A quasi-monotonicity formula and partial regularity for borderline solutions to a parabolic equation

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www.elsevier.com/locate/anihpc

A quasi-monotonicity formula and partial regularity for borderline solutions to a parabolic equation

Gao-Feng Zheng

Department of Mathematics, Huazhong Normal University, Wuhan 430079, PR China Received 1 July 2009; received in revised form 15 May 2010; accepted 22 June 2010

Available online 14 July 2010

Abstract

A quasi-monotonicity formula for the solution to a semilinear parabolic equationu_t=u+V (x)|u|^p⁻¹u,p > (N+2)/(N−2) inΩ×(0, T )with 0-Dirichlet boundary condition is obtained. As an application, it is shown that for some suitable global weak solutionu and any compact setQ⊂Ω×(0, T ) there exists a close subsetQ⊂Qsuch thatu is continuous inQand the (N−_p₋⁴₁)-dimensional parabolic Hausdorff measureH^(N⁻^p−1⁴ ⁾(Q\Q)ofQ\Qis finite.

Keywords:Quasi-monotonicity formula; Partial regularity; Borderline solutions; Semilinear parabolic equations; Potential

1. Introduction

In this paper, we are interested in the following semilinear parabolic problem

⎧⎨

⎩

u_t=u+V (x)|u|^p⁻¹u inΩ×(0, T ), u(x, t )=0 on∂Ω×(0, T ), u(x,0)=u₀(x) inΩ,

(1.1)

whereΩ⊂R^N(N3) is a bounded, smooth domain,p >^N_N⁺₋²₂,u₀∈L^∞(Ω), and the potentialV ∈C¹(Ω)¯ satisfies V (x)cfor some positive constantcand allx∈Ω. It is well known that for anyu0∈L^∞(Ω)problem (1.1) has a unique local in time solution. Specially, if theL^∞-norm of the initial datum is small enough, then (1.1) has a global, classical solution, while the solution to (1.1) ceases to exist after some timeT >0 and limt↑Tu(·, t )L∞(Ω)= ∞ provided that the initial datumu0is large in some suitable sense. In the latter case we call the solution uto (1.1) blowing up in finite time andT the blow-up time.

WhenV ≡1, problem (1.1) is one of the parabolic problems that have been studied extensively in the past. See for example, [1,2,4,8,10–17,19]. Consider (1.1) with initial data of the formu₀=λϕwhereλis a positive number andϕ is a fixed non-negative function inL^∞(Ω)which does not vanish almost everywhere. For largeλ, the energy ofλϕis negative, so the (maximal) solution,u_λ, blows up in finite time. Whenλis small, the solution is global and decays to zero at infinity. It is natural to set

E-mail address:gfzheng@mail.ccnu.edu.cn.

doi:10.1016/j.anihpc.2010.07.001

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λ^∗(ϕ)≡sup

λ >0: The solutionu_λsatisfyingu_λ(0)=λϕis global and decays to zero ast→ ∞ , and define

U_ϕ(x, t )≡lim

λ↑λ^∗u_λ(x, t ). (1.2)

Note that by the maximum principle,u_λ is monotone increasing andU_ϕ coincides withu_λ∗on the maximal interval of existence of the latter. In the following we shall not distinguish U_ϕ andu_λ∗, and we call it (positive) borderline solution.

In [24], Ni, Sacks and Tavantzis had examined the properties of this borderline solution for some range of p.

Among other things, they have proven the following result under the assumption thatΩ is convex:

Forp2^∗=(N+2)/(N−2),N3,u_λ∗is a global,L¹-solution to(1.1), which must be unbounded.

The definition of an L¹-solution will be given in Section 1. There was little progress on the critical and supercritical case since then. Considering global,L¹-solutions for radially symmetric and decreasing initial data in a ball, Galaktionov and Vazquez [11] have proven the following results:

(1) Whenp=2^∗,uλ^∗remains bounded for all time and tends to zero uniformly away from the origin ast→ ∞and (2) whenp∈(p^∗, p), wherep=(N−4)/(N−10), forN11andp= ∞for3N10,uλ^∗blows up in finite

time.

Later, Mizoguchi [21] shows that u_λ∗ blows up in finite time for all supercritical p, that is, the upper bound (N−4)/(N−10)in (2) can be removed. When 2^∗< p <p, where˜ p˜ is the Joseph–Lundgren exponent given by

˜

p=1+4/(N−4−2√

N−1)ifN11 andp˜= ∞ifN10, it is shown in Fila, Matano and Poláˇcik [9] that the blow-up times ofu_λ∗ form a finite set, which in some cases is a singleton. More information on the corresponding Cauchy problem can be found in [22] and [23].

Recently, we [6] have proven that when Ω is convex the borderline solutionuλ∗ blows up in finite time and it decays to zero uniformly after some finite time. Moreover, we have established partial regularity theorem for this borderline solution, i.e., there exists a closed setSinΩ×(0,∞), whose distance to the boundary ofΩ×(0,∞)is greater than a positive number and which satisfiesH^(N⁻^p⁻⁴¹⁾(S)=0, so thatuis continuous inΩ×(0,∞)\S. Here H^s(E)denotes thes-dimensional Hausdorff measure of the setEwith respect to the parabolic metric.

The main purpose of this paper is to improve these results in [6] for more generalV, i.e., we will establish the following theorems.

Theorem 1.1.Consider(1.1)whereΩ is convex. For any positive, borderline solutionuthere exists a closed setS inΩ×(0,∞), whose distance to the boundary ofΩ×(0,∞)is greater than a positive number and which satisfies H^(N⁻^p⁴⁻¹⁾(S)=0, so thatuis continuous inΩ×(0,∞)\S.

Theorem 1.2.Consider(1.1)whereΩ is convex. Any positive borderline solution must blow up in finite time. After some time, it becomes uniformly bounded and decays to zero ast goes to infinity.

These results are definitely not a direct consequence of those of [6]. Due to the appearance of the potentialV, some extra works should be done. The novelty is to establish a quasi-monotonicity formula for the rescaled local energy and to get the estimates forL^p⁺¹-norm of the solution in terms of local energy. WhenV =1, this quasi-monotonicity formula is almost trivial. WhenV =1, it is not easy. There is a “bad” term

Ω_s

∂V¯

∂s

|w|^p⁺¹ψ²ρ dy

involved in the derivative of the local energyE[w](see e.g. Section 3 for the definition). To eliminate this difficulty, we use a trick similar to what was used in [15] to get the blow-up rate estimate for (1.1). Notice that in [15] the basic assumption is 1< p < (N+2)/(N−2)while in this paper we always assumep > (N+2)/(N−2). Actually, we can get the main estimates for local energy for allp >1 in this paper. To explain more, it is easy to see that

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d

dsE[w](s)−1 2

Ω_s

w²_sψ²ρ dy+C

Ω_s

∂V¯

∂s

|w|^p⁺¹ψ²ρ dy+Cϕ(s),

whereϕ is an integrable function on[0,∞)such that _τ^∞e^sϕ(s) dsCe⁻^τ. Since ^∂_∂s^V^¯ can be written as ∇V (x)· ye⁻^s/2, the integral _Ω

s|^∂_∂s^V^¯||w|^p⁺¹ψ²ρ dycan be controlled bye⁻^s/2 _Ω

s|y||w|^p⁺¹ψ²ρ dy. The question is how to estimate the integral _Ω

s|y||w|^p⁺¹ψ²ρ dy. To this end, we introduce E2k[w](s)=1

2

Ω_s

|∇w|²+βw²

|y|^2kψ²ρ dy− 1 p+1

Ω_s

V¯|w|^p⁺¹|y|^2kψ²ρ dy, k∈N.

First, we establish some rough estimates forE2k[w]using the fact that ^∂_∂s^V^¯ = ∇V (x)·ye⁻^s/2= ∇V (x)·(x− ¯x)is bounded, i.e.,

E2k[w](s)M_ke^2λs, ∞ 0

e⁻^2λs

Ωs

|∇w|²|y|^2kψ²ρ dy dsN_k,

for allk∈Nands0. HereM_k, N_kare positive constants depending onk.

Second, by the mathematical induction and|^∂_∂s^V^¯| = |∇V (x)·ye⁻^s/2|C|y|e⁻^s/2, we can improve our estimates by at most finite steps to get

E2k[w](s)M_ke^αs, ∞ 0

e⁻^αs

Ωs

for someα∈(0,1/2).

Finally, we get the quasi-monotonicity formula E[w](s)+1

4 s τ

Ω_s

w²_sψ²ρ dy dsE[w](τ )+C₃e⁻^δτ, ∀s > τs.

Hereδ∈(0,1/2)is a constant. Consequently, we obtain a lower bound estimate E[w](s)−C₄e⁻^δs, ∀ss

and s τ

Ωs

|w|^p⁺¹ψ²ρ dy dsC

1+(s−τ ) η

E[w](τ )+C₅e⁻^δτ ,

whereη(s)=s+s^1/2. With these estimates in hands, we prove our main theorems as in [6].

The monotonicity formula plays an important role in the partial regularity theories. See for example, Struwe’s work [25] on harmonic map heat flow and Caffarelli, Nirenberg amd Kohn’s work [3] on Navier–Stokes equations. For more discussion on local monotonicity formulas, please refer to Ecker [7].

Throughout the paper we will denote byCa constant that does not depend on the solution itself. And it may change from line to line. AndK₁, K₂, . . .,L₁, L₂, . . .,M₁, M₂, . . .,N₁, N₂, . . .are positive constants depending onp, N, Ω, a lower bound ofV,V_C¹₍_Ω)_¯ and the initial energyE[w₀]. Here and hereafterw₀(y)=w(y, s).

2. Preliminaries

Recall that anL¹-solution to (1.1) is a functionuinC([0, T );L¹(Ω))so thatf (x, u)∈L¹(Q_T),Q_T =Ω×(0, T ), and satisfies

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t s

Ω

uφ_t+uφ+f (x, u)φ dx dt−

Ω

uφ|^tsdx=0,

for allφ∈C²(QT),φ=0 on∂Ω×(0, T )and 0s < tT, here and hereafterf (x, u)=V (x)|u|^p⁻¹u. We are more interested in a stronger notion of weak solution. A functionuinC([0, T );L²(Ω))is called anH¹-solution to (1.1) if∇u,ut∈L²(QT),uf (x, u)∈L¹(QT)and

t s

Ω

u_tφ+ ∇u· ∇φ−f (x, u)φ

dx dt=0, (2.1)

holds for allφ∈C([0, T ), H₀¹(Ω))and 0s < t < T. AnL¹- orH¹-solution is called a globalL¹- orH¹-solution respectively if it is anL¹- orH¹-solution inΩ×(0, T )for everyT >0.

For anH¹-solution its energy E(t )=E

u(t )

=1 2

Ω

|∇u|²dx−

Ω

F (x, u) dx,

is well defined for a.e.t. HereF (x, u)= ₀^uf (x, t ) dt. AnH¹-solution is called an energy-decreasing solution if it also satisfies the energy inequality

E(t )+ t s

Ω

u²_tdx dtE(s), (2.2)

for a.e.t > s, includings=0 in[0, T ).

The following theorem is established in [6]. See also e.g., [4], [9] and [20].

Theorem 2.1.

(a) Letube a global, energy-decreasing solution to(1.1). There exists a positive constantC depending onε₀,C₀,

|Ω|and the initial energyE₀such that (1) ess inftE(t )−C;

(2) u_tL²(Ω×(0,∞))C;

(3) u(t )L²C,∀t;

(4) |u(t )L²− u(s)L²|C|t−s|^1/2,∀t, s;and

(5) theL⁴(0, T;H¹(Ω))-norm ofu and theL²(0, T;L¹(Ω))-norm of uf (·, u)are bounded byC(1+T )for everyT >0.

(b) (Compactness)Let{u_k}be a sequence of global, energy-decreasing solutions to(1.1)whereu_k(0)converges to someu₀ inH₀¹(Ω). Suppose that the initial energies ofu_k are uniformly bounded from above. There exists a subsequence{u_k_j}and a functionusuch that

u_k_j →u inC

[0, T );L²(Ω) ,

∇u_k_j∇u, u_k_j_t u_t inL²(Q_T), F (·, uk_j) F (·, u), uk_jf (·, uk_j) uf (·, u) inL¹(QT),

for everyT >0. Consequentlyuis a global,H¹-solution to(1.1)withu(0)=u₀. Moreover, if it is known that u_k_jf (·, u_k_j)→uf (·, u)inL¹(Ω)for a.e.t, then

∇u_k_j→ ∇u inL²(Q_T), F (·, u_k_j)→F (·, u) inL¹(Q_T),

for everyT >0anduis also a global, energy-decreasing solution.

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A stationary solutionwof (1.1) is called stable if there exists a ballB centered atw inL^∞∩H₀¹(Ω)such that every solution to (1.1) starting inside this ball stays in the ball for all subsequent time. Let

U(w)≡

u₀∈L^∞∩H₀¹(Ω): The solution of (1.1) starting atu₀belongs to the ballBat some finite time . It is routine to verify thatU=U(w)is an open, connected subset ofL^∞∩H₀¹(Ω). The boundary ofU,∂U, is non- empty. For any boundary pointwthere exists a sequence{u^k₀}inU converging towinL^∞∩H₀¹(Ω). Since everyu^k₀ generates a globalH¹-solution, Theorem 2.1 asserts that the maximal solution starting atu0can be extended to be a global,H¹-solution. Uniqueness of this global solution is not known generally. We shall call any global,H¹-solution starting at a boundary point ofU aborderline solution.

It is easy to see that 0 is a non-negative, stable stationary solution to (1.1). For any non-negativeϕ∈L^∞∩H₀¹(Ω) which does not vanish identically, the solution of (1.1) withu(0)=λϕ,u_λ, belongs toU for smallλ >0. Sinceu_λ blows up in finite time for largeλ, we can find someλ^∗such thatu_λbelongs toU for allλ < λ^∗, andλ^∗ϕlies on∂U. By the comparison principleu_λconverges monotonically tou_λ∗asλ↑λ^∗.

The monotone convergence theorem implies that

λlim↑λ^∗

Ω

F

x, u_λ(x) dx=

Ω

F

x, u_λ∗(x) dx.

Theorem 2.1(b) is applicable to conclude that this positive borderline solution is also energy-decreasing.

In order to get the lower bound estimates for our energy functionals, we need the following

Lemma 2.2.Lety,z,g andhbe smooth functions on[0,∞). Supposey, g andh are non-negative and for some positive constantsα,KandL,

t+τ

t

g(s) dsK(1+τ ), ∀t, τ >0;

∞ 0

e⁻^αsh(s) dsL.

If for some positive constantsa, b, q >1, the differential inequalities y(s)−az(s)+by^q(s)−g(s),

z(s)αz(s)+h(s) hold on[0,∞), then

z(s)−2Le^αs for alls0.

Proof. Suppose the conclusion is not true. Then there exists ans₁0 such thatz(s₁)e⁻^αs¹+2L <0. From the second differential inequality, we see that

d ds

e⁻^αsz(s)

e⁻^αsh(s).

So for allss₁,

e⁻^αsz(s)−e⁻^αs¹z(s1)L, i.e.,e⁻^αsz(s) <−L.

Therefore, forss₁,

y(s)aLe^αs−g(s)+by^q(s).

Then we deduce that

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y(s) s s1

aLe^ατ−g(τ ) dτ+b

s s1

y^q(τ ) dτ

aL

s s₁

e^ατdτ−K(s−s₁+1)+b s s₁

y^q(τ ) dτ.

It is easy to check that there exists ans₂> s₁, such thataL _s^s

1e^ατdτ−K(s−s₁+1) >0 for alls > s₂. Therefore for alls > s₂,

y(s)b s s₁

y^q(τ ) dτ.

And then the quantity _s^s

1y^q(τ ) dτwill blow up in finite time. But this is impossible. So the lemma is proved. 2 3. Local energy estimates and quasi-monotonicity formula

Suppose in this section thatuis a global classical solution. Let(x,¯ t )¯ ∈Ω×(0,∞)be a fixed point. We introduce the self-similar scaling

w(y, s)=(¯t−t )^βu(x¯+y

¯ t−t , t )

withs= −log(¯t−t ),β=_p₋¹₁. Ifusolves (1.1), thenwsatisfies w_s−w+1

2y· ∇w+βw−V

¯

x+ye⁻^s/2

|w|^p⁻¹w=0 inΩ_s×(s,∞)

whereΩ_s= {y: x¯+ye⁻^s/2∈Ω},s= −logt. We may assume¯ t¯=1 for simplicity as in [15] so that we assume s=0. Here and hereafter we will always denoteV (x¯+ye⁻^s/2)byV (y, s).¯

By introducing a weight functionρ(y)=exp(−^|^y₄^|²), we can rewrite the equation as the divergence form:

ρw_s= ∇ ·(ρ∇w)−βρw+ ¯V|w|^p⁻¹wρ inΩ_s×(0,∞). (3.1) Fix a positive numberRand letψ (y, s)=φ(e⁻^s/2|y|/R)whereφ(r)is the function that is equal to 1 forr1/2, to 0 forr1 and linear betweenr=1 and 1/2. The local energy ofwis given by

E[w](s)=1 2

Ωs

|∇w|²ψ²ρ dy+β 2

Ωs

w²ψ²ρ dy− 1 p+1

Ωs

V¯|w|^p⁺¹ψ²ρ dy.

Note that the local energy depends on(x, t )andR. Notice that this kind of local energies were firstly introduced by Giga, Matsui and Sasayama in [15,16]. In these papers,ψ=ψ (y)was a cutoff function of a fixed ball. However, in this paper,ψ=ψ (y, s)is a cutoff function of moving balls at times. In other words, the functionψis a function of two variables in our case, but one variable in their definition.

Calculating the derivative ofE[w](s)and noting thatws|∂Ω_s = −¹₂y· ∇wwe have d

dsE[w](s)= −

Ω_s

w_s²ψ²ρ dy−1 4

∂Ω_s

|∇w|²(y·γ )ψ²ρ dσ− 1 p+1

Ω_s

∂V¯

∂s|w|^p⁺¹ψ²ρ dy

−2

Ωs

∇w· ∇ψ ψ wsρ dy+

Ωs

|∇w|²+βw²− 2V¯

p+1|w|^p⁺¹

ψ ψsρ dy

−1

2

Ωs

w_s²ψ²ρ dy−1 4

∂Ωs

|∇w|²(y·γ )ψ²ρ dσ− 1 p+1

Ωs

∂V¯

∂s|w|^p⁺¹ψ²ρ dy

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+2

Ω_s

|∇w|²|∇ψ|²ρ dy+

Ω_s

|∇w|²+βw²+ 2V¯

p+1|w|^p⁺¹

ψ|ψ_s|ρ dy

or 1 2

Ω_s

w²_sψ²ρ dy−d

dsE[w](s)−1 4

∂Ωs

|∇w|²(y·γ )ψ²ρ dσ+2

Ω_s

|∇w|²|∇ψ|²ρ dy

+

Ωs

|∇w|²+βw²+ 2V¯

p+1|w|^p⁺¹

ψ|ψ_s|ρ dy

+ 1

2(p+1)

Ωs

∇ ¯V ·y|w|^p⁺¹ψ²ρ dy.

Let us takeR <dist(x, ∂Ω)so that the boundary integrals above vanish. Using the estimates

|∇w|²=(t−t )

p+1 p−1|∇u|²,

|w| =(t−t )^p−1¹ |u|,

|∇ψ| = φe⁻^s/2

R

2e⁻^s/2

R χA_R, and

|ψ_s| = φe⁻^s/2

2R |y|

e⁻^s/2

R |y|χ_A_R, A_R=B_R(x)¯ \B_R/2(x),¯ we can find a constantCwhich depends onN, Randssuch that

exp

N+2

2 − 2

p−1

s+ 1 R²exp

N+2

2 −p+1 p−1−1

s

exp

−R² 16e^s

Ce⁻^2s. And then by the estimates foru, we get

∞ τ

e^s

Ωs

|∇w|²|∇ψ|²+

|∇w|²+w²+ |w|^p⁺¹ ψ|ψs|

ρ dy ds

Ce⁻^τ

¯

t 0

BR(x)¯

|∇u|²+u²+ |u|^p⁺¹ dx dt

Ce⁻^τ.

In the last inequality above the constantCalso depends ont¯. Denote ϕ(s)=

Ωs

|∇w|²|∇ψ|²+

|∇w|²+w²+ |w|^p⁺¹ ψ|ψ_s|

ρ dy.

Then we have dE[w]

ds −1 2

Ω_s

w_s²ψ²ρ dy+ 1 2(p+1)

Ω_s

∇ ¯V ·y|w|^p⁺¹ψ²ρ dy+Cϕ(s), (3.2)

where ∞ τ

e^sϕ(s) dsCe⁻^τ. (3.3)

Firstly, we have the following rough estimates for the local energy.

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Lemma 3.1.There exists a constantCdepending onN,R,p,t¯, the lower bound ofV,V_C¹₍_Ω)_¯ andE[w](s)such that E[w](s)Ce^λs, for allss,

whereλ=_7(p¹⁶₋₁₎^d_d²

1, andd₁,d₂are constants such thatV (x)d₁>0andsup_x_∈_Ω|∇V (x)|diam(Ω)d₂. Proof. We see from (3.1) that

1 2

d ds

Ω_s

w²ψ²ρ dy=

Ω_s

wwsψ²ρ dy+

Ω_s

w²ψ ψsρ dy

= −

Ωs

|∇w|²ψ²ρ dy−

Ωs

βw²ψ²ρ dy+

Ωs

V¯|w|^p⁺¹ψ²ρ dy

+

Ωs

w²ψ ψ_sρ dy−2

Ωs

∇w∇ψ wψρ dy

= −2E[w] + p−1 p+1

Ωs

¯

V|w|^p⁺¹ψ²ρ dy+

Ωs

w²ψ ψ_sρ dy−2

Ωs

∇w∇ψ wψρ dy. (3.4)

Notice thatV¯ is bounded below byd₁. By (3.4), using Young’s inequality, we have

−2E[w] +p−1 p+1d1

Ωs

|w|^p⁺¹ψ²ρ dy−2E[w] +p−1 p+1

Ωs

V¯|w|^p⁺¹ψ²ρ dy

=

Ωs

ww_sψ²ρ dy+

Ωs

ww_sψ²ρ dy+ε 2

Ωs

w²ψ²ρ dy+C(ε)

Ωs

|∇w|²|∇ψ|²ρ dy

Ωs

ww_sψ²ρ dy+ε 2

Ωs

|w|^p⁺¹ψ²ρ dy+C

+C(ε)ϕ(s)

ε

Ωs

w²_sψ²ρ dy+ε

Ωs

|w|^p⁺¹ψ²ρ dy+C(ε)

1+ϕ(s) .

Here we have used the inequality abε

a²+b^p⁺¹

+C(ε), p >1, ∀ε >0.

So we obtain that

Ωs

|w|^p⁺¹ψ²ρ dy2c(p, d1)E[w] +η

Ωs

|w|^p⁺¹ψ²ρ dy+η

Ωs

w_s²ψ²ρ dy+C(p, d₁, η)

1+ϕ(s) .

Here and hereafter we will denote _(p^p₋⁺_1)d¹

1 byc(p, d1)andC(p, d1, η)denotes a constant depending onp, d1, η >0 and may be different at each occurrence. Takeη <1/8 and we hence have

Ωs

|w|^p⁺¹ψ²ρ dy2c(p, d1)

1−η E[w] + η 1−η

Ωs

w²_sψ²ρ dy+C(p, d₁, η)

1+ϕ(s)

16c(p, d1)

7 E[w] +2η

Ωs

w_s²ψ²ρ dy+C(p, d₁, η)

1+ϕ(s)

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α16c(p, d1)

7 E[w] +2αη

Ωs

w²_sψ²ρ dy+αC(p, d₁, η)

1+ϕ(s)

, (3.5)

for allα1. Choosingηsmall further such that 2αηd2<1/4, we get

Ω_s

|w|^p⁺¹ψ²ρ dyα16c(p, d1)

7 E[w] + 1 4d2

Ω_s

w_s²ψ²ρ dy+C(α)

1+ϕ(s)

, (3.6)

where sup_y_∈_Ω_s|∇ ¯V||y| =sup_x_∈_Ω|∇V||x− ¯x|d2. By (3.2), (3.6), we have for any fixedα1, d

dsE[w](s)−1 4

Ωs

w²_sψ²ρ dy+αλE[w](s)+C(α)

1+ϕ(s) .

Therefore, we obtain that d

dsE[w](s)−1 4

Ω_s

w²_sψ²ρ dy+μE[w](s)+C(μ)

1+ϕ(s)

, (3.7)

for allμλ. In particular, we have d

ds

e⁻^λsE[w](s) +1

4e⁻^λs

Ωs

w_s²ψ²ρ dyC1e⁻^λs

1+ϕ(s)

. (3.8)

It follows thatE[w](s)Ce^λsdue to (3.3).

In order to get the lower bound ofE[w](s), we need to estimate the last two terms in (3.4) firstly. For anyε >0, we have

Ω_s

w²ψ ψsρ dy−2

Ω_s

Ωs

w²|ψ ψ_s|ρ dy+2

Ωs

|∇w|²|∇ψ|²ρ dy ¹

2

Ωs

w²ψ²ρ dy ¹

2

Ωs

w²|ψ ψ_s| + |∇w|²|∇ψ|² ρ dy+

Ωs

w²ψ²ρ dy

ϕ(s)+ε

Ω_s

|w|^p⁺¹ψ²ρ dy+C(ε)

Ω_s

ψ²ρ dy

ϕ(s)+ε

Ω_s

|w|^p⁺¹ψ²ρ dy+C(ε).

Now by (3.4), the above estimate and Jensen’s inequality, if we sety(s)= _Ω_sw²ψ²ρ dy, then we have y(s)−4E[w] +Cy^p+1² (s)−C

ϕ(s)+1

. (3.9)

SinceC2=C1 ∞

0 e⁻^λs(1+ϕ(s)) ds <∞, applying Lemma 2.2 forz(s)=E[w](s), we getE[w](s)−2C2e^λs. So the lemma follows. 2

To get some refined estimates forE[w], we introduce E2k[w] =1

2

Ω_s

|∇w|²+βw²

|y|^2kψ²ρ dy− 1 p+1

Ω_s

V¯|w|^p⁺¹|y|^2kψ²ρ dy, k∈N.

HereN= {0,1,2,3, . . .}. For these energy functionals, by straightforward calculation, we can obtain the following identities.

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Proposition 3.2.

1 2

d ds

Ωs

w²|y|^2kψ²ρ dy= −2E2k[w] +p−1 p+1

Ωs

V¯|w|^p⁺¹|y|^2kψ²ρ dy

+

Ω_s

k

N+2k−2−1 2|y|²

w²|y|^2k⁻²ψ²ρ dy

+

Ωs

w²|y|^2kψ ψsρ dy−

Ωs

∇w∇ ψ²

w|y|^2kρ dy

+k

Ωs

y· ∇ ψ²

w²|y|^2k⁻²ρ dy. (3.10)

Proposition 3.3.

d

dsE2k[w] = −

Ω_s

w_s²|y|^2kψ²ρ dy−2k

Ω_s

ψ²ρ(y· ∇w)ws|y|^2k⁻²dy

−

Ω_s

∇w∇ ψ²

ws|y|^2kρ dy− 1 p+1

Ω_s

∂V¯

∂s|w|^p⁺¹|y|^2kψ²ρ dy

+

Ω_s

|∇w|²+βw²− 2

p+1V¯|w|^p⁺¹

ψ ψ_s|y|^2kρ dy. (3.11)

Denote ϕ_2k(s)=

Ωs

|∇w|²+w²+ |w|^p⁺¹

ψ|ψ_s| + |∇w|²|∇ψ|²

|y|^2kρ dy+2k

Ωs

w²ψ|∇ψ||y|^2k⁻¹ρ dy.

As before, we can find a constantCdepending onN,R,p,k, andt¯such that ∞

τ

e^sϕ2k(s) dsCe⁻^τ.

It is easy to see from (3.10) that

Ωs

ww_s|y|^2kψ²ρ dy= −2E2k[w] +p−1 p+1

Ωs

V¯|w|^p⁺¹|y|^2kψ²ρ dy

+

Ωs

k

N+2k−2−1 2|y|²

w²|y|^2k⁻²ψ²ρ dy

−

Ωs

∇w∇ ψ²

w|y|^2kρ dy+k

Ωs

y· ∇ ψ²

w²|y|^2k⁻²ρ dy.

So

p−1 p+1

Ωs

V¯|w|^p⁺¹|y|^2kψ²ρ dy

Ωs

ww_s|y|^2kψ²ρ dy+2E2k[w]

−

Ωs

k

N+2k−2−1 2|y|²

w²|y|^2k⁻²ψ²ρ dy

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+

Ω_s

∇w∇ ψ²

w|y|^2kρ dy−k

Ω_s

y· ∇ ψ²

w²|y|^2k⁻²ρ dy

Ωs

|w||ws||y|^2kψ²ρ dy+2E2k[w] +ϕ2k(s)

+k 2

Ωs

w²|y|^2kψ²ρ dy+

Ωs

∇w∇ ψ²

w|y|^2kρ dy

Ωs

|w||w_s||y|^2kψ²ρ dy+2E2k[w]

+

1+k 2

Ωs

w²|y|^2kψ²ρ dy+2ϕ2k(s).

We have used Cauchy’s inequality in the last inequality and the fact thatN+2k−2>0 in the second inequality.

Making use of the inequality abε

a²+b^p⁺¹

+C(ε), p >1, ∀ε >0, we have

Ω_s

|w||w_s||y|^2kψ²ρ dyε

Ω_s

|w|^p⁺¹|y|^2kψ²ρ dy+ε

Ω_s

w²_s|y|^2kψ²ρ dy+C(ε, k).

Applying Young’s inequality we obtain that

1+k 2

Ωs

w²|y|^2kψ²ρ dyε

Ωs

|w|^p⁺¹|y|^2kψ²ρ dy+C(ε, k).

Therefore,

Ω_s

|w|^p⁺¹|y|^2kψ²ρ dy2c(p, d1)E2k[w] +η

Ω_s

w²_s|y|^2kψ²ρ dy

+η

Ωs

|w|^p⁺¹|y|^2kψ²ρ dy+C(p, d1, k)

1+ϕ2k(s) ,

i.e.,

Ωs

|w|^p⁺¹|y|^2kψ²ρ dy2c(p, d1)

1−η E2k[w] + η 1−η

Ωs

w_s²|y|^2kψ²ρ dy+C(p, d₁, k, η)

1+ϕ_2k(s)

α16c(p, d1)

7 E2k[w] +2αη

Ωs

w²_s|y|^2kψ²ρ dy+C(α)

1+ϕ_2k(s)

for allα1 andη <1/8.

Choosingηsmall further such that 2αηd2<1/4, we get that for allα1,

Ωs

w^p⁺¹|y|^2kψ²ρ dyα16c(p, d1)

7 E2k[w] + 1 4d2

Ωs

w_s²|y|^2kψ²ρ dy+C(α)

1+ϕ_2k(s)

. (3.12)

Now it is easy to see from Young’s inequality that

−2k

Ωs

ψ²ρ(y· ∇w)w_s|y|^2k⁻²dyε int_Ω_sw²_s|y|^2kψ²ρ dy+C(ε)

Ωs

|∇w|²|y|^2k⁻²ψ²ρ dy,

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and

−

Ω_s

∇w∇ ψ²

ws|y|^2kρ dyε

Ω_s

w²_s|y|^2kψ²ρ dy+C(ε)

Ω_s

|∇w|²|∇ψ|²|y|^2kρ dy.

So by (3.11), the above inequalities, Hölder’s inequality and (3.12) we have d

dsE2k[w]−1 2

Ω_s

w_s²|y|^2kψ²ρ dy+C

Ω_s

|∇w|²|y|^2k⁻²ψ²ρ dy

− 1 p+1

Ω_s

∂V¯

∂s|w|^p⁺¹|y|^2kψ²ρ dy+Cϕ2k(s)

−3

8

Ωs

w_s²|y|^2kψ²ρ dy+μE2k[w] +C(μ)

1+ϕ2k(s)

+C(μ)

Ωs

|∇w|²|y|^2k⁻²ψ²ρ dy, (3.13)

for allμλ. Herek1.

On the other hand, by (3.10), Hölder’s inequality, Young’s inequality and Jensen’s inequality we have 1

2 d ds

Ωs

w²|y|^2kψ²ρ dy−2E2k[w] −C

Ωs

w²|y|^2kψ²ρ dy+C

Ωs

|w|^p⁺¹|y|^2kψ²ρ dy−Cϕ_2k(s)

−2E2k[w] +C

Ω_s

|w|^p⁺¹|y|^2kψ²ρ dy−C

1+ϕ_2k(s)

−2E2k[w] −C

1+ϕ_2k(s) +C

Ω_s

w²|y|^2kψ²ρ dy ^p⁺¹

2

. (3.14)

With these crucial inequalities, (3.13), (3.14), in hands, we can get the following rough estimates.

Lemma 3.4.For anyk∈N, there exist positive constantsLk,Mk,andNk, such that the following estimates hold:

−L_ke^2λsE2k[w](s)M_ke^2λs, ∞

0

e⁻^2λs

Ωs

for alls0. Hereλ=_7(p¹⁶₋₁₎^d_d²₁, andd1,d2are constants such thatV (x)d1>0andsup_x_∈_Ω|∇V (x)|diam(Ω) d2.

Proof. Let{λ_k}^∞_k₌₀⊂(λ,2λ)be a strictly increasing sequence. It suffices to show the following estimates:

−Lke^λ^k^sE2k[w](s)Mke^λ^k^s, (3.15)

∞ 0

e⁻^λ^k^s

Ωs

|∇w|²|y|^2kψ²ρ dy dsN_k. (3.16)

We prove these estimates by induction.