Life Span of Solutions for a Semilinear Heat Equation with Initial Data Non-Rarefied at $\infty$

0.4 MB·English

by  Zhiyong Wang

Life Span of Solutions for a Semilinear
Heat Equation with Initial Data
5 Non-Rarefied at ∞ ∗
1
0
2
Zhiyong Wang†, Jingxue Yin‡

n
School of Mathematical Sciences, South China Normal University,
a
J Guangzhou, 510631, PR China

3
1 January 14, 2015

]
P
A
Abstract
.
h
t We study the Cauchy problem for a semilinear heat equation with
a initial data non-rarefied at ∞. Our interest lies in the discussion of the
m
effect of the non-rarefied factors on the life span of solutions, and some
[ sharp estimates on the life span is established.

1 MSC: 35K05; 35B44
v
6 Keywords: Life span; Heat equation; Non-rarefied
5
8
2 1 Introduction
0
.
1 Consider the Cauchy problem
0
5 (cid:40)u =∆u+|u|p−1u, (x,t)∈Rn×(0,+∞),
1 t (1)
: u(x,0)=φ(x), x∈Rn,
v
i
X where p>1, and φ is a non-negative, bounded and continuous function in Rn,
r which is not identically equal to zero.
a Itiswellknownthatthesolutionu(x,t)of (1)mayblowupinfinitetimeT∗,
that is lim (cid:107)u(·,t)(cid:107) (Rn) = ∞, see [4, 7]. The finite time T∗ is popularly
L∞
t→T∗−
called to be the life span, which depends heavily on the exponent p and the
properties of the initial datum, and its study has arisen much attention during
recent years, see for example [8, 6, 9, 10, 12] and references therein.
∗ThisworkispartiallysupportedbyNSFC(No. 11371153),SpecializedResearchFundfor
theDoctoralProgramofHighEducationalDepartmentofChina.
†E-mailaddress: [email protected].
‡Correspondingauthor. E-mailaddress: [email protected].
1
It is worthy of mentioning that the properties of the initial data in some
neighbourhood of ∞ are shown to be crucial factors affecting the life span of
solutions. In this respect, for φ(x) ≡ λψ(x) with ψ(x) ∈ C(Rn)∩L∞(Rn),
ψ(x) ≥ 0, and ψ(x) is positive in some neighbourhood of ∞, the asymptotic
behaviour of the life span T seems to be interesting. Indeed, Lee and Ni [8]
λ
proved that if liminfψ(x) = k > 0, then C λ1−p ≤ T ≤ C λ1−p for some
1 λ 2
|x|→∞
positive constants C and C which are independent of λ; while Gui and Wang
1 2
[6] further showed that if lim ψ(x) = k, then lim λp−1T = 1 k1−p. More
λ p−1
|x|→∞ λ→0+
recently, Yamauchi [18] has found that as long as the initial datum φ(x) is
positive in some conic neighbourhood of ∞, the solution does blow up in a
finite time T∗, and some elegant estimates on the life span are given.
The purpose of the present paper is to characterize the role of rarefaction
propertiesoftheinitialdataat∞,thatistheeffectonthelifespanofsolutions
of (1). We will show that as long as ∞ is not rarefied, although it is permitted
for initial data to have zeros in any conic neighbourhood of ∞, the solutions
will blow up definitely. Of course, we are interested in estimating on the life
span of solutions from the point of view of density analysis.
Beforestatingourmainresultsweshouldrecallorintroducesomenotations
and conceptions. We begin with the definition of rarefaction point (see [15,
p. 162]): A point x is a point of rarefaction of the set E if
0
(cid:0) (cid:1)
mes B(x ,r)∩E
0
lim =0,
(cid:0) (cid:1)
r→0+ mes B(x0,r)
where mes(F) is the Lebesgue measure of the set F, and B(x,r) is the ball
centeredatxwithradiusr. Itisreasonabletosaythat∞isapointofrarefaction
of the set E if
(cid:0) (cid:1)
mes B(0,r)∩E
lim =0.
(cid:0) (cid:1)
r→+∞ mes B(0,r)
Alternatively, ∞ is called to be a rarefaction point of a non-negative function φ
if for any α > 0 it is a point of rarefaction of the set {x | φ(x) ≥ α}. We have
the following theorem.
Theorem 1.1. If ∞ is not a rarefaction point of the initial datum φ, then
the solution of (1) blows up in finite time T∗ with
1 (cid:16) (cid:17)1−p
T∗ ≤ inf αD(α) , (2)
p−1α>0
where
(cid:0) (cid:1)
mes {x|φ(x)≥α}∩B(0,r)
D(α):=limsup . (3)
(cid:0) (cid:1)
r→+∞ mes B(0,r)
Inwhatfollows, wedonotintendtogiveaprooffortheabovetheorem, but
prefer to present a stronger version of Theorem 1.1 in the following settings.
2
Let α,r >0, denote
(cid:0) (cid:1)
mes B(x,r)∩{y |φ(y)≥α}
D(α;r):= sup ,
(cid:0) (cid:1)
x∈Rn mes B(x,r)
and define
D(α):=limsupD(α;r). (4)
r→+∞
Wearenowinapositiontopresentthemainresultofthefollowingtheorem.
Theorem 1.2. Suppose that there exists α > 0 such that D(α) > 0. Then
the solution of (1) blows up in finite time T∗ with
1 (cid:16) (cid:17)1−p
T∗ ≤ αD(α) . (5)
p−1
We shall give some comments on Theorem 1.2. First, by the definition of
D(α)andD(α),Theorem1.2impliesTheorem1.1directly. Second,byasimple
comparison argument we obtain the lower bound of the life span:
1
(cid:107)φ(cid:107)1−p ≤T∗.
p−1 L∞(Rn)
Thus Theorem 1.2 can show that the minimal time blow-up occurs1 for some
initial data (for the details see the example in Section 3). Finally, we point out
that to prove Theorem 1.2 we take good advantage of basic properties of the
heat kernel, so we believe that the argument is general and can be applied to
similarproblemsposedonmanifolds,e.g.,thehyperbolicspace,seeRemark2.2
and Remark 2.3.
This paper is organized as follows. In Section 2 we give the proof of Theo-
rem 1.2. Subsequently, we show that Theorem 1.2 implies the main results of
Yamauchi [18], in Section 3.
2 Proof of Theorem 1.2
Before proving Theorem 1.2 we first introduce the basic properties of the heat
kernel in Rn, and give a lemma on the life span of the solutions of (1).
The solution of (1) can be written as (see [3, p.51, (17)]):
(cid:90) (cid:90) t(cid:16)(cid:90) (cid:17)
u(x,t)= g(x,y,t)φ(y)dy+ g(x,y,t−s)|u|p−1u(y,s)dy ds.
Rn 0 Rn
1If the blow-up time of the solution of the problem (1) is the same as that of the ode:
u(cid:48) = up with the initial value u(0) = (cid:107)φ(cid:107)L∞(Rn), we say that the minimal time blow-up
occurs,formoredetailsseeYamauchi[18]andreferencestherein.
3
Here g is the heat kernel in Rn, which is a function of the distance of x,y ∈Rn
and the time t, that is,
(cid:0) (cid:1)
g(x,y,t)=k d(x,y),t . (6)
Wesummarizesomeelementarypropertiesoftheheatkernelinthefollowing
lemma.
Lemma 2.1 ([5, Section 1.3 and 2.7]). Let x,y,z ∈Rn, and s,t>0.
(i) g(x,y,t)=g(y,x,t);
(ii) g(Tx,Ty,t)=g(x,y,t), where T is an isometry in Rn;
(iii) Semigroup property
(cid:90)
g(x,y,t)g(y,z,s)dy =g(x,z,s+t); (7)
Rn
(iv) Conservation of probability
(cid:90)
g(x,y,t)dy =1. (8)
Rn
Remark 2.1. In Rn it is well known that g(x,y,t) = 1 exp(cid:0)−|x−y|2(cid:1).
(4πt)n/2 4t
In this paper we take T to be the translation T(x)=x+x for fixed x ∈Rn.
0 0
Remark 2.2. In Lemma 2.1 the properties (i) – (iii) are standard for the
heat kernel and (iv) is satisfied for the manifold that is complete and with
Ricci curvature bounded below (see [2, Theorem 5.2.6]). The property (6),
used only to derive the uniform estimate (22), is satisfied for the homogeneous
spaces. Thus the proofs of Theorem 2 depend only on the basic properties of
the heat kernel, which do not depend on the explicit expression of the heat
kernel. Therefore the proofs can be extended to the problems posed on other
Riemannian manifolds, e.g., on the hyperbolic space2.
Nowweshallgiveanaprioriestimateonthelifespan,whichwasessentially
introduced by Weissler [17] for proving the blow-up of the non-trivial positive
solutions of (1) in the critical case, see also [7, Chapter 5]. For the convenience
of the reader, we prefer to represent it here.
Lemma 2.2. Let p > 1, and φ ≥ 0 in L∞(Rn) is not identically zero.
Supposethatuisasolutionoftheproblem (1)withtheinitialvalueφin[0,T∗).
Then
1 (cid:16) (cid:90) (cid:17)1−p
sup g(z,y,t)φ(y)dy ≥t for t∈(0,T∗). (9)
p−1 z∈Rn Rn
In particular, we have the following upper bound estimate on the life span T∗:
(cid:110) 1 (cid:16) (cid:90) (cid:17)1−p (cid:111)
T∗ ≤sup T >0| sup g(z,y,t)φ(y)dy ≥t for t∈(0,T) .
p−1 z∈Rn Rn
2Forblow-upproblemsofsemilinearheatequationsonthehyperbolicspace,seetherecent
worksofBandle,PozioandTesei[1]andWangandYin[16].
4
Proof. Take z ∈ Rn and fix 0 < t < T∗. Let t ∈ [0,t]. Since φ(x) ≥ 0, by the
comparison principle it follows that u(x,t)≥0. Thus we have
(cid:90) (cid:90) t(cid:90)
u(x,t)= g(x,y,t)φ(y)dy+ g(x,y,t−s)up(y,s)dyds. (10)
Rn 0 Rn
Multiplying (10) by g(x,z,t−t) and integrating with respect to x over Rn, we
obtain
(cid:90) (cid:90) (cid:90)
g(x,z,t−t)u(x,t)dx= g(x,z,t−t) g(x,y,t)φ(y)dydx
Rn Rn Rn
(cid:90) (cid:90) t(cid:90)
+ g(x,z,t−t) g(x,y,t−s)up(y,s)dydsdx.
Rn 0 Rn
By (i) and (iii) in Lemma 2.1, and by Fubini’s theorem,
(cid:90) (cid:90)
g(z,x,t−t)u(x,t)dx= g(z,y,t)φ(y)dy
Rn Rn
(cid:90) t(cid:90)
+ g(z,y,t−s)up(y,s)dyds.
0 Rn
(cid:82)
Since g(z,y,t−s) ≥ 0 and g(z,y,t−s)dy = 1, by Jensen’s inequality, the
Rn
above equation implies
(cid:90) (cid:90)
g(z,x,t−t)u(x,t)dx≥ g(z,y,t)φ(y)dy
Rn Rn
(cid:90) t(cid:16)(cid:90) (cid:17)p
+ g(z,y,t−s)u(y,s)dy ds. (11)
0 Rn
Denote the right hand side of (11) by G(t):
(cid:90) (cid:90) t(cid:16)(cid:90) (cid:17)p
G(t):= g(z,y,t)φ(y)dy+ g(z,y,t−s)u(y,s)dy ds.
Rn 0 Rn
We have G(t)>0 for t∈[0,T∗) and
(cid:90)
G(0)= g(z,y,t)φ(y)dy. (12)
Rn
Differentiating G(t) with respect to t, by (11) we have the inequality
(cid:16)(cid:90) (cid:17)p
G(cid:48)(t)= g(z,y,t−t)u(y,t)dy ≥Gp(t);
Rn
that is
G−p(t)G(cid:48)(t)≥1. (13)
5
Integrating (13) with respect to t over [0,t], we obtain
1 (cid:16) (cid:17)
G1−p(t)−G1−p(0) ≥t.
1−p
Thus
1
G1−p(0)≥t, (14)
p−1
since p>1. Then the lemma follows from (12) and (14).
Now we are in a position to prove Theorem 1.2.
Proof of Theorem 1.2. By the assumption of the theorem, for any ε > 0 there
exist z ∈Rn and r >0 for k =1,2,..., such that
k k
lim r =+∞, (15)
k
k→∞
and
(cid:0) (cid:1)
mes B(z ,r )∩{y |φ(y)≥α}
k k
≥D(α)−ε, (16)
ω rn
n k
where ω is the volume of n-dimensional unit ball.
n √
Take r¯ = r . We claim the following lemma, which will be proved in the
k k
end of this section.
Lemma 2.3. For any δ ∈(0,1), there exists K ∈N such that for k >K
(cid:90)
sup 1 (y)g(x,y,T∗)φ(y)dy
B(x,r¯k)
x∈B(zk,rk−r¯k) B(zk,rk)
≥ (r1−−r¯δ)n(cid:104)α(cid:0)D(α)−ε(cid:1)rkn−(cid:107)φ(cid:107)L∞(Rn)(cid:0)rkn−(rk−2r¯k)n(cid:1)(cid:105). (17)
k k
Here 1 is the characteristic function of the set E.
E
Lemma 2.3 implies that for k >K
(cid:90) (cid:90)
sup g(z,y,T∗)φ(y)dy ≥ sup 1 (y)g(x,y,T∗)φ(y)dy
z∈Rn Rn x∈B(zk,rk−r¯k) B(zk,rk) B(x,r¯k)
≥ (r1−−r¯δ)n(cid:104)α(cid:0)D(α)−ε(cid:1)rkn−(cid:107)φ(cid:107)L∞(Rn)(cid:0)rkn−(rk−2r¯k)n(cid:1)(cid:105). (18)
k k
Let k →∞. (15) and (18) show
(cid:90)
sup g(z,y,T∗)φ(y)dy ≥(1−δ)α(cid:0)D(α)−ε(cid:1);
z∈Rn Rn
and since δ and ε were arbitrary, we obtain
(cid:90)
sup g(z,y,T∗)φ(y)dy ≥αD(α). (19)
z∈Rn Rn
Now the theorem follows from (19) and Lemma 2.2 immediately.
6
Finally, we should give the proof of Lemma 2.3.
Proof of Lemma 2.3. It is sufficient to show that there exists K satisfies the
following property: for k >K there exists x ∈B(z ,r −r¯ ) such that
k k k k
(cid:90) 1 (y)g(x ,y,T∗)φ(y)dy ≥ 1−δ (cid:104)α(cid:0)D(α)−ε(cid:1)rn
B(xk,r¯k) k (r −r¯ )n k
B(zk,rk) k k
−(cid:107)φ(cid:107)L∞(Rn)(cid:0)rkn−(rk−2r¯k)n(cid:1)(cid:105). (20)
For any k =1,2,···, we define a sequence of functions F :B(z ,r )→R+ as
k k k
(cid:90)
F (x):= 1 (y)g(x,y,T∗)φ(y)dy.
k B(x,r¯k)
B(zk,rk)
Then F is a non-negative, continuous and bounded function. Consider the
k
integral of F (x) over B(z ,r −r¯ ). By Fubini’s theorem, we have
k k k k
(cid:90) (cid:90)
F (x)dx= 1 (x)F (x)dx
k B(zk,rk−r¯k) k
B(zk,rk−r¯k) B(zk,rk)
(cid:90) (cid:16)(cid:90) (cid:17)
= 1 (x) 1 (y)g(x,y,T∗)φ(y)dy dx
B(zk,rk−r¯k) B(x,r¯k)
B(zk,rk) B(zk,rk) (21)
(cid:90) (cid:16)(cid:90) (cid:17)
= 1 (x)1 (y)g(x,y,T∗)dx φ(y)dy
B(zk,rk−r¯k) B(x,r¯k)
B(zk,rk) B(zk,rk)
(cid:90)
= I (y)φ(y)dy,
k
B(zk,rk)
where
(cid:90)
I (y):= 1 (x)1 (y)g(x,y,T∗)dx.
k B(zk,rk−r¯k) B(x,r¯k)
B(zk,rk)
Since g(x,y,t) is a function of the distance between x and y for fixed t; and
since
(cid:90)
g(x,y,T∗)dx=1 for y ∈Rn,
Rn
it follows that for fixed δ >0 there exists R>0 such that
(cid:90) (cid:90)
1 (x)g(x,y,T∗)dx= g(x,y,T∗)dx≥1−δ (22)
B(y,r¯k)
Rn B(y,r¯k)
√
for any y ∈Rn and any r¯ >R. Since r¯ = r and r →∞ as k →∞, there
k k k k
exists K such that
r¯ >R for k >K.
k
Thus by (22) we obtain
(cid:90)
1 (x)g(x,y,T∗)dx≥1−δ for k >K. (23)
B(y,r¯k)
Rn
7
Inthefollowingproofweshallalwaystakek >K. Moreover,fory ∈B(z ,r −
k k
2r¯ ), it is easily seen that
k
1 (x)=1 (y) and B(y,r¯ )⊂B(z ,r −r¯ ). (24)
B(y,r¯k) B(x,r¯k) k k k k
Then by (24) and (22) we obtain, for y ∈B(z ,r −2r¯ ),
k k k
(cid:90)
I (y)= 1 (y)g(x,y,T∗)dx
k B(x,r¯k)
B(zk,rk−r¯k)
(cid:90)
= 1 (x)g(x,y,T∗)dx
B(y,r¯k)
B(zk,rk−r¯k)
(cid:90)
= 1 (x)g(x,y,T∗)dx≥1−δ.
B(y,r¯k)
Rn
Thus by the above inequality and the definition of I we have
k
(cid:40)
1−δ, y ∈B(z ,r −2r¯ ),
I (y)≥ k k k (25)
k
0, y ∈B(z ,r )\B(z ,r −2r¯ ).
k k k k k
By (21) and (25) we obtain
(cid:90) (cid:90)
F (x)dx= I (y)φ(y)dy
k k
B(zk,rk−r¯k) B(zk,rk)
(cid:90) (cid:90)
= I (y)φ(y)dy+ I (y)φ(y)dy
k k
B(zk,rk−2r¯k) B(zk,rk)\B(zk,rk−2r¯k)
(cid:90) (cid:90)
≥ I (y)φ(y)dy ≥(1−δ) φ(y)dy,
k
B(zk,rk−2r¯k) B(zk,rk−2r¯k)
which, together with the continuity of F (x), implies that there exists x ∈
k k
B(z ,r −r¯ ) such that
k k k
(cid:82)
(1−δ) φ(y)dy
F (x )≥ B(zk,rk−2r¯k) . (26)
k k ω (r −r¯ )n
n k k
Since (16) implies that
(cid:90) (cid:90)
φ(y)dy ≥α dy
B(zk,rk) B(zk,rk)∩{y|φ(y)≥α}
≥αmes(cid:0)B(z ,r )∩{x|φ(x)≥α}(cid:1)≥α(cid:0)D(α)−ε(cid:1)ω rn,
k k n k
we obtain
(cid:90) (cid:90) (cid:90)
φ(y)dy = φ(y)dy− φ(y)dy
B(zk,rk−2r¯k) B(zk,rk) B(zk,rk)\B(zk,rk−2r¯k)
≥α(cid:0)D(α)−ε(cid:1)ωnrkn−(cid:107)φ(cid:107)L∞(Rn)ωn(cid:0)rkn−(rk−2r¯k)n(cid:1). (27)
8
Then by (26) and (27)
Fk(xk) ≥ (r1−−r¯δ)n(cid:104)α(cid:0)D(α) − ε(cid:1)rkn − (cid:107)φ(cid:107)L∞(Rn)(cid:0)rkn − (rk − 2r¯k)n(cid:1)(cid:105),
k k
which is exactly (20) by the definition of F . This completes the proof.
k
Remark 2.3. Since in the hyperbolic space Hn the volume of the ball of
radius r is σn−1(Sn−1)(cid:82)rsinhn−1ηdη, where σn−1(Sn−1) is the surface area of
0
the n−1 Euclidean sphere of radius 1, see [13, p. 79]; and since the heat kernel
in the hyperbolic space satisfies all the corresponding properties in Lemma 2.1
and (6), we can modify the above proofs to adapt the similar problem posted
in the hyperbolic space, say, replacing the ∆ in (1) with the Laplace-Beltrami
operator in Hn.
3 Proving Theorem 3.1 by Theorem 1.2
In this section we shall show that Theorem 1.2 implies the previous results of
Yamauchiin[18]. TostateYamauchi’sresultspreciselywerecallsomenotations
√
in [18]. For ξ(cid:48) ∈Sn−1 and δ ∈(0, 2), we set conic neighbourhood Γ (δ):
ξ(cid:48)
Γξ(cid:48)(δ)={η ∈Rn\{0}|(cid:12)(cid:12)ξ(cid:48)− |ηη|(cid:12)(cid:12)<δ},
and set
S (δ)=Γ (δ)∩Sn−1. (28)
ξ(cid:48) ξ(cid:48)
Define
φ (x(cid:48))=liminfφ(rx(cid:48)) for x(cid:48) ∈Sn−1.
∞
r→+∞
Yamauchi proved the following theorem.
Theorem 3.1 ([18, Theorem 1 and Theorem 2]). Let n ≥ 2. Assume that
there exist ξ(cid:48) ∈ Sn−1 and δ > 0 such that essinf φ (x(cid:48)) > 0. Then
x(cid:48)∈Sξ(cid:48)(δ) ∞
the classical solution for (1) blows up in finite time, and the blow-up time is
estimated as
1 (cid:16) (cid:17)1−p
T∗ ≤ essinf φ (x(cid:48)) . (29)
p−1 x(cid:48)∈Sξ(cid:48)(δ) ∞
Let n = 1. Assume that max{liminfφ(x),liminfφ(x)} > 0. Then the classical
x→+∞ x→−∞
solution for (1) blows up in finite time, and the blow-up time is estimated as
1 (cid:16) (cid:17)1−p
T∗ ≤ max{liminfφ(x),liminfφ(x)} . (30)
p−1 x→+∞ x→−∞
It is easily seen that Theorem 3.1 implies the upper bounded estimate of
Gui and Wang in [6] immediately; that is, if lim ψ(x) = k, then λp−1T ≤
λ
|x|→∞
9
1 k1−p. However,Theorem3.1cannotbeappliedtosomesimplecases,which
p−1
can be illustrated by the following example.
a
Take a sequence {a }∞ by a = k!. We see that lim k = 0. Using
k k=1 k k→∞ak+1
the sequence {a }∞ we can construct a function Φ ∈ C(Rn), which satisfies
k k=1
0≤Φ(x)≤1 and
(cid:40)
0, |x|∈[a + 1,a − 1],
Φ(x)= 2k−1 4 2k 4
1, |x|∈[a ,a ].
2k 2k+1
By the definition of D(α) in (3) it follows that
mes(cid:0){x|φ(x)≥1}∩B(0,a )(cid:1) an −an
D(1)≥limsup 2k+1 ≥limsup 2k+1 2k =1.
mes(cid:0)B(0,a )(cid:1) an
k→+∞ 2k+1 k→+∞ 2k+1
Thus for the initial datum u (x) = Φ(x), by Theorem 1.1 the life span T∗ of
0
the solution u can be estimated by
T∗ ≤ 1 (cid:0)1D(1)(cid:1)1−p = 1 .
p−1 p−1
Since v(x,t) = (cid:0)1−(p−1)t(cid:1)p−11, which blows up at T = 1 , is an upper
p−1
solution of (1) with v(x,0) ≥ u(x,0), by the comparison principle it follows
that T∗ ≥ T = 1 . Thus T∗ = 1 , which shows that the minimal time
p−1 p−1
blow-up occurs for the initial datum Φ(x).
Now we use Theorem 1.2 to prove Theorem 3.1.
Proof of Theorem 3.1. We shall prove the case n=1 and n≥2, respectively.
(i) n=1. Let us assume that
A=liminfφ(x)≥liminfφ(x).
x→+∞ x→−∞
Then for any ε>0 there exists R>0 such that
φ(x)≥A−ε for x>R.
Hence D(A−ε)=1. By Theorem 1.2 we obtain
1
T∗ ≤ (A−ε)1−p.
p−1
Since ε>0 was arbitrary, it follows
1
T∗ ≤ A1−p.
p−1
The proof for n=1 is finished.
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