Semilinear parabolic equation in <span class="mathjax-formula">$ {\mathbf{R}}^{N}$</span> associated with critical Sobolev exponent

Semilinear parabolic equation in R ^N associated with critical Sobolev exponent

Ryo Ikehata

, Michinori Ishiwata

, Takashi Suzuki

aDepartment of Mathematics, Faculty of Education, Hiroshima University, Higashi-Hiroshima 739-8524, Japan bDepartment of Mathematical Science, Common Subject Division, Muroran Institute of Technology, Muroran 050-8585, Japan

cDepartment of Mathematics, Graduate School of Science, Osaka University, Toyonaka 560-0043, Japan Received 27 February 2008; received in revised form 8 December 2009; accepted 15 December 2009

Available online 7 January 2010

Abstract

We consider the semilinear parabolic equationu_t−u= |u|^p−1uon the whole spaceR^N,N3, where the exponentp= (N+2)/(N−2)is associated with the Sobolev imbeddingH¹(R^N)⊂L^p+1(R^N). First, we study the decay and blow-up of the solution by means of the potential-well and forward self-similar transformation. Then, we discuss blow-up in infinite time and classify the orbit.

©2010 Elsevier Masson SAS. All rights reserved.

MSC:35K55

Keywords:Parabolic equation; Critical Sobolev exponent; Cauchy problem; Stable and unstable sets; Self-similarity

1. Introduction

The purpose of the present paper is to classify the asymptotic behavior of the solution to the semilinear parabolic equation

u_t−u= |u|^p−1u inR^N×(0, T ) (1)

with

u|t=0=u0 inR^N, (2)

whereN3 andp=(N+2)/(N−2), the critical exponent associated with the Sobolev imbedding. In the previous work [9,8,11,23], we studied the long time behavior of the solution defined on the bounded domain in connection with

* Corresponding author.

E-mail addresses:ikehatar@hiroshima-u.ac.jp (R. Ikehata), ishiwata@mmm.muroran-it.ac.jp (M. Ishiwata), takashi@math.sci.osaka-u.ac.jp (T. Suzuki).

1 The author is partially supported by JSPS Grant-in-aid for Scientific Research (C) #19540184.

2 The author is partially supported by JSPS Grant-in-Aid for Young Scientists B #19740081.

3 The author is partially supported by JSPS Grant-in-aid for Scientific Research (B) #20340034.

0294-1449/$ – see front matter ©2010 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2010.01.002

the stable and unstable sets introduced by [22,20]. We have shown that if the orbit enters in the stable set, then the solution exists globally in time [11]. If the orbit enters in the unstable set, then the solution blows up in finite time [9].

The other case is called “floating”. Thus, the orbit is floating, by definition, if it never enters in the stable set nor the unstable set. If the domain is star-shaped, then the orbit floating globally in time must blow up in infinite time, while the floating orbit blowing up in finite time never exists under the additional assumption of the domain and the initial value, that is, the convexity and symmetry [23].

In contrast with these cases on the bounded domain, there is a family of stationary solutions to (1)–(2) concerning the whole spaceR^N. We have the lack of the Poincaré inequality also. These differences are made clear by the forward self-similar transformation and the non-existence of the self-similar solution [5,14]. Consequently, we can classify the rate ofu(t )_∞ast↑ +∞for the solutionu=u(·, t )global in time.

More precisely, ifu=u(x, t )is the solution, then

v(y, s)=(1+t )^1/(p−1)u(x, t ), t=e^s−1, x=(1+t )^1/2y (3) satisfies

v_s+Lv= |v|^p−1v+ 1

p−1v inR^N×(0, S) (4)

with

v|s=0=u₀ inR^N, (5)

whereS=log(1+T )and Lf = −f−1

2y· ∇f.

Since

Lf = −1

K∇ ·(K∇f ) holds for

K(y)=e^|y|2/4,

problem (4)–(5) is associated with the Hilbert spaceL²(K), the set of measurable functionsf =f (y)defined inR^N such that

f2,K=

R^N

f (y)²K(y) dy 1/2

<+∞.

We also take H^m(K)=

f∈L²(K)D^αf ∈L²(K)for any multi-indexαin|α|m , wherem=1,2, . . .. It is a Hilbert space provided with the norm

fH^m(K)=

|α|m

D^αf ²

2,K

1/2

.

ThisLis realized as a self-adjoint operator inL²(K)associated with the bilinear form AK(u, v)=

R^N

∇u(y)· ∇v(y)K(y) dy defined foru, v∈H¹(K)through the relation

AK(u, v)=(Lu, v)_K, u∈D(L)⊂H¹(K), v∈H¹(K),

where

(u, v)K=

R^N

u(y)v(y)K(y) dy,

see [13], for the general theory of the bilinear form. The domain D(L) of this operator is the set of v∈L²(K) satisfyingLv∈L²(K), and we haveD(L)=H²(K), see Lemma 2.1 of [14]. It holds also thatLis positive self- adjoint and has the compact inverse, and in particular, the set of normalized eigenfunctions ofL forms a complete ortho-normal system inL²(K). The first eigenvalueλ₁ofLis given byλ₁=N/2, and hence the Poincaré inequality

λ₁v²_2,K∇v²_2,K, v∈H¹(K), (6)

is valid, see Proposition 2.3 of [5].

We have λ₁=N

2 > λ≡ 1

p−1=N−2 4 and therefore, the operator

A=L− 1 p−1

inL²(K)is also positive self-adjoint with the domainD(A)=H²(K). The fractional powersA^α and the semigroup {e^{−t A}}t0 are thus defined inL²(K), where α∈ [0,1]. These structures guarantee the well-posedness of (4)–(5) locally in time. Later, we shall show the following fact.

Proposition 1.1.Eachu₀∈H¹(K) admitsT >0 such that (4)–(5) has a unique solution v∈C([0, T];H¹(K)) satisfying the following:

1. v∈C((0, T];D(A^ν)).

2. v(s)=e^−sAu₀+_t

0e^{−(s−σ )A}|v(σ )|^p−1v(σ ) dσ. 3. lims↓0s^ν−12A^νv(s)2,K=0,

where

ν

⎧⎨

⎩

=N/(N+2) (N5),

∈(2/3,1) (N=4),

=4/5 (N=3).

We callv=v(·, s) theH¹(K)-solution, or simply thesolutionto (4)–(5). Proposition 1.1 assures local in time unique existence of the solution u=u(·, t ) to (1)–(2) for u0∈H¹(K). Also, this v=v(·, s) coincides with the solution discussed in [14]. Henceforth,Tm(K)∈(0,+∞]denotes the supremum ofT such that the solutionv=v(·, s) to (4)–(5) exists fors∈ [0, T]. The local existence timeT assured by Proposition 1.1, however, is not estimated from below byu₀H¹(K). Consequently, we cannot conclude

lim sup

s↑Tm(K)

v(s)

H¹(K)= +∞

fromTm(K) <+∞(cf. Theorem 1.10(ii) of [14]), similarly to the case of the bounded domain, see [9]. Henceforth, we putT =Tm(K)∈(0,+∞]for simplicity.

In this paper, we show that the orbit made from the above mentioned solution is classified in the following way.

We emphasize that the solutionu=u(·, t )may be sign-changing.

Theorem 1.Ifu=u(·, t )is the solution to(1)–(2)withu₀∈H¹(K)andp=^N_N⁺₋²₂,N3, then we have the following alternatives.

1. T = +∞andlim sup_t↑+∞t^N/2u(t )_∞<+∞.

2. T = +∞andlimt↑+∞t^(N−2)/4u(t )_∞= +∞.

3. T <+∞andlimt↑Tu(t )_∞= +∞.

Ifu0=u0(x)0,u0≡0, the first case is refined asT = +∞with u(t )

∞∼t^−N/2 ast↑ +∞. (7)

Here are some comments.

1. There is an analogous result concerning sub-critical nonlinearities [15]. Thus, ifu₀=u₀(x)0 and 1+_N² <

p <^N_N⁺₋²₂, then we have the following alternatives for the solutionu=u(·, t )to (1)–(2);

(a) T = +∞and (7).

(b) T = +∞andu(t )∞∼t^−1/(p−1)ast↑ +∞. (c) T <+∞.

The second case of this result indicates the decay rate with that of the self-similar solution. Our Theorem 1 is, actually, associated with the non-existence of the self-similar solution for the critical Sobolev exponent [14], see Proposition 5.1 below. The second case of Theorem 1, thus, may be called “type II rate” att = +∞because 1/(p−1)=(N−2)/4 forp=^N_N⁺₋²₂.

2. The second case of Theorem 1 arises if the orbit is floating globally in time in the rescaled variables (3). Such a case occurs actually ifu₀=u₀(x)is a non-trivial stationary solution. In fact, in contrast with the case of the bounded star-shaped domain provided with the Dirichlet boundary condition, problem (1)–(2) admits a family of non-trivial stationary solutions, normalized by

−U=U^p, 0< U U (0)=1 inR^N, (8)

see [1,3]. As for the solution converging uniformly to 0 in infinite time, however, there remains two possibilities – the first and the second cases of Theorem 1.

3. Positive radially symmetric solutions have been studied in detail. Particularly, analogous results to Theorem 1 are obtained [21], in accordance with the threshold of the modulus of the initial value for the blow-up of the solution. Its proof, however, uses the intersection comparison principle and is different from ours. The blow-up profile is also known by [12]. Thus,T = +∞implies the existence of limt↑+∞u(t )_∞=α∈ [0,+∞]for a suitable family of radially symmetric positive initial values. Ifα=0, the second case of Theorem 1 arises. It holds, furthermore, that

u(t )= u(t )

∞U u(t ) ^N2−2

∞ · +o(1) ast↑ +∞inH˙¹(R^N), where

H˙¹ R^N

=

v∈L^N−22N

R^N ∇v∈L² R^NN

andU=U (y) >0 is the normalized non-trivial stationary solution defined by (8).

Our proof of Theorem 1 is involved by the imbedding theorem concerningH¹(K). Henceforth,L^q(K)denotes the Banach space composed of measurable functionsf=f (y)defined inR^Nsuch that

fq,K=

R^N

f (y)^qK(y) dy 1/q

<+∞

forq∈ [1,∞)and f_∞,K=ess sup

y∈R^N

f (y)<+∞

forq= ∞. The spaceL^∞(K)=L^∞(R^N)is thus compatible to the other spaces, i.e.,

qlim↑∞fq,K= f_∞,K, f ∈L¹(K)∩L^∞ R^N

. (9)

Although the inclusion

L^p(K)⊂L^q(K) (1q < p∞) fails, we have

H¹(K)⊂L^2∗(K)

for 2^∗=2N/(N−2)=p+1. More precisely, Corollary 4.20 of [5] guarantees the following fact, regarded as a Sobolev–Poincaré inequality.

Proposition 1.2.It holds that

S₀v²_p+1,K+λ_∗v²_2,K∇v²_2,K, v∈H¹(K), (10) whereλ_∗=max(1, N/4)andS0stands for the Sobolev constant:

S₀=inf

∇v²₂v∈C^∞₀ R^N

, vp+1=1 .

We introduce the functionals J_K(v)=1

2∇v²_2,K−λ

2v²_2,K− 1

p+1v^p_p⁺₊¹_1,K and

I_K(v)= ∇v²_2,K−λv²_2,K− v^p_p⁺₊¹_1,K

defined forv∈H¹(K), whereλ=1/(p−1). Then, the potential depth ofJ_KinH¹(K)is defined by d₀=inf

sup

μ>0

J_K(μv)v∈H¹(K)\ {0} ,

and it holds that d₀=

1

2− 1

p+1

S^(p₀ ^+1)/(p−1)= 1

NS₀^N/2>0. (11)

Here and henceforth, · q indicates the standardL^q norm onR^N. The stable and unstable sets to (4) are defined by

W_K=

v∈H¹(K)J_K(v) < d₀, I_K(v) >0

∪ {0}

and

VK=

v∈H¹(K)JK(v) < d0, IK(v) <0 , respectively.

Theorem 1 is proven by the study of the above defined stable and unstable sets. First, if the orbit enters in the stable set, thenv(·, s)converges to 0 in infinite time.

Proposition 1.3.Ifv=v(·, s)is the solution to(4)–(5)satisfyingv(s0)∈WK for somes0∈ [0, T ), then it holds that T = +∞and

∇v(s) ²

2,K=O e^−αs

(12) ass↑ +∞, whereα∈(0,1).

If the orbit enters in the unstable set, on the contrary, thenv(·, s)blows up in finite time; the following proposition implies Theorem 1.10(i) of [14].

Proposition 1.4. If the solution v=v(·, s) to(4)–(5) satisfiesv(s₀)∈V_K for somes₀∈ [0, T ), then it holds that T <+∞.

The orbit{v(s)}is, thus,floatingin the other case than the ones treated in Propositions 1.3–1.4, i.e.,v(s) /∈(WK∪ VK)fors∈ [0, T ). In spite of the possibility of the floating orbit blowing up in finite time, the following proposition is sufficient to classify the orbit to (1)–(2) as in Theorem 1.

Proposition 1.5.If the orbit{v(s)}is floating globally in time, then it holds that

s↑+∞lim v(s)

∞,K= +∞. (13)

This paper is composed of six sections. Section 2 takes preliminaries. We confirm inequality (11) and Proposi- tion 1.1. In Section 3, we prove Propositions 1.3 and 1.4 by the study of stable and unstable sets. Section 4 describes theL^q(K)-theory ofL. Using this, we study the floating orbit and show Proposition 1.5 in Section 5. The proof of Theorem 1 is completed in Section 6.

2. Preliminaries

This section is devoted to the preliminaries. First, we show (11), noting that sup

μ>0

J_K(μv)=J_K(μv)|μ=μ^∗

holds for μ^∗=

∇v²_2,K−λv²_2,K v^p_p⁺₊¹_1,K

1/(p−1)

and v∈H¹(K)\ {0}. Then, it follows that

sup

μ>0

J_K(μv)= p−1 2(p+1)

∇v²_2,K−λv²_2,K v²_p+1,K

(p+1)/(p−1)

and hence d₀=

1

2− 1

p+1

S_λ^{(p+1)/(p−1)},

where S_λ=inf

∇v²_2,K−λv²_2,K v²_p+1,K

v∈H¹(K)

.

We have, however, λ= 1

p−1 =N−2 4 ,

and therefore,Sλ=S0by Theorem 4.10 and Lemma 4.11 of [5]. This means (11).

Next, we confirm the operator theoretical feature ofLinL²(K)described in the previous section. It is actually involved by the Schrödinger operator with harmonic oscillator;

K^1/2LK^−1/2=H≡ −+V (y), where

V (y)=N 4 +|y|²

16 .

ThisHis associated with the bilinear form

A(u, v)=

R^N

∇u(y)· ∇v(y)+V (y)u(y)v(y) dy

defined foru, v∈H₁¹(R^N)through the relation A(u, v)=(H u, v)

foru∈D(H )⊂H₁¹(R^N)andv∈H₁¹(R^N), where H₁¹

R^N

= v∈H¹

R^N |y|v∈L² R^N

and( , ) denotes theL² inner product. It is realized as a positive self-adjoint operator inL²(R^N)with the compact inverse, because the inclusionH₁¹(R^N)⊂L²(R^N)is compact. Now we show the following lemma.

Lemma 2.1.The domain ofH inL²(R^N)is given by D(H )=H₂²

R^N

≡ v∈H²

R^N |y|²v∈L² R^N

.

Proof. The inclusionH₂²(R^N)⊂D(H )is obvious. We show thatv∈H₁¹(R^N)with

−v+V v=g∈L² R^N

(14) impliesv∈H²(R^N). In fact, from the proof of Lemma 2.1 of [14] we have

R^N

(v)²+V|∇v|²

dyC_Ng²₂ (15)

with a constant CN>0 determined by N. This implies v∈H²(R^N), and then V v∈L²(R^N)by (14). We obtain D(H )⊂H₂²(R^N)and the proof is complete. 2

For later arguments, we describe the proof of (15) in short. First, (14) implies, formally, that

R^N

(v)²+V v·(−v) dy=

R^N

g(−v) dy

and

R^N

V v·(−v) dy=

R^N

V|∇v|²+1

2∇V · ∇v²

dy=

R^N

V|∇v|²−1 2v²V

dy.

Then, (15) holds by

R^N

(v)²+V|∇v|² dy N

16v2+ g2· v2.

To justify these calculations, we note thatv∈H_loc² (R^N)follows from (14). Next, takingϕ∈C₀^∞(R^N)such that 0ϕ1, suppϕ⊂

|y|<2

, ϕ=1 for|y|1,

we multiply−v·ϕnby (15), whereϕn(y)=ϕ(y/n). Then, (14) is obtained by makingn→ +∞. Henceforth, such argument of justification will not be described explicitly.

Lemma 2.1 establishes the operator theoretical profiles ofLas is desired; it is realized as a positive self-adjoint operator inL²(K)with the compact inverse and the domainD(L)=H²(K). Here, we note the following lemma.

Lemma 2.2.The multiplicationK^1/2induces the isomorphisms L²(K)∼=L²

R^N

, H¹(K)∼=H₁¹ R^N

, H²(K)∼=H₂² R^N

.

Proof. It is obvious thatK^1/2:L²(K)→L²(R^N)is an isomorphism. To confirm thatK^1/2:H¹(K)→H₁¹(R^N)is an isomorphism, we takev=K^1/2u∈L²(R^N), givenu∈H¹(K). Since

∇v=K^1/2∇u+y

4v, (16)

it holds that K|∇u|²=

∇v−y 4v

²= |∇v|²−y

2v· ∇v+|y|²

16 v²= |∇v|²−1

4y· ∇v²+|y|² 16 v². Using

−1 4

R^N

y· ∇v²dy=1 4

R^N

(∇ ·y)v²dy=N 4

R^N

v²dy,

we obtain

R^N

|∇u|²K dy=

R^N

|∇v|²+ N

4 + 1 16|y|²

v²

dy,

and therefore, yv∈L²(R^N)^N and∇v∈L²(R^N)^N. This meansv=K^1/2u∈H₁¹(R^N). Given v∈H₁¹(R^N), con- versely, we takeu=K^−1/2v∈L²(K). Then,∇u∈L²(K)^Nfollows from (16).

To show that K^1/2:H²(K)→H₂²(R^N) is an isomorphism, first, we take u∈H²(K) and put v=K^1/2u∈ L²(R^N). Sinceu∈H¹(K), it holds thatv∈H₁¹(R^N)and henceyv∈L²(R^N)^N. This means

y_iu∈L²(K) (i=1,2, . . . , N ). (17)

It also holds that∇u∈H¹(K)^Nand hence y_i ∂u

∂yj ∈L²(K) (i, j=1,2, . . . , N ) (18)

follows similarly. The right-hand side of

∂

∂yj

(y_iu)=δ_iju+y_i ∂u

∂yj

,

therefore, belongs toL²(K), and henceyu∈H¹(K)^N follows from (17). This implies

|y|²u∈L²(K) (19)

similarly to (17) again, which means|y|²v∈L²(R^N). Finally, the right-hand side of

∂²v

∂y_i∂y_j =√ K ∂²u

∂y_i∂y_j +

√K 4 y_i ∂u

∂y_j +

√K

16 y_iy_ju+

√K 4 y_j ∂u

∂y_i

belongs toL²(R^N)by (18) and (19). This meansv∈H²(R^N)and hencev=K^1/2u∈H₂²(R^N).

Givenv∈H₂²(R^N), conversely, we takeu=K^−1/2v∈L²(K). Sincev∈H₁¹(R^N), it holds thatu∈H¹(K). The right-hand side of

∂²u

∂yi∂yj =K^−1/2 ∂²v

∂yi∂yj −K^−1/2 4 y_i ∂v

∂yj +K^−1/2

16 y_iy_jv−K^−1/2 4 y_j∂v

∂yi

,

on the other hand, belongs toL²(K)because ofv∈H₂²(R^N)and the following lemma. Then, we obtainu=K^−1/2v∈ H²(K)and the proof is complete. 2

Lemma 2.3.Ifv∈H₂²(R^N), then it holds that yi

∂v

∂y_j ∈L² R^N

(i, j=1, . . . , N ).

Proof. Henceforth,Cdenotes a generic large positive constant. It suffices to show y· ∇v2Cv_H²

2(R^N) (20)

forv∈C₀^∞(R^N), where v_H²

2(R^N)=

|α|2

D^αv ²

2+ y^αv ²

2

^1/2 .

This inequality is obtained by

R^N

y_i² ∂v

∂y_i 2

dy= −

R^N

∂

∂y_j

y_i²∂v

∂y_j

v dy= −

R^N

2δijy_i ∂v

∂y_jv+y_i²∂²v

∂y_j²v

dy

2y_iv2· ∂v

∂y_j

2

+ y_i²v

2· ∂²v

∂y_j²

2

,

and the proof is complete. 2

Lemma 2.4.The operatorA=L−_p−¹₁ inL²(K), positive, self-adjoint, and with the compact inverse inL²(K), is provided with the following properties:

1. D(A) →L^q(K)for

q

⎧⎨

⎩

=2^∗∗ (N5),

∈ [2,∞) (N=4),

∈ [2,∞] (N=3), where2^∗∗=2N/(N−4).

2. v2p,KCA^N/(N+2)v2,K forv∈D(A^N/(N+2)), whereC >0is a constant.

3. lims↓0s^γA^γe^−sAv2,K=0, whereγ >0andv∈L²(K).

Proof. The last fact is a consequence of the general theory, becauseAis a positive self-adjoint operator inL²(K), see [9]. To show the first and the second facts, we use the relation derived from the interpolation theory, see [7], that is,

D A^θ

=

L²(K), H²(K)

θ≡H^2θ(K), K^1/2:H^2θ(K)→

L² R^N

, H₂² R^N

θ is an isomorphism, whereθ∈(0,1).

In fact, we have H^2N/(N+2)

R^N

= L²

R^N , H²

R^N

N/(N+2)→L^2p R^N

, L^p(K), L^q(K)

θ=

K^1/pL^p R^N

, K^1/qL^q R^N

θ=K^1/rL^r R^N

=L^r(K), K^1/(2p):H^2p(K)→L^2p

R^N

is an isomorphism, and therefore,

D

A^N/(N+2)

= H^2N/(N+2)(K)∼=K^−1/2 L²

R^N , H₂²

R^N

N/(N+2)

→K^−1/2 L²

R^N , H²

R^N

N/(N+2)=K^−1/2H^2N/(N+2) R^N →K^−1/2L^2p

R^N∼=K^{−1/2+1/(2p)}L^2p(K) →L^2p(K).

This means the second case.

The first case is proven as follows. IfN=3, we haveH²(R^N) →L^r(R^N)for 2r∞. Then, it holds that D(A) = H²(K)∼=K^−1/2H₂²

R^N

→K^−1/2H² R^N →K^−1/2L^r

R^N

→L^q(K) for 2< qr,r >2 andr=q=2, and therefore,

H²(K) →L^q R^N for any 2q∞.

IfN=4, we haveH²(R^N)⊂L^r(R^N)for anyr∈ [2,∞). Then, it holds that D(A) = H²(K)∼=K^−1/2H₂²

R^N

→K^−1/2H² R^N →K^−1/2L^r

R^N

→L^q(K) for 2< qr,r >2 andr=q=2, and therefore,

H²(K) →L^q(K)

for any 2q <∞. Finally, ifN5, it holds that H²(K)=K^−1/2H₂²

R^N

→K^−1/2L^2∗∗

R^N

=K^{−(1/2)+(1/2∗∗)}L^2∗∗(K) →L^2∗∗(K).

The proof is complete. 2

Once Lemma 2.4 is proven, Proposition 1.1 is shown similarly to [9]. It suffices to use f (u)−f (v)

2,KC

u2p,K+ v2p,K

p−1

u−v2p,K

forf (u)= |u|^p−1u. The following proposition is also proven similarly.

Proposition 2.1.Ifu0∈H¹(K)andv=v(·, s)denotes the solution to(4)–(5), then we have the following properties:

1. t

sv_r(r)²_2,Kdr+J_K(v(t ))=J_K(v(s))fort, s∈ [0, S).

2. s∈ [0, S)→J_K(v(s))is monotone decreasing.

3. v(s₀)∈W_K(resp.V_K)⇒v(s)∈W_K (resp.V_K)fors∈ [s₀, S).

4. ¹_2ds^dv(s)²_2,K+I_K(v(s))=0fors∈ [0, S).

The H¹(K)-solutionv=v(·, s)to (4)–(5) is regarded as theH¹(K)-solutionu=u(·, t )to (1)–(2) through the transformation (3). It is also theL¹-mild solution of [16], and is provided with the following properties.

Proposition 2.2.Ifu₀∈H¹(K), we have d

dtJ u(t )

= − ut(t ) ²

2 and d

dt u(t ) ²

2=I u(t ) for the solutionu=u(·, t )to(1)–(2), where

J (u)=1

2∇u²2− 1

p+1u^p_p⁺₊¹₁ and I (u)= ∇u²2− u^p_p⁺₊¹₁. 3. Stable and unstable sets

In this section we prove Propositions 1.3 and 1.4.

Lemma 3.1.Ifv=v(·, s)is the solution to(4)–(5)satisfyingv(s₀)∈W_Kfor somes₀∈ [0, T ), then it holds that v(s) ^p+1

p+1,K(1−γ ) ∇v(s) ²

2,K−λ v(s) ²

2,K

(21)

fors∈ [s₀, T ), whereγ∈(0,1)andλ=_p−¹₁.

Proof. First, Proposition 1.2 implies S₀^(p+1)/2v^p_p⁺₊¹_1,K

∇v²2,K−λ_∗v²2,K

(p+1)/2

∇v²_2,K−λv²_2,K

∇v²_2,K−λ_∗v²_2,K(p−1)/2

forv∈H¹(K). Next, ifv∈WK it holds that J_K(v)= p−1

2(p+1)

∇v²_2,K−λv²_2,K + 1

p+1I_K(v)

p−1

2(p+1)

∇v²_2,K−λv²_2,K

p−1

2(p+1)

∇v²_2,K−λ_∗v²_2,K

0.

Letv=v(·, s)be the solution to (4)–(5) satisfyingv(s0)∈WK. Thenv(s)∈WK holds fors∈ [s0, T )by Proposi- tion 2.1. Using the above inequalities, we obtain

S₀^2∗/2 v(s) ^p+1

p+1,K

2(p+1) p−1 J_K

v(s)^(p−1)/2 ∇v(s) ²

2,K−λ v(s) ²

2,K

2(p+1) p−1 J_K

v(s₀)^(p−1)/2 ∇v(s) ²

2,K−λ v(s) ²

2,K

for 2^∗=p+1. Since d₀= p−1

2(p+1)S^(p₀ ^+1)/(p−1)> J_K v(s₀)

,

it holds that

γ≡1−S₀^−2∗/2

2(p+1) p−1 J_K

v(s₀)^(p−1)/2

∈(0,1).

Then, inequality (21) follows. 2

Proof of Proposition 1.3. First, we showT = +∞, assumingv(s₀)∈W_K for somes₀∈ [0, T ). In fact, ifv∈W_Kis the case, then

v^p_p⁺₊¹₁<∇v²2,K−λv²2,K< 2

p+1v^p_p⁺₊¹_1,K+2d0

and hence

v^p_p⁺₊¹₁< S₀^n/2

byp=ⁿ_n⁺₋²₂ andd₀=¹_nS^n/2. This implies

∇v²_2,K−λv²_2,K< 2

p+1v^p_p⁺₊¹_1,K+2d0< S₀^n/2. Sincev(s0)∈W_K, we obtain

lim sup

s↑T

∇v(s) ²

2,K−λ v(s) ²

2,K

< S₀^n/2, lim sup

s↑T

v(s) ^p+1

p+1< S₀^n/2 (22)

and hence lim sup

t↑T

∇v(s)

2,K<+∞ (23)

by Proposition 1.2.

Inequality (23) then implies S₀ u(t ) ²

p+1 ∇u(t ) ²

2= ∇v(s) ²

2 ∇v(s) ²

2,KC

with a constantC >0 independent oft∈ [0, T0), whereT0=e^T −1, and therefore,

tlim↑T0

J u(t )

>−∞

forJ (u)=¹₂∇u²₂−_p+¹₁u^p_p⁺₊¹₁. Then, the blow-up analysis guarantees lim sup

t↑T0

u(t ) ^p+1

p+1S₀^N/2

similarly to the bounded domain case, see [11], and therefore, lim sup

s↑T

v(s) ^p+1

p+1,Klim sup

s↑T

v(s) ^p+1

p+1=lim sup

t↑T₀

u(t ) ^p+1

p+1S₀^N/2, a contradiction to (22). Thus,T = +∞follows.

The proof of (12) is similar to that of [9]. First,v∈WK implies J_K(v) p−1

2(p+1)(λ_∗−λ)v²2,K (24)

with 0< λ=_p¹₋₁ < λ_∗ by (22) and Lemma 2.1. Ifv=v(·, s)is the solution satisfyingv(s₀)∈W_K for somes₀∈ [0,+∞), then it holds that

s₁

s

IK

v(r) dr=1

2 v(s) ²

2,K− v(s1) ²

2,K

1

2 v(s) ²

2,K p+1

(p−1)(λ_∗−λ)J_K v(s)

(25) by Proposition 2.1 and (24), wheres0ss1. Since Lemma 3.1 implies

γ ∇v(s) ²

2,K−λ v(s) ²

2,K

I_K v(s) byv(s)∈W_K, it follows that

JK

v(s)

= p−1

2(p+1) ∇v(s) ²

2,K−λ v(s) ²

2,K

+ 1 p+1IK

v(s)

p−1

2γ (p+1)+ 1 p+1

I_K

v(s)

. (26)

Inequalities (25) and (26) imply +∞

s

JK

v(r)

drMJK

v(s)

with a constantM >0 independent ofs∈ [s₀,+∞). Then, inequality (12) is obtained by Komornik’s method, see [17], and the proof is complete. 2

Lemma 3.2.Ifv=v(·, s)is anH¹(K)-solution to(4)–(5), satisfyingv(s₀)∈V_K for somes₀∈ [0, T ), then there is δ >0such that

v(s) ^p+1

p+1,K(1+δ) ∇v(s) ²

2,K−λ v(s) ²

2,K

(27)

fors∈ [s₀, T ).

Proof. Ifv∈V_K, we haveI_K(v) <0, and therefore, 2(p+1)

p−1 d₀{∇v²_2,K−λ_∗v²_2,K}^{(p+1)/(p−1)} {v^p₂∗⁺,K¹}^2/(p−1)

{∇v²_2,K−λv²_2,K}^{(p+1)/(p−1)}

{∇v²_2,K−λv²_2,K}^2/(p−1) = ∇v²2,K−λv²2,K (28) by Lemma 1.2, (11), andλ_∗> λ=1/(p−1). Proposition 2.1, on the other hand, implies

−I_K v(s)

=p−1

2 ∇v(s) ²

2,K−λ v(s) ²

2,K

−(p+1)JK

v(s)

p−1

2 ∇v(s) ²

2,K−λ v(s) ²

2,K

−(p+1)JK

v(s0)

=p−1

2 ∇v(s) ²

2,K−λ v(s) ²

2,K

−(p+1)(1−₀)d₀ (29) forss₀, where

₀=1−J_K(v(s₀)) d₀ >0.

We shall show p−1

2 ∇v(s) ²

2,K−λ v(s) ²

2,K

−(p+1)(1−₀)d₀ δ ∇v(s) ²

2,K−λ v(s) ²

2,K

(30)

forδ=(p−1)0/2>0. Then, (27) will follow from (29).

For this purpose, we use p−1

2 ∇v(s) ²

2,K−λ v(s) ²

2,K

−(p+1)(1−₀)d₀−δ ∇v(s) ²

2,K−λ v(s) ²

2,K

= p−1

2 −δ ∇v(s) ²

2,K−λ v(s) ²

2,K

−(p+1)(1−₀)d₀

=p−1 2 · 1

d0·J_K

v(s₀) ∇v(s) ²

2,K−λ v(s) ²

2,K

−(p+1)JK

v(s₀)

=p−1 2 · 1

d₀·JK

v(s0) ∇v(s) ²

2,K−λ v(s) ²

2,K−2(p+1)d0

p−1

.

The right-hand side of the above inequality is non-negative by (28), and therefore, (30) follows. 2

Proof of Proposition 1.4. The argument of [9] for the case of the bounded domain is not valid here, because L^p+1(K)⊂L²(K)does not arise. To avoid this difficulty, we suppose the contrary,T = +∞. It holds thatv(s)∈V_K fors∈ [s₀,+∞)by Proposition 2.1, and hence

−I_K v(s)

δ ∇v(s) ²

2,K−λ v(s) ²

2,K

δ N

2 − 1

p−1 v(s) ²

2,K

by (6) and Lemma 3.2. This means 1

2· d

ds v(s) ²

2,Kδ1 v(s) ²

2,K

by Proposition 2.1, where δ1=δ

N

2 − 1

p−1

>0, and therefore,