On solutions of space-fractional diffusion equations by means of potential wells

On solutions of space-fractional diffusion equations by means of potential wells

Dedicated to Professor Tibor Krisztin on the occasion of his 60th birthday

Yongqiang Fu

and Patrizia Pucci

1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P.R. China

2Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, 06123 Perugia, Italy

Received 30 June 2016, appeared 12 September 2016 Communicated by Jeff R. L. Webb

Abstract. In this paper, we study the initial boundary value problem of space-fractional diffusion equations. First, we introduce a family of potential wells. Then we show the existence of global weak solutions, provided that the initial energyJ(u0)is positive and less than the potential well depthd. Finally, we establish the vacuum isolating and blow up of strong solutions.

Keywords: potential wells, space-fractional wave equations, global solutions, fractional Sobolev spaces.

2010 Mathematics Subject Classification: 35R11, 35A15, 45K05.

1 Introduction

There exist several natural phenomena that cannot be modeled by partial differential equations based on ordinary calculus, since they depend on the so-called memory effect. In order to take account of this dependence, we may use fractional differential calculus. Fractional differential equations have gained considerable importance due to their applications in various sciences, such as physics, mechanics, chemistry, engineering, etc. In recent years, there has been a significant development in fractional differential equations which may be ordinary or partial, see for examples [8–10,15,16,18,21,34,35] and the references therein.

A space–time fractional diffusion-wave equation is obtained from the classical diffusion or wave equation by replacing the first or second order time derivatives and second order space derivatives by fractional derivatives, see for examples [12,17]. We can describe space–time fractional diffusion-wave equations with three space variables as

C0D^α_tu=χ ∂^βu

∂|x|^β + ^∂

γu

∂|y|^γ + ^∂

δu

∂|z|^δ

, (1.1)

BCorresponding author. Email: patrizia.pucci@unipg.it

whereχis a positive constant,^C₀D^α_t is the Caputo derivative of orderαand∂^β/∂|x|^β,∂^γ/∂|y|^γ and∂^δu/∂|z|^δ are symmetric Riesz derivatives of ordersβ,γandδ, respectively. If β=γ=δ, symmetric Riesz derivatives can be treated as fractional Laplace operators. Equation (1.1) yields different diffusion-wave equations for various values of the parametersα, β, γ andδ.

Precisely,

(1) Classical diffusion equationα=1, β= γ=δ=2.

(2) Time-fractional diffusion equation 0< α<1,β=γ=δ =2, see for examples [23,24,36].

(3) Space-fractional diffusion equationα= 1, either 0 < β,γ ≤ 2 0< δ <2, or 0 < β < 2, 0< γ,δ≤2, or 0< β,δ ≤2, 0<γ<2, see for examples [6].

(4) Space–time fractional diffusion equation 0< α< 1, either 0< β,γ ≤ 2, 0 < δ < 2, or 0< β<2, 0<γ,δ ≤2 or 0<β,δ ≤2, 0<γ<2, see for examples [7,19,29].

(5) Classical wave equationα=β=γ= δ=2.

(6) Time-fractional wave equation 1<α<2,β=γ= δ=2, see for examples [22,27,33].

(7) Space-fractional wave equation α = 2, either 0 < β,γ ≤ 2 ,0 < δ < 2, or 0 < β < 2, 0< γ,δ≤2, or 0< β,δ ≤2, 0<γ<2, see for examples [2].

(8) Space–time fractional wave equation 1 < α < 2, either 0 < β,γ ≤ 2, 0 < δ < 2, or 0< β<2, 0<γ,δ ≤2, or 0<β,δ≤2, 0<γ<2, see for examples [5,14].

In this paper, we study the space-fractional diffusion problem:











u_t+ (−4)^su=|u|^p−1u, x ∈_Ω, t >_{0, (1.2)}

u(x, 0) =u₀(x), x ∈_Ω, (1.3)

u(x,t) =0, x ∈_R^N\_Ω, t≥0, (1.4)

where Ω ⊂ _R^N is a smooth bounded domain, N > 2s, and p satisfies 1 < p ≤ 2^∗_s −1 = (N+2s)/(N−2s).

A suitable stationary fractional Sobolev space for (1.2)–(1.4) isX0(_Ω)which consists of all functionsu∈ H^s(_R^N)withu= 0 a.e. inR^N\Ω. We refer to Section2for further details and recall that the use of the spaceX₀(_Ω)to find solutions of nonlinear fractional elliptic problems was begun in [30].

LetTbe the existence time of the solutionufor the problem (1.2)–(1.4), whereTmay be∞.

We say thatu∈ L^∞(0,T;X₀(_Ω)), withu_t∈ L²(0,T;L²(_Ω)), is aweak solutionof (1.2)–(1.4) if Z _t

0

Z

Ωu_τ(x,τ)ϕ(x,τ)dxdt+C(N,s)

Z _t

0

Z

R^N×_R^N

(u(x,τ)−u(y,τ))(ϕ(x,τ)−ϕ(y,τ))

|x−y|^{N+2s dxdydτ}

=

Z _t

0

Z

Ω|u(x,τ)|^p−1u(x,τ)ϕ(x,τ)dxdτ for any ϕ∈ L¹(0,∞;X₀(_Ω))and any t∈[0,T), and

u(x, 0) =u₀(x)∈ X₀(_Ω)_.

Here and in the following

1 C(N,s) =

Z

R^N

1−cosξ₁

|ξ|^{N+2s dξ.}

If a weak solutionubelongs toC(0,T;X₀(_Ω)), we call uastrong solution.

In order to find solutions of (1.2)–(1.4), we use the potential well theory, see for examples [11,13,20,25,28] and the references therein. All the results obtained for the problem (1.2)–(1.4) are still valid if we replace the equation (1.2) by

ut+ (−4)^su= f(u),

provided that f satisfies the following conditions first introduced in [25]:

(f₁) f ∈C¹(_R)and f(0) = f⁰(0) =0;

(f₂) f is monotone increasing inR, and is convexR⁺, concaveR⁻;

(f₃) (p+1)F(u) ≤ u f(u), |u f(u)| ≤ γ|F(u)|, where 2 < p+1 ≤ γ < _∞ if N = 1, 2, and 2< p+1≤γ≤2N/(N−2s) =2^∗_s if N ≥3, and F(u) =Ru

0 f(s)ds.

For example, concerning a global existence theorem, i.e. Theorem 4.1 in Section4, the key functionals associated to problem (1.2)–(1.4) are

J(u) = ^C(N,s) 2 kuk²_X

0(_Ω)− ¹

p+1kuk^p+1

L^p+1(_Ω), I(u) =C(N,s)kuk²_X

0(_Ω)− kuk^p+1

L^p+1(_Ω). If we replace |u|^p−1u by f(u)which satisfies (f₁)–(f₃), then we should replace the key func- tionals by

J(u) = ^C(N,s) 2 kuk²_X

0(_Ω)−

Z

ΩF(u)dx, I(u) =C(N,s)kuk²_X

0(_Ω)−

Z

Ω f(u)udx.

After the replacement, Theorem4.1in Section4is still valid.

The paper is organized as follows. In Section2, we provide notations and some facts con- cerning fractional Sobolev spaces which shall be used later. In Section3, we introduce a family of potential wells in order to study the space-fractional diffusion equations. In Section 4, we obtain the existence of global weak solutions. In Section 5, we establish the phenomenon of vacuum isolating and blow up for strong solutions.

2 Preliminaries

Let s ∈ (0, 1) and 2s < N. The fractional Laplace operator for a function ϕ ∈ C₀^∞(_R^N) is defined pointwise by

(−4)^sϕ(x) = ^C(N,s) 2

Z

R^N

ϕ(x)−ϕ(x+y)−ϕ(x−y)

|y|^{N+2s dy,}

1 C(N,s) =

Z

R^N

1−cosξ₁

|ξ|^{N+2s dξ,} for all x∈_R^N.

The fractional Sobolev spaceH^s(_R^N)_{is set as} H^s(_R^N) =

(

u∈ L²(_R^N) : |u(x)−u(y)|

|x−y|^N2+s ∈ L²(_R^N×_R^N) )

,

endowed with the norm kuk_Hs(_R^N)=

kuk²_L2(RN)+

Z Z

R^N×_R^N

|u(x)−u(y)|²

|x−y|^{N+2s dxdy} 1/2

. Let

X₀(_Ω) =ⁿu∈ H^s(_R^N):u=0 a.e. inR^N\_Ω^o. In the sequel we take

kuk_X0(Ω) = _{Z Z}

R^N×_R^N

|u(x)−u(y)|²

|x−y|^{N+2s dxdy} 1/2

as norm on X₀(_Ω). It is easily seen that X₀(_Ω) = X₀(_Ω),k · k_X0(Ω) is a Hilbert space with inner product

hu,vi_X0(Ω)=

Z Z

R^N×_R^N

(u(x)−u(y))(v(x)−v(y))

|x−y|^{N+2s dxdy.}

Sinceu∈X₀(_Ω), we know that the norm and inner product can be extended to allR^N×_R^N. Denote by

0< λ₁<λ₂<· · · <λ_n< · · ·

the distinct eigenvalues and e_k the eigenfunction corresponding toλ_kof the elliptic eigenvalue problem:







(−4)^su=λu, inΩ, u=0, inR^N\_Ω.

(2.1) Concerning the eigenvalue of the problem (2.1), by [31] we have fork∈_N

λ_k = ^C(N,s)

2 min

u∈_Pk\{0}

kuk²_X

0(_Ω)

kuk²_L2(Ω)^, where

P1= X₀(_Ω) and for allk ≥2

Pk =ⁿu∈ X₀(_Ω) : hu,e_ji_X0(Ω) =0 for allj=1, 2, . . . ,k−1o .

For the readers’ convenience, we recall the main embedding results for the fractional Sobolev spaces, see [3] for details.

Lemma 2.1. LetΩbe bounded domain. Then

(1)the embedding X₀(_Ω)→L^p(_R^N)is compact for any p ∈[1, 2^∗_s); (2)the embedding X0(_Ω),→L^2∗s(_R^N)is continuous.

Let 1 ≤ p < _∞and let X be a Banach space. The space L^p(0,T;X)denotes the space of L^p-integrable functions from[0,T)intoXwith the norm

kuk_Lp(0,T;X) = _{Z T}

0

ku(·,t)k^p_Xdt 1/p

.

If p = ∞, the space L^∞(0,T;X)is the space of essentially bounded functions from [0,∞)into Xwith norm

kuk_L∞(_0,∞;X)= sup

t∈[0,T)

ku(·,t)k_X.

The spaceC(_0,T;X)consists of all functionsufrom[_0,T)_{intoXsuch that}kuk_Xis continuous on [0,T). See for example [32] for facts concerning this kind of spaces. In this paper we take eitherX= L²(_Ω), or X=L^p+1(_Ω), or X= X₀(_Ω).

3 Potential wells in variational stationary setting

For simplicity, in this section we consider the problem (1.2)–(1.4) in stationary case. In fact, if we replaceuin this section byu(t)for any t∈[0,T), all the facts are still valid.

We define J(u) = ^C(N,s)

2 kuk²_X

0(_Ω)− ¹

p+₁kuk^p+1

L^p+1(_Ω), I(u) =C(N,s)kuk²_X

0(_Ω)− kuk^p+1

L^p+1(_Ω)

and the potential well

W = {u∈X₀(_Ω) : I(u)>0, J(u)<d} ∪ {0}, where

d= inf

u∈X0(_Ω) u6=0

sup

λ≥0

J(λu).

It is easy to see that J(λu)attains its maximum, with respect toλ, at

λ^∗ =





C(N,s)kuk²_X

0(_Ω)

kuk^p+1

L^p+1(_Ω)





1/(p−1)

.

Normalizinguso thatλ^∗=1, i.e.C(N,s)kuk²_X

0 = kuk^p+1

L^p+1(_Ω), we get d=infJ(u)

subject tou∈ X₀(_Ω),kuk_X0(Ω)6=0, I(u) =0.

From [31], we know that the problem







(−4)^sw=|w|^p−1w, inΩ,

w=0, inR^N\_Ω.

(3.1)

admits a nontrivial solution. Then for allλ>0, the functionv=λ

1

1−pwis a solution of







(−4)^sv=λ|v|^p−1v, inΩ,

v=0, inR^N\_Ω.

Define

S_p+1 = inf

u∈X0(_Ω) u6=0

C(N,s)kuk_X0(Ω) kuk_Lp+₁(_Ω)

.

The Euler equation for this homogeneous variational problem is (−4)^su=λ|u|^p−1u,

where λ is a Lagrange multiplier. Therefore, the nontrivial solution w of (3.1) attains the infimum, that is

Sp+1= ^C(N,s)kwk_X0(Ω) kwk_Lp+1(_Ω)

. On the other hand, by the definition of solution for (3.1),

C(N,s)kwk²_X

0(_Ω)=kwk^p+1

L^p+1(_Ω). Thus

d =kwk^p+1

L^p+1

p−1 2(p+1)^. Therefore

Sp+₁=C(N,s)¹²kwk

p−1 2

L^p+1(_Ω) =C(N,s)¹²

2(_p+₁) p−1 d

₂₍^p_p⁻₊¹₁₎

and

d= ^p−1

2(p+₁)^C(N,s)^p

+1 1−pS

2(p+1) p−1

p+1 . Furthermore, for problem (1.2)–(1.4) andδ∈ (0, 1)we define

J_δ(u) = ^δ

2C(N,s)kuk²_X0 − ¹

p+1kuk^p+1

L^p+1(_Ω), (3.2) d(δ) = (1−δ)[(p+1)δ]^{p−21 S}

2p+1

2C(N,s)

!^p_p⁺₋¹₁

. (3.3)

From the definition ofS_p+1, it is easy to get the following lemmas.

Lemma 3.1. If J(u)≤d(δ), then (i) J_δ(u)>0if and only if

0< kuk_X0(Ω)<





(p+1)δS^p_p⁺₊₁¹ 2C(N,s)^p





1/(p−1)

; (ii) J_δ(u)<0if and only if

kuk_X0(Ω) >





(p+1)δS^p_p⁺₊¹₁ 2C(N,s)^p





1/(p−1)

. Lemma 3.2. If J(u) =d(δ), then J_δ(u) =0if and only if

kuk_X0(Ω) =





(p+1)δS^p_p⁺₊¹₁ 2C(N,s)^p





1/(p−1)

.

Lemma 3.3. The function d=d(δ)has the following properties on the interval[0, 1]. (i) d(₀) =d(₁) =0;

(ii) d takes the maximum value atδ₀=2/(p+1)and d(δ₀) =d;

(iii) d is increasing on[0,δ₀]and decreasing on[δ₀, 1];

(iv) for any given e ∈ (0,d), the equation d(δ) = e has exactly two solutions δ₁ ∈ (0,δ0) and δ₂∈(δ₀, 1).

Theorem 3.4. d(δ) =minJ(u)subject to u∈X₀(_Ω),kuk_X0(Ω)6=0, J_δ(u) =0.

First, it is easy to show that J(u)≥ d(δ)_{, when}u ∈ X₀(_Ω)_, kuk_X0(Ω) 6=0, J_δ(u) =_{0, and,} in view of the definition of S_p+1, this concludes the proof of Theorem3.4.

Corollary 3.5. d=d(δ₀) =minJ(u)subject to u∈ X₀(_Ω),kuk_X0(Ω) 6=0, I(u) =0.

The proof of Corollary4.2is an immediate consequence of Theorem 3.4 and the fact that J_δ0(u) =0 is equivalent to I(u) =0.

Let us define the followingfamily of potential wellsfor allδ∈ (0, 1) W_δ ={u∈ X₀(_Ω) : J_δ(u)>0, J(u)<d(δ)} ∪ {0},

W_δ =W_δ∩∂W_δ ={u∈ X₀(_Ω) : J_δ(u)≥0, J(u)≤d(δ)} ∪ {0}. ClearlyW_δ0 =W. In addition, let us introduce for allδ ∈(0, 1)

Vδ ={u∈ X0(_Ω) : Jδ(u)<0, J(u)<d(δ)},

V_δ =V_δ∪∂V_δ = {u∈X₀(_Ω) : J_δ(u)≤0, J(u)≤d(δ)}, V ={u∈ X₀(_Ω): I(u)<0,J(u)<d},

B_δ =







u∈ X₀(_Ω) : kuk_X0(Ω) <





(p+1)δS^p_p+⁺¹₁ 2C(N,s)^p





1/(p−1)



 ,

B_δ =B_δ∪∂B_δ =







u∈X₀(_Ω) : kuk_X0(Ω) ≤





(p+1)δS^p_p⁺₊¹₁ 2C(N,s)^p





1/(p−1)



 ,

B^c_δ =







u∈ X₀(_Ω) _: kuk_X

0(_Ω) >





(p+₁)δS^p_p+⁺¹₁ 2C(N,s)^p





1/(p−1)



 .

Furthermore,V_δ0 =V.

Letu∈X₀(_Ω)\ {0}, with J(u)≤C(N,s)kuk²_X

0/2. Then for any givenδ∈ (0, 1)such that 0< kuk_X0 <(₁−δ)¹²





(p+₁)δS^p_p⁺₊¹₁ 2C(N,s)^p





1/(p−1)

, we get J(u)<d(δ)and J_δ(u)>0. Consequently,

B_δ ⊂Wδ, with δ= (1−δ)^p−21δ.

From this and Lemma3.1, it is immediate to get the following theorem.

Theorem 3.6. B_δ ⊂W_δ ⊂ B_δ, V_δ ⊂ B^c_δ. Corollary 3.7. B_δ

0 ⊂W ⊂ B_δ0, V ⊂ B^c_δ

0, where

B_δ0 =

u∈X₀(_Ω):kuk_X

0(_Ω)<C(N,s)^1−ppS

p+1 p−1

p+1

, δ₀ =

p−1 p+1

^p−₂¹ 2 p+1 andδ₀ =2/(p+1)by Lemma3.3.

In view of Lemma3.3, we obtain the following lemma.

Lemma 3.8. (i)If0<δ⁰ <δ⁰⁰ ≤δ₀, then W_δ⁰ ⊂W_δ⁰⁰. (ii)Ifδ₀ ≤δ⁰ <δ⁰⁰ <1, then V_δ⁰⁰ ⊂V_δ⁰.

It is easy to prove the following lemma by contradiction.

Lemma 3.9. Assume that0< J(u)<d for some u∈X₀(_Ω), and thatδ₁ <δ₂ are the two solutions of the equation d(δ) = J(u). Then J_δ(u)does not change sign forδ∈ (δ₁,δ2).

4 Existence of global weak solutions

In this section we study the global existence of weak solutions for the problem (1.2)–(1.4). Via the results on eigenfunctions of fractional Laplace operators established in [31], we are able to apply the Galërkin method and we construct finite-dimensional Galërkin approximations for the problem (1.2)–(1.4). In particular, we present a priori estimates, which allow us to pass to the limit and to obtain the desired weak solution u of (1.2)–(1.4). Indeed, u verifies the conditions of initial data and belongs to the family of potential wells.

Theorem 4.1. Let u₀ ∈ X₀(_Ω). Suppose that 0 < J(u₀) < d, δ₁ < δ₂ are the two solutions of equation d(δ) = _J(_u0) _{and Jδ2}(_u0) > 0. Then problem (1.2)–(1.4) admits a global weak solution u∈ L^∞(0,∞;X₀(_Ω)), with ut ∈ L²(0,∞;L²(_Ω))and u(·,t)∈W_δforδ ∈(δ₁,δ₂)and t∈_R⁺₀. Proof. Fixu₀∈ X₀(_Ω)such that 0< J(u₀)<d, d(δ_i) = J(u₀),i=1, 2, andJ_δ2(u₀)>0.

By [31], the sequence{e_k}_kof eigenfunctions corresponding to the sequence{λ_k}_{kof eigen-} values of the fractional Laplace operator (−4)^s is an orthonormal basis of L²(_Ω) and an orthogonal basis ofX₀(_Ω). Let

um(x,t) =

∑

g_jm(t)e_j(x), m=1, 2, . . . , be the Galërkin approximate solutions of the problem (1.2)–(1.4) satisfying

(u_mt,e_j)_L2(_Ω)+C(N,s)hu_m,e_ji_X0(Ω)=

Z

Ω|u_m|^p−1u_me_jdx, (4.1) u_m(·, 0) =

∑

a_je_j →u₀ in X₀(_Ω). (4.2) Substitutingu_m into (4.1)–(4.2), we get

g⁰_jm+λ_jg_jm =

∑

|g_jl|^p−1g_jl Z

Ω|e_l|^p−1e_le_jdx, (4.3) g_jm(0) =a_j, j=1, . . . ,m. (4.4)

According to standard ordinary differential equations theory (for example Wintner’s theo- rem), problem (4.3)–(4.4) admits a solutiong_jm of classC¹([0,T])for eachm. Multiplying (4.1) byg⁰_jm(t), summing forjand integrating with respect tot, we have

Z _t

0

kumτ(·,τ)k_L2(_Ω)dτ+J(um(·,t)) = J(um(·, 0)). (4.5) Since J_δ2(u₀)>0 impliesku₀k_X0(Ω) 6=0, an argument similar to the proof of Lemma3.9gives J_δ(u0)>0 for all δ∈ (δ₁,δ2). Furthermore, J(u0) =d(δ₁)implies that the initial valueu0 is in W_δ for all δ ∈(δ₁,δ₂). Hence, for any fixedδ ∈(δ₁,δ₂), the inequality J_δ2(u₀)>0 implies that J_δ(u_m(·, 0)) > 0 and J(u_m(·, 0)) < d(δ), provided that m is sufficiently large. This happens for all δ ∈ (δ₁,δ2) by virtue of Lemma 3.8(i). Therefore, we may assume without loss of generality, that u_m(·, 0)∈W_δ for all allδ∈ (δ₁,δ₂)andm.

We claim thatu_m(·,t)∈ W_δ for all δ ∈ (δ₁,δ₂), allm and all t > 0. Otherwise there exist δ ∈ (δ₁,δ2), m andt0 > 0 such thatum(·,t0) ∈ ∂W_δ, i.e. either (i) J(um(·,t0)) = d(δ), or (ii) J_δ(u_m(·,t₀)) =0 andku_m(·,t₀)k_X0(Ω)6=0. From (4.5), we get

J(um(·,t))≤ J(um(·, 0))<d(δ) for allt>0.

Hence the case(i)is impossible. If(ii)occurs, then by Theorem3.4we get J(u_m(·,t₀))≥d(δ), which is also impossible. This completes the proof of the claim.

Thus, Lemmas3.1and3.3(ii)imply that for allt >0 andm

ku_m(·,t)k_X0(Ω)<





(p+₁)δS^p_p⁺₊¹₁ 2C(N,s)^p





1/(p−1)

<





(p+₁)δ₂S^p_p⁺₊¹₁ 2C(N,s)^p





1/(p−1)

,

ku_m(·_,t)k_Lp+1(_Ω) ≤ ^C(N,s) S_p+1

ku_m(·_,t)k_X

0(_Ω) < (p+1)δ2S²_p+1 2C(N,s)

!1/(p−1)

, Z _t

0

ku_mτ(·_,τ)k²_L2(Ω)dτ<2d(δ)≤2d(δ₀)_,

being δ ∈ (δ₁,δ₂). Then by the weak^∗ compactness of bounded sets in L^∞(_0,∞;L^p+1(_Ω)) and in L^∞(0,∞;X₀(_Ω)) and the weak compactness of bounded sets in L²(0,∞;L²(_Ω)), we conclude that there exists a subsequence of (u_m)_m – still denoted by(u_m)_m – such that

um →uweakly^∗ in L^∞(0,∞;L^p+1(_Ω))and inL^∞(0,∞;X₀(_Ω)), and

umt →ut weakly inL²(0,∞;L²(_Ω)). From (4.2) we have thatu(·, 0) =u₀in X₀(_Ω).

Integrating (4.1) with respect totand lettingm→∞, we obtain that for eachw_j, Z _t

0

Z

Ωu_τ(x,τ)w_j(x)dxdτ+C(N,s)

Z _t

0

hu(·,τ)_,w_ji_X0(Ω)dτ

=

Z _t

0

Z

Ω|u(x,τ)|^p−1u(x,τ)w_j(x)dxdτ and furthermore for any v∈X0(_Ω),

Z _t

0

Z

Ωuτ(x,τ)v(x)dxdτ+C(N,s)

Z _t

0

hu(·,τ),vi_X0(Ω)dτ=

Z _t

0

Z

Ω|u(x,τ)|^p−1u(x,τ)v(x)dxdτ.

Differentiating with respect tot, we have Z

Ωu_t(x,t)v(x)dx+C(N,s)hu(·,t),vi_X0(Ω) =

Z

Ω|u(x,t)|^p−1u(x,t)v(x)dx.

For any ϕ∈ L¹(0,∞;X₀(_Ω)), lettingv(x) = ϕ(x,t), with t fixed, and integrating with respect tot, we conclude thatu is a weak solution of problem (1.2)–(1.4).

Sinceum(·,t)∈Wδ for allδ ∈ (δ₁,δ₂)_{, allmand allt} ∈ _R⁺₀, we get thatu(·,t)∈Wδ for all δ∈ (δ₁,δ₂)and allt ∈_R⁺₀. This completes the proof.

In view of the facts that I(u0)> 0 implies J_δ(u0) >0 and that J_δ2(u0)≥ J_δ0(u0)thanks to their definitions and Lemma3.3(iv), we get at once

Corollary 4.2. If in Theorem4.1the assumption Jδ₂(_u0)> ₀is replaced by I(_u0)> _{0, i.e. u0} ∈W, then the result of Theorem4.1continues to hold.

Next we consider the problem (1.2)–(1.4), under the critical conditions I(u₀) ≥ _{0 and} J(u₀) =d.

Theorem 4.3. Let u₀ ∈ X₀(_Ω). Suppose that J(u₀) = d and I(u₀) ≥ 0, then problem(1.2)–(1.4) admits a global weak solution u∈ L^∞(0,∞;X0(_Ω)), with ut ∈ L²(0,∞;L²(_Ω))and u(·,t)∈W for all t∈_R0⁺.

Proof. Fixu0∈ X0(_Ω), with J(u0) =dandI(u0)≥0.

Let λ_m = 1−1/mand u_0m = λ_mu₀, m ∈ N. Consider the problem (1.2), (1.4), with the initial condition

u(·, 0) =u0m. (4.6)

FromI(_u0)≥0 we have

I(u0m) =λ²_mC(N,s)ku₀k²_X

0(_Ω)−λ_m^p+1ku₀k^p+1

L^p+1(_Ω)

= λ²_mI(u₀) + (λ²_m−λ_m^p+1)ku₀k^p+1

L^p+1(_Ω) >0, J(u_0m) = ^C(N,s)

2 ku_0mk²_X

0(_Ω)− ¹

p+1ku_0mk^p+1

L^p+1(_Ω)

= 1

2− ¹ p+1

C(N,s)ku0mk²_X

0(_Ω)+ ¹

p+1I(u0m)>0, J(u_0m) = J(λ_mu₀)< J(u₀) =d.

By Theorem4.1for eachm∈ N, problem (1.2), (1.4), under (4.6), admits a global weak solution u_m ∈ L^∞(0,∞;X₀(_Ω)), withu_mt ∈L²(0,∞;L²(_Ω))andu_m(·,t)∈W for all t∈_R⁺₀.

The fact thatu_m(·_,t)∈W for allt ∈_R⁺₀ implies that for allt∈_R⁺₀ J(u_m(·,t))≥

1 2 − ¹

p+1

C(N,s)ku_m(·,t)k²_X

0(_Ω)+ ¹

p+1I(u_m(·,t))

≥ ^p−1

2(p+1)^C(N,s)ku_m(·,t)k²_X

0(_Ω)

and, furthermore, for allt ∈_R⁺₀,

kum(·,t)k_X0(Ω)≤

2d p+₁ (p−1)C(N,s)

1/2

, ku_m(·,t)k_Lp+1(_Ω) ≤ ^C(N,s)

S_p+1

ku_m(·,t)k_X0(Ω) ≤ ¹ S_p+1

2d(p+1)C(N,s) p−₁

1/2

and

Z _t

0

ku_mτ(·,τ)k²_L2(Ω)dτ<2d.

Then by the weak^∗ compactness of bounded sets in L^∞(0,∞;L^p+1(_Ω))and L^∞(0,∞;X₀(_Ω)) and by the weak compactness of bounded sets in L²(0,∞;L²(_Ω)), we conclude that problem (1.2)–(1.4) admits a global weak solutionu ∈ L^∞(0,∞;X0(_Ω)), withut ∈ L²(0,∞;L²(_Ω))and u(·,t)∈W for allt∈_R⁺₀. This completes the proof.

5 Vacuum isolating and blow up of strong solutions

Let T be the existence time of any solution u of the problem (1.2)–(1.4). In the following, similar to (4.5), we assume that

Z _t

0

kuτ(·,τ)k_L2(_Ω)dτ+_J(_u(·,t))≤ J(_u0) _(5.1) for all t∈[0,T).

Theorem 5.1. Let u₀ ∈ X₀(_Ω). Fix e ∈ (0,d)and let δ₁ < δ₂ be the two solutions of the equation d(δ) =e.Then for any strong solution u of the problem(1.2)–(1.4), with initial energy J(u0) =e,

(i)u(·,t)belongs to W_δ for allδ∈ (δ₁,δ₂)and t∈[0,T), provided that I(u₀)>0.

(ii)u(·,t)belongs to V_δ for allδ ∈(δ₁,δ₂)and t∈ [0,T), provided that I(u₀)<0.

Proof. Fixu₀ ∈X₀(_Ω),e∈(0,d), letδ₁ <δ₂be the two solutions of the equationd(δ) =e, and fix a strong solutionuof the problem (1.2)–(1.4), with initial energy J(u₀) =e.

(i) Corollary 4.2 and the proof of Theorem 4.1 give that u₀ ∈ W_δ for any δ ∈ (δ₁,δ₂). Assume by contradiction that there exists some t₀ ∈ (0,T) such that u(·,t₀) ∈ ∂W_δ for some δ∈(δ₁,δ₂), i.e. either J(u(t₀)) =d(δ)_or J_δ(u(t₀)) =_{0 and}ku(·_,t₀)k_X

0(_Ω)6=0. By (5.1), J(u(t))≤ J(u₀)<d(δ) fort∈(0,T), (5.2) so that J(u(·,t₀)) = d(δ)is impossible. While, if J_δ(u(t₀)) = 0 and ku(·,t₀)k_X0(Ω) 6= 0 occur, then Theorem3.4gives that J(u(t₀))≥ d(δ), which contradicts (5.2). This completes the proof of case (i).

(ii)The assumption I(u0)<0 implies that J_δ(u₀)<

δ 2− ¹

p+1

kuk^p+1

L^p+1(_Ω)<0

for all δ ∈ (δ₁,δ₀) _{by Lemma 3.3}(ii) _and (iv). Since the sign of J_δ(u₀) does not change for δ ∈ (δ₁,δ₂) by Lemma3.9, then J_δ(u₀) < 0 for allδ ∈ (δ₁,δ₂). This fact and J(u₀) < d(δ)for allδ∈ (δ₁,δ₂)giveu₀∈V_δ forδ∈(δ₁,δ₂).

Assume now by contradiction that there exist some t₀ ∈ (_0,T) _andδ ∈ (δ₁,δ₂)_{such that} u(·,t₀)∈∂V_δ, i.e. either J(u(·,t₀)) = d(δ)or J_δ(u(·,t₀)) = 0. From (5.2), the case J(u(·,t₀)) = d(δ) is impossible. Suppose next that t₀ is the smallest t such that J_δ(u(·,t₀)) = 0, then J_δ(u(·_,t))<_{0 for all} t∈[_0,t₀). From (5.2) and Lemma3.1we have

ku(·,t)k_X0(Ω)>





(p+1)δS^p_p⁺₊¹₁ 2C(N,s)^p





1/(p−1)

=κ_δ for all t∈[0,t₀)

and furthermore ku(·,t₀)k_X0(Ω) ≥κ_δ. Thus Theorem3.4implies that J(u(·,t₀))≥ d(δ), which contradicts (5.2) and completes the proof of(ii)_.

From Theorem4.3 and Lemma3.9we obtain the following theorem.

Theorem 5.2. Let u₀, e and δ_i, i= 1, 2, be as stated in Theorem 5.1. Then for any strong solution u of problem(1.2)–(1.4), with initial energy J(u₀)satisfying0< J(u₀)≤ e,

(i)u(·,t)belongs to W_δfor allδ ∈(δ₁,δ₂)and t∈[0,T), provided that I(u₀)>0;

(ii)u(·,t)belongs to V_δ for allδ∈(δ₁,δ₂)and t∈[0,T), provided that I(u₀)<0.

Corollary 5.3. Let u₀, e andδ_i, i=1, 2, be as stated in Theorem5.1. Then for any strong solution u of problem(1.2)–(1.4), with initial energy J(u₀)satisfying0< J(u₀)≤ e,

(i)u(·,t)belongs to W_δ1 for all t∈ [0,T), provided that I(u₀)>0;

(ii)u(·,t)belongs to V_δ2 for all t∈[0,T), provided that I(u₀)<0.

Proof. From (5.1)

J(u(t))≤ d(δ₁) (ord(δ₂)) for allt∈ [0,T).

Fix t ∈ [0,T). Letting δ → δ₁ (or δ → δ₂) in J_δ(u(t)) > 0 (or J_δ(u(t)) < 0) for the case (i) (or case (ii)), we have J_δ1(u(t)) ≥ 0 (or J_δ2(u(t)) ≤ 0) for all t ∈ [0,T). This completes the proof.

From Corollary5.3and Lemma3.1we get the following theorem.

Theorem 5.4. Let u₀, e and δ_i, i= 1, 2, be as stated in Theorem 5.1. Then for any strong solution u of problem(1.2)–(1.4), with initial energy J(u0)satisfying0< J(u0)≤ e,

(i)u(·,t)lies inside the ball B_δ1 for all t∈[0,T), provided u₀∈ B_δ0; (ii)u(·,t)lies outside the ball B_δ2 for all t∈[0,T), provided u₀ ∈B_δ^c

0; where B_δ is the open ball of X₀(_Ω), with special radius, defined in Section3.

The result of Theorem5.4 shows that for any givene ∈(0,d), there exists a corresponding vacuum region

Ve=







w∈ X0(_Ω):





(p+1)S_p^p+₊¹₁ 2C(N,s)^{p δ1}





1/(p−1)

<kwk_X0(Ω)<





(p+1)S^p_p⁺₊¹₁ 2C(N,s)^{p δ2}





1/(p−1)





for the set of strong solutions of the problem (1.2)–(1.4), with initial energy J(u₀) _satisfying 0< J(u₀)≤ e, i.e. there are no strong solutionsu such thatu(·,t)∈ Ve for all t ∈ [0,T). The vacuum regionV_e becomes bigger whene decreases to 0.

Let us next consider the limit casee =_0.

Theorem 5.5. Let u₀ ∈ X₀(_Ω). Then any nontrivial strong solution u of (1.2)–(1.4), with initial energy J(u₀)≤0, is such that u(·,t)lies outside the ball B₁ for all t∈[0,T), where

B₁ =







u∈X₀(_Ω) _: kuk_X

0(_Ω) <





(p+₁)S^p_p⁺₊¹₁ 2C(N,s)^p





1/(p−1)



 ,

introduced in Section3.

Proof. Fix a nontrivial strong solutionuof (1.2)–(1.4), with initial energy J(u₀)≤0.

Inequality (5.1) gives J(u(·,t))≤0 fort ∈[_0,T)_{. Thus by} (p+1)C(N,s)

2 ku(·_,t)k²_X

0(_Ω)≤ ku(·_,t)k^p+1

L^p+1(_Ω) ≤

C(N,s) S_p+1

ku(·_,t)k_X0 p+1

,