality: for a 0, b 0 and ε > 0, (a + b)p εa p + (1 − ε)bp , we have U p (x, t)
p p b0 − 1 0 (t − t) 1−p |x0 | (1−p)β exp pA|x 0 |α 2b0 b0 + 1 p p |x 0 |α−1 |x − x 0 | pω + M0 f0 exp pA|x|α . 2b0 a
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From (2.4), (2.12) and the above estimates, we get p 1 1 b0 − 1 0 2α−2−1/β LU − + |x | (t 0 − t) 1−p |x 0 | (1−p)β exp A|x 0|α 1 − p C exp(Ca) 2 2α−2 α−2 + Aα(Aα + N − 2)|x| − (Aα) |x| |x 0 |α−1 |x − x 0 | pω × M0 f0 exp A|x|α a CM0 2α−2 |x 0 |α−1 |x − x 0 | pω |x| exp A|x|α − a a 0 α−1 |x | |x − x 0 | pω 1 + b0 M0 f0 (Aα)2 |x|2α−2 exp A|x|α . + 2 a
(2.14)
From (2.14), first choose a large, then fix l0 > 2a large enough to obtain LU > 0 in Ba/|x 0 |α−1 (x 0 ) × (0, t 0 ].
(2.15)
By standard comparison theorems, u − U must attain maximum on the parabolic boundary of the cylinder Ba/|x 0 |α−1 (x 0 ) × [0, t 0 ]. Since on ∂Ba/|x 0|α−1 (x 0 ) × [0, t 0 ], u U , but 0 < u(x 0 , t 0 ) = u(x 0 , t 0 ) − U (x 0 , t 0 ), there must exist some point x¯ ∈ Ba/|x 0 |α−1 (x 0 ) for which 1 1 (t 0 ) 1−p |x 0 | (1−p)β exp A|x 0 |α U (x, ¯ 0) < u(x, ¯ 0) α ¯ C exp A|x 0|α , M0 exp A|x|
(2.16)
where C depends only on the data. Hence from (2.16), the definition of S(t; u) and the arbitrarity of x 0 ∈ S(t 0 ; u) it easily follows assertion (2.6). ✷ Now we pass to an extinction phenomenon Theorem 2.2. Assume p and u0 (x) as in Theorem 2.1, and let u ∈ Eα,M0 be a solution of (1.1). If (a) for α = 2, b(x, t) b0 (2A)2 |x|2f 1−p (x) + b1 ,
(2.17)
(b) for α > 2, 1−p
b(x, t) b0 (Aα)2 M0
|x|2α−2f 1−p (x) + b1 ,
where b0 > 1 and b1 > 0, then u vanishes identically after some time T0 > 0. Proof. For α = 2, by (2.8) we have ω u(x, t) (1 + ε0 )f (x) 1 − β0 t|x|2 + ,
(x, t) ∈ RN × [0, τ ],
(2.18)
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0 in RN for some constant M 0 . For β1 > 0 to be fixed, set and hence u(x, τ ) M 1 1−p 1−p U (x, t) = M − β (t − τ ) 1 + . 0
From (2.17), one has LU > 0 in RN × (τ, ∞), provided that β1 is chosen small enough. Then by comparison (see Theorem 3.2 of Section 3) 1 1−p 1−p u(x, t) U (x, t) = M − β (t − τ ) 1 + 0
in RN × [τ, ∞),
1−p /β1 . which implies u(x, t) ≡ 0, t τ + M 0 For α > 2, by Theorem 2.1 and the definition of Eα,M0 , there exist M1 and l1 > l0 such that u(x, t) M1
in RN × [1, ∞),
u(x, t) = 0 for |x| l1 , t 1.
(2.19)
For β2 > 0 to be fixed, set 1−p 1 V (x, t) = M1 − β2 (t − 1) +1−p . From (2.18), we have LU > 0 in RN × (1, ∞), provided that β2 is chosen small enough. By (2.19), similarly as in above we can get 1−p
u(x, t) ≡ 0,
t 1+
M1 . β2
✷
The following theorem shows that in a certain sense condition (2.3) is sharp. Theorem 2.3. Assume α = 2 and 0 < p < 1, and let u ∈ C(RN × [0, T0 )) ∩ C 2,1 (RN × (0, T0 )) (0 < T0 ∞) be a solution of (1.1) with u0 (x) = f (x) = f0 exp(A|x|2). If
0 b(x, t) (2A)2 |x|2 + 2NA f 1−p (x), then ISS does not occur for u. Proof. Set V (x, t) = f (x). A straightforward calculation leads to
LV = − (2A)2 |x|2 + 2NA V + b(x, t)V p 0. Then by comparison (see Theorem 3.2 of Section 3), we have u(x, t) V (x, t) = f (x) in RN × [0, T0 ), whence the result.
✷
Remark 2.2. For the case α > 2, since the comparison principle is inadequate when 0 1−p b(x, t) (Aα)2 |x|2α−2M0 f 1−p (x), the counter example is left open.
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3. Existence and comparison principle We begin with a global existence result. Theorem 3.1. If 0 < p 1, and b(x, t) ∈ C α0 ,α0 /2 (RN × [0, ∞)) (α0 ∈ (0, 1)) satisfies
b(x, t) (Aα)2 |x|2α−2 + Aα(N + α − 2)|x|α−2 f 1−p (x), (3.1) then there exists a solution u(x, t) of (1.1) in the sense of Definition 2.1 with 0 u(x, t) f (x) in RN × [0, ∞).
(3.2)
Proof. For any large n, let un (x, t) be the unique solution of the approximated problem p−1 u = 0 in B (0) × (0, n2 ], (3.3) n n (un )t − ∆un + b(x, t)|un | on ∂Bn (0) × (0, n2 ], un (x, t) = 0 (3.4) un (x, 0) = u0 (x)ψn (x) in Bn (0), (3.5) where ψn (x) is a smooth cutoff function in Bn (0) satisfying 0 ψn 1, ψn (x) = 0 on ∂Bn (0) and ψn (x) = 1 in Bn−1 (0). Clearly, we have un (x, t) 0 in Bn (0) × [0, n2 ]. To estimate the upper bound of un (x, t) consider the function V (x, t) = f (x). From the hypotheses one has
LV = − (Aα)2 |x|2α−2 + Aα(N + α − 2)|x|α−2 f + b(x, t)f p 0, and hence, by comparison, un (x, t) V (x, t) = f (x)
in Bn (0) × [0, n2 ].
(3.6)
From (3.6), according to the well-known interior estimates of solutions and their continuity module (see, for instance, [1,2,4,9,12]), it follows that for any l 1 and n l + 4, we have un C 2+γ0 ,1+γ0 /2 (Bl (0)×[1/ l 2,l 2 ] Cl
(3.7)
un (x, t) − un (x , t ) ωl |x − x | + |t − t |1/2
(3.8)
and
for all (x, t), (x , t ) ∈ Bl (0) × [0, l 2 ], where the positive constant γ0 is independent of n and l; Cl and ωl (r) are independent of n, and ωl (r) tends to zero as r ↓ 0. The above estimates together with Arzela’s lemma and a diagonal argument imply that there exists u(x, t) ∈ C(RN × [0, ∞)) ∩ C 2,1 (RN × (0, ∞)) such that, after extracting a subsequence if necessary, lim un (x, t) = u(x, t) f (x)
n→∞
in RN × [0, ∞),
and u(x, t) solves (1.1) in the classical sense.
✷
Next, when b(x, t) is subject to some conditions, we establish a comparison theorem as follows:
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Theorem 3.2 (Comparison principle). Assume 0 < p 1. Let u− ∈ Eα,M0 , 0 u+ ∈ C(RN × [0, ∞)) ∩ C 2,1 (RN × (0, ∞)) satisfy − Lu 0 in RN × (0, ∞), Lu+ 0 in RN × (0, ∞), and u− (x, 0) u+ (x, 0) in RN . (a) If α = 2 and b(x, t) 0, then u− (x, t) u+ (x, t)
in RN × [0, ∞).
(b) If α > 2 and b(x, t) satisfies, in addition to b(x, t) 0, the following
1−p 1 b(x, t) (Aα)2 |x|2α−2 + Aα(N + α − 2 + θ0 )|x|α−2 M0 f (x) p in {|x| l1 } × [0, ∞), where θ0 > 0, l1 1, then u− (x, t) u+ (x, t)
(3.9)
in RN × [0, ∞).
Proof. The assertion for α = 2 was established in [13, Theorem 2.2]. Here we only give the proof for α > 2. Suppose the contrary, then there exists a point (x 0 , t 0 ) ∈ RN × (0, ∞) such that u− (x 0 , t 0 ) − u+ (x 0 , t 0 ) = 2a > 0.
(3.10)
Set
w(x, t) = u− (x, t) − u+ (x, t) + a ,
c(x, t) = (Aα)2 |x|2α−2 + Aα(N + α − 2 + θ0 )|x|α−2 χ{|x|l1 } (x),
where χ{·} (x) is the characteristic function of {·}. From the hypotheses we see that
(3.11) wt − ∆w + b(x, t) (u− )p − (u+ + a)p 0. By (3.9), (3.11), in view of the elementary inequality p(A − B) for A B > 0, (3.12) A1−p and proceeding similarly as in the proof of Theorem 2.2 in [13], we deduce that in the weak sense Ap − B p
(w+ )t − ∆w+ + c(x, t)w+ 0.
(3.13)
Now choose a cutoff function ω(x) ∈ satisfying 0 ω 1, ω(x) = 1 in 0 0 B1/2 (x ) and ω(x) = 0 on ∂B1 (x ). For any integer n > l1 such that B2 (x 0 ) ⊂ Bn (0), let ϕ(x, t) be the solution of the problem := ϕt + ∆ϕ − c(x, t)ϕ = 0 in Bn (0) × [0, t 0 ), (3.14) Lϕ ϕ=0 on ∂Bn (0) × [0, t 0 ), (3.15) ϕ(x, t 0 ) = ω(x) in Bn (0). (3.16) C02 (B1 (x 0 ))
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It is obvious that 0 ϕ(x, t) 1. Moreover, we have Claim. There exist constants M1 and M2 depending only on the data, θ0 , l1 , x 0 and t 0 such that exp{−A|x|α } ϕ(x, t) M1 , (x, t) ∈ Bn (0) × [0, t 0 ], (3.17) (1 + |x|2 )k1 and
α ∂ϕ M2 exp(−An ) ∂ν nk2
on ∂Bn (0) × [0, t 0 ],
(3.18)
provided n is large enough. Here α + N − 2 θ0 θ0 + , k2 = N − 1 + ; 2 4 2 ν is the outward unit normal to ∂Bn (0). k1 =
Suppose for the moment that this claim has been proved. From w(x, 0) < 0 and the continuity of w, there is τn ∈ (0, 1/n] such that w+ (x, t) = 0 in Bn (0) × [0, τn ]. We take ϕ(x, t) as the test function in the weak inequality (3.13), taking into account (3.14)–(3.16) and (3.18), to obtain w+ (x, t 0 ) w+ (x, t 0 )ϕ(x, t 0 ) B 1 (x 0 )
Bn (0)
2
t 0
t 0
w+ (ϕt + ∆ϕ − cϕ) − τn Bn (0)
t 0
∂ϕ ∂ν
τn ∂Bn (0)
=−
w+
w+
∂ϕ Cn−θ0 /2 , ∂ν
τn ∂Bn (0)
provided n is large enough; here C depends only on the data, M0 , θ0 , l1 , x 0 and t 0 . Letting n → ∞, we get w+ (x, t 0 ) 0, B 1 (x 0 ) 2
whence w+ (x, t 0 ) ≡ 0 in B1/2 (x 0 ), which contradicts (3.10). ✷ Proof of the claim. For λ0 > 1 to be fixed, set ψ(x, t) = M
exp(λ0 (t 0 − t)) exp(−A|x|α ) , (1 + |x|2 )k1
where M = (1 + (|x 0 | + 2)2 )k1 exp{A(|x 0 | + 2)α }. A quick calculation gives
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− λ0 + Aα(N + α − 2)|x|α−2 + c(x, t) ψ Lψ
4k1(1 + k1 ) + (Aα)2 |x|2α−2 + 4k1 Aα|x|α−2 + ψ. 1 + |x|2 From the definition of c(x, t), it is easily seen that Lψ 0, provided λ0 is large enough. Hence, by comparison, ϕ(x, t) ψ(x, t) M1
exp(−A|x|α ) (1 + |x|2)k1
in (x, t) ∈ Bn (0) × [0, t 0 ],
for some constant M1 depending only on the data, θ0 , l1 , x 0 and t 0 . = n1−α , in (Bn (0) \ B (0)) × [0, t 0 ] We turn now to the proof of (3.18). Denote R n−R consider the function α (n − |x|) exp(−An ) , V (x, t) = ϕ(x, t) − M k n2 where M is a positive constant to be fixed. One easily see that LV 0 in (Bn (0) \ Bn−R(0)) × (0, t 0 ), which implies that V attains or |x| = n. By (3.17), the definition of R its maximum only on t = t 0 , or |x| = n − R, large enough, we have V (x, t 0 ) 0, x ∈ Bn (0) \ B (0), V (x, t) 0 on and fixing M n−R ∂Bn−R(0) × [0, t 0 ] and V (x, t) = 0 on ∂Bn (0) × [0, t 0 ], and hence sup (Bn (0)\Bn−R(0))×[0,t 0]
V (x, t) = 0.
Therefore, ∂V 0 on ∂Bn (0) × [0, t 0 ], ∂ν which yields α ∂ϕ exp(−An ) on ∂Bn (0) × [0, t 0 ]. −M ∂ν nk2 On the other hand, it is clear that
∂ϕ 0 on ∂Bn (0) × [0, t 0 ]. ∂ν Thus inequality (3.18) immediately follows from (3.19) and (3.20). ✷
(3.19)
(3.20)
Along the lines of the proof of Theorem 3.2, one can easily see the validity of the following: Theorem 3.3. For 0 < T0 ∞, 0 < R0 , assume p and b(x, t) as in Theorem 3.2. Let R0 (0)) × (0, T0 ]) be nonnegative and u± ∈ C((RN \ BR0 (0)) × [0, T0]) ∩ C 2,1 ((RN \ B satisfy − N (0)) × (0, T ), R0 0 Lu 0 in (R \ B + R0 (0)) × (0, T0 ), Lu 0 in (RN \ B − u (x, 0) u+ (x, 0), x ∈ RN \ BR0 (0),
J.-J. Li / J. Math. Anal. Appl. 281 (2003) 382–394
393
and u− (x, t) u+ (x, t),
|x| = R0 , t ∈ [0, T0 ].
u−
∈ Eα,M0 . Then u (x, t) u (x, t) in RN \ BR0 (0) × [0, T0 ].
Assume further that −
+
From Theorems 3.1 and 3.2, we have Corollary 3.1. If 0 < p < 1 and b(x, t) ∈ C α0 ,α0 /2 (RN × [0, ∞)) (α0 ∈ (0, 1)) satisfies (a) for α = 2,
b(x, t) (2A)2 |x|2 + 2NA f 1−p (x),
(b) for α > 2, b(x, t)
1 (Aα)2 |x|2α−2 + Aα(N + α − 2 + θ0 )|x|α−2 f 1−p (x), p
then problem (1.1) is uniquely solvable in Eα,1 . The following result shows the sharpness of condition (3.1) for the case α = 2. Theorem 3.4. If 0 p 1, α = 2 and
0 b(x, t) (1 − ε1 )(2A)2|x|2 + 2NA f 1−p (x),
(3.21)
where ε1 ∈ (0, 1), then there is no global solution in E2,M0 with u0 (x) = f (x). Proof. Suppose the contrary, then there exists a global solution u(x, t) ∈ E2,M0 to problem (1.1). For any β > 0, we set A
V (β) (x, t) = f0 e 1−βt |x| , 2
(x, t) ∈ RN × [0, 1/β).
By (3.21) we compute
2 p βA 2A 2NA 2 (β) 2 V (β) + b(x, t) V (β) |x| V − |x| + LV = 1 − βt 1 − βt (1 − βt)2
2 2A βA 2 2 V (β) 0, |x| − ε |x| 1 1 − βt (1 − βt)2 provided β 4Aε1 . Then by comparison, (β)
A
u(x, t) V (β) (x, t) = f0 e 1−βt
|x|2
in RN × [0, 1/β),
which implies limt →1/β u(x, t) = ∞, x = 0, a contradiction. ✷ Remark 3.1. The counter example for the case α > 2, when b(x, t) satisfies
0 b(x, t) (1 − ε1 ) (Aα)2 |x|2α−2 + Aα(N + α − 2)|x|α−2 f 1−p (x), is left open since the lack of a comparison theorem.
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J.-J. Li / J. Math. Anal. Appl. 281 (2003) 382–394
Acknowledgments The author wishes to express his thanks to Professor A.S. Kalashnikov, who brought the author’s attention to the present subject and encouraged him through fruitful discussions and valuable suggestions. This work was carried out during the author’s visit to Moscow State University, to which he is deeply grateful for its hospitality. The author is also greatly indebted to China Scholarship Council for providing this visit for him. The author would like to thank the referees for their very helpful comments and suggestions.
References [1] M. Bertsch, A class of degenerate diffusion equations with a singular nonlinear term, Nonlinear Anal. 7 (1983) 117–127. [2] E. DiBeneditto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients, Ann. Scuola Norm. Sup. 13 (1986) 485–535. [3] L.C. Evans, B.F. Knerr, Instantaneous shrinking of the support of non-negative solutions to certain nonlinear parabolic equations and variational inequalities, Illinois J. Math. 23 (1979) 153–166. [4] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice–Hall, 1964. [5] A.M. Il’in, A.S. Kalashnikov, O.A. Oleinik, Second order linear equations of parabolic type, Russian Math. Surveys 17 (1962) 1–143. [6] A.S. Kalashnikov, Some problems of the qualitative theory of nonlinear second-order parabolic equations, Russian Math. Surveys 42 (1987) 169–222. [7] A.S. Kalashnikov, On quasilinear degenerating parabolic equation with singular lower-order terms and growing initial values, Differentsial’nye Uravneniya 29 (1993) 999–1009. [8] A.S. Kalashnikov, Instantaneous shrinking of the support for solutions to certain parabolic equations and systems, Rend. Mat. Accad. Lincei Ser. 9 8 (1997) 263–272. [9] R. Kersner, Degenerate parabolic equations with general nonlinearities, Nonlinear Anal. 4 (1980) 1043– 1062. [10] R. Kersner, F. Nicolosi, The nonlinear heat equation with absorption: effects of variable coefficients, J. Math. Anal. Appl. 170 (1992) 551–566. [11] R. Kersner, A. Shishkov, Instantaneous shrinking of the support of energy solutions, J. Math. Anal. Appl. 198 (1996) 729–750. [12] O.A. Ladyzenskaja, N.N. Ural’cevar, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968. [13] J. Li, Instantaneous shrinking of the support of solutions to certain parabolic equations with unbounded initial data, Nonlinear Anal. 48 (2002) 1–12. [14] M. Ughi, On the porous media equation with either source or absorption, Cuad. Inst. Mat. Beppo Levi 22 (1991) 1–72.