Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations

Theorem 1.1

Let conditions (a) − (d) hold and u₀ ∈ X₀. If J(u₀) < d and I(u₀) > 0, then problem (1.1) admits a global weak solution u ∈ L^∞(0, ∞; X₀) with u_t ∈ L²(0, ∞; L²(Ω)) and u(t) ∈ W for 0 ⩽ t < ∞. The weak solution is unique if it is bounded. Moreover, there exists δ₁ ∈ (0, 1) such that the following estimates hold:

1. if p < 2, then

   ∥u∥L2(Ω)2⩽∥u0∥L2(Ω)2−p−(2−p)(1−δ1)C∗−pt+22−p,∀t⩾0,

   where (x)₊ := max{x, 0}, which implies u vanishes in finite time;

2. if p = 2, then

   ∥u(t)∥L2(Ω)2⩽∥u0∥L2(Ω)2e−2(1−δ1)C∗−2t,∀t⩾0;

3. if p > 2, then

   ∥u(t)∥L2(Ω)2⩽(1(p−2)(1−δ1)C∗−pt+∥u0∥L2(Ω)2−p)2p−2,∀t⩾0.

Theorem 1.2

Let conditions (a) − (d) hold and u₀ ∈ X₀. If J(u₀) < d and I(u₀) < 0, then there exists a finite time T^* such that the weak solution of problem (1.1) blows up in the sense of

Next, we are concerned with the case of the critical initial energy (J(u₀) = d).

Theorem 1.3

Let conditions (a) − (d) hold and u₀ ∈ X₀. If J(u₀) = d and I(u₀) ⩾ 0, then problem (1.1) admits a global weak solution u ∈ L^∞(0, ∞; X₀) with u_t ∈ L²(0, ∞; L²(Ω)) and u(t) ∈ W ∪ ∂ W for 0 ⩽ t < ∞. The weak solution is unique if it is bounded. If I(u) > 0 for 0 < t < ∞, then there exists t₀ > 0 and δ₁ ∈ (0, 1) such that the following estimates hold:

1. if p < 2, then

   ∥u∥L2(Ω)2⩽∥u(t0)∥L2(Ω)2−p−(2−p)(1−δ1)C∗−pt+22−p,∀t⩾t0,

   which implies u vanishes in finite time;

2. if p = 2, then

   ∥u(t)∥L2(Ω)2⩽∥u0∥L2(Ω)2e−2(1−δ1)C∗−2t,∀t⩾0;

3. if p > 2, then

   ∥u(t)∥L2(Ω)2⩽(1(p−2)(1−δ1)C∗−pt+∥u0∥L2(Ω)2−p)2p−2,∀t⩾0.

   If there exists t^* > 0 such that I(u) > 0 for 0 < t < t^* and I(u(t^*)) = 0, then there exists a weak solution u(x, t) which vanishes in finite time t^*.

Theorem 1.4

Let conditions (a) − (d) hold, u₀ ∈ X₀. If J(u₀) = d and I(u₀) < 0, then there exists a finite time T^* such that the weak solution of problem (1.1) blows up in the sense of (1.6).

To state the main results regarding the high initial energy (J(u₀) > d), we introduce the following sets:

Theorem 1.5

Let conditions (a) − (d) hold and u₀ ∈ X₀. If J(u₀) > d, then the following statements hold:

1. If u₀ ∈ 𝓝₊ and ∥u₀∥_L²(Ω) ⩽ λ_J(u0), then u₀ ∈ 𝓖₀.

2. If u₀ ∈ 𝓝₋ and ∥u₀∥_L²(Ω) ⩾ Λ_J(u0)λ_J(u0), then u₀ ∈ 𝓑.

The following theorem indicates that there exists blow-up solutions to problem (1.1) for any high initial energy.

Theorem 1.6

Let conditions (a) − (d) hold and u₀ ∈ X₀. For any M > d, then there exists u_M ∈ 𝓝₋ such that J(u_M) = M and u_M ∈ 𝓑.

This paper is organized as follows. Some necessary definitions and properties of the functional spaces used in this paper are given in Section 2. In Section 3, we define some notations and give some results about potential well theory. Section 4 and Section 5 are devoted to the cases J(u₀) < d and J(u₀) = d, respectively. In Section 6, we consider the case J(u₀) > d.

Lemma 2.1

[5, Theorem 2.4] For any s > 0, C0∞ (Ω) of smooth functions with compact support is a dense subspace of X₀.

Lemma 2.2

[5, Theorem 6.5] Let {u_k} be a bounded sequence in X₀. Then there exists u ∈ L^β(ℝ^N) such that up to a subsequence, u_k → u strongly in L^β(Ω) as k → ∞ for any β ∈ [1, ps∗ ).

At the end of this section, we list some properties about the nonlinear term in the problem (1.1). These lemma and corollary were obtained in [23].

Lemma 2.3

If f satisfy conditions (a)–(c). Then

1. ∣F(u)∣ ⩽ A∣u∣^r for some A > 0 and all u ∈ ℝ.

2. F(u) ⩾ B∣u∣^q+1 for some B > 0 and ∣u∣ ⩾ 1.

3. u(uf′(u) − f(u)) ⩾ 0, the equality holds only for u = 0.

Corollary 2.1

Under the conditions of Lemma 2.3, we have

1. ∣uf(u)∣ ⩽ rA∣u∣^r, ∣f(u)∣ ⩽ rA∣u∣^r−1for all u ∈ ℝ.

2. uf(u) ⩾ (q + 1)B∣u∣^q+1 for ∣u∣ ⩾ 1.

Lemma 3.1

The depth of the potential well W is positive.

Proof

Fix u ∈ 𝓝, from Corollary 2.1(1) and the embedding X₀ into L^r(Ω), we get

which implies ∥u∥X0⩾1ArC∗r1r−p. By the condition (c), we have

Therefore d⩾q+1−pp(q+1)1ArC∗rpr−p>0. □

Lemma 3.2

If I(u) > 0, then J(u) > 0.

Proof

Assuming that J(u) ⩽ 0, by a direct computation, we get

According to condition (c), we get

It is obvious from p < q + 1 that there does not exist ∥u∥X0p satisfying (3.1). Therefore, this contradiction implies the assumption J(u) ⩽ 0 does not hold.□

For any δ > 0, define the modified functional and Nehari manifold as follows:

The corresponding modified potential well and its corresponding set are defined respectively by

where d(δ)=infu∈NδJ(u) is the potential depth of W_δ.

Define

and the (open) sublevels of J

Obviously, by the definition of J(u), 𝓝, J^ς and d, we get

For ς > d, define

Obviously, λ_ς is non-increasing and Λ_ς is non-decreasing.

Lemma 3.3

For any u ∈ X₀ with ∥u∥_X0 ≠ 0, we have

1. limλ→0+J(λu)=0,limλ→+∞ J(λu) = −∞.

2. There exists a unique λ^* = λ^*(u) > 0 such that dJ(λu)dλ|λ=λ∗=0. J(λu) is increasing on 0 < λ ⩽ λ^*, decreaing on λ^* ⩽ λ < ∞ and take its maximum at λ = λ^*.

3. I(λu) > 0 on 0 < λ ⩽ λ^*, I(λu) < 0 on λ^* ⩽ λ < ∞ and I(λ^*u) = 0.

Proof

1. It follows from the definition of J(u), condition (c) and Corollary 2.1(1) that

   J(λu)=λpp∥u∥X0p−∫ΩF(λu)dx⩾λpp∥u∥X0p−1q+1∫Ωλuf(λu)dx⩾λpp∥u∥X0p−Arλrq+1∥u∥Lr(Ω)r.

   Obviously, assertion (1) follows from r > p.

2. Clearly, J(λu) = 0 at λ = 0, and J(λu) > 0 for any λ∈0,q+1Arp∥u∥X0p∥u∥Lr(Ω)r1r−p. By a direct computation, we get

   dJ(λu)dλ=λp−1∥u∥X0p−∫Ωf(λu)udx.

   Therefore, there exists a λ^* > 0 such that

   dJ(λu)dλ|λ=λ∗=(λ∗)p−1∥u∥X0p−∫Ωf(λ∗u)udx=0.

   Moreover, combining with condition (d) and ∥u∥_X0 ≠ 0, we get

   d2J(λu)dλ2|λ=λ∗=(p−1)(λ∗)p−2∥u∥X0p−∫Ωf′(λ∗u)u2dx=1(λ∗)2(p−1)(λ∗)p∥u∥X0p−∫Ωf′(λ∗u)(λ∗u)2dx=1(λ∗)2(p−1)∫Ωf(λ∗u)λ∗udx−∫Ωf′(λ∗u)(λ∗u)2dx<0. (3.2)

   Next, we prove that λ^* = λ^*(u) is uniquely determined. Assume that there exists two roots λ₁, λ₂ of dJ(λu)dλ=0. That is

   dJ(λu)dλ|λ=λ1=0,d2J(λu)dλ2|λ=λ1<0;dJ(λu)dλ|λ=λ2=0,d2J(λu)dλ2|λ=λ2<0.

   If there exists a λ₃ with λ₁ < λ₃ < λ₂ such that J(λ₃u) is the minimum of J(λu) in [λ₁, λ₂], then

   dJ(λu)dλ|λ=λ3=0,d2J(λu)dλ2|λ=λ3⩾0,

   which is a contradiction. Therefore, we conclude that J(λu) gets its minimum in [λ₁, λ₂] at either λ₁ or λ₂. If the first case happens, then there exists a σ > 0 such that J(λu) at [λ₁, λ₁+σ] is not only maximum, but also minimum. That is, J(λu) is a constant. Therefore, we obtain

   dJ(λu)dλ|λ=λ1=0,d2J(λu)dλ2|λ=λ1=0,

   which contradicts (3.2). We conclude that J(λ₁u) is not the minimum. By the same argument, J(λ₂u) also is not the minimum. So we complete the proof of (2).

   Next, we will prove (3). By the definition of I(u), we get

   dJ(λu)dλ=λp−1∥u∥X0p−∫Ωf(λu)udx=I(λu)λ.

   It is obvious that (3) holds from (2).□

Lemma 3.4

For any u ∈ X₀ and y(δ)=(δArC∗r)1r−p, we have

1. If 0 ⩽ ∥u∥_X0 ⩽ y(δ), then I_δ(u) ⩾ 0.

2. If I_δ(u) < 0, then ∥u∥_X0 > y(δ).

3. If I_δ(u) = 0, then ∥u∥_X0 = 0 or ∥u∥_X0 ⩾ y(δ).

Proof

1. We get

   ∫Ωuf(u)dx⩽rA∥u∥Lr(Ω)r⩽ArC∗r∥u∥X0r⩽ArC∗r(y(δ))r−p∥u∥X0p

   from Corollary 2.1(1) and the embedding from X₀ into L^r(Ω). So it implies I_δ(u) ⩾ 0.

2. It easily follows from (1).

3. If ∥u∥_X0 = 0, then I_δ(u) = 0. If I_δ(u) = 0, ∥u∥_X0 ≠ 0, then from Corollary 2.1(1) and the embedding from X₀ into L^r(Ω), we have

   δ∥u∥X0p=∫Ωuf(u)dx⩽rA∥u∥Lr(Ω)r⩽ArC∗r∥u∥X0r−p∥u∥X0p,

   which implies ∥u∥_X0 ⩾ y(δ).□

Lemma 3.5

The function d(δ) satisfies the following properties:

1. limδ→0+d(δ)=0,limδ→+∞ d(δ) = −∞.

2. d(δ) is strictly increasing on 0 < δ ⩽ 1, strictly decreasing on δ ⩾ 1 and take its maximum d = d(1) at δ = 1.

Proof

(1) For any u ∈ X₀ with ∥u∥_X0 ≠ 0 and δ > 0, there exists a unique λ = λ(δ) such that I_δ(λu) = 0 from Lemma 3.3(3). Furthermore, we get

It is obvious that

where

We have the following two claims:

1. η(λ) is strictly increasing on (0, +∞).

   Indeed, under the condition (d), we get

   η′(λ)=1λp+1∫Ω(λu)2f′(λu)dx−∫Ω(p−1)λuf(λu)dx>0,

   which yields Claim 1.

2. limλ→0+η(λ)=0,limλ→+∞ η(λ) = +∞.

   By Corollary 2.1(1), we get

   η(λ)=1λp−1∫Ωuf(λu)dx⩽rAλr−p∥u∥Lr(Ω)r.

   Therefore, limλ→0+ η(λ) = 0. For sufficiently large λ, we obtain

   η(λ)=1λp−2∫Ωλuf(λu)dx⩾(q+1)Bλp−2∥λu∥Lq+1(Ω)q+1=(q+1)Bλq−p+3∥u∥Lq+1(Ω)q+1

   by Corollary 2.1(2). It is obvious that limλ→+∞ η(λ) = +∞. This completes the proof of Claim 2.

   Therefore, it follows from Claim 1, Claim 2 and (3.2) that λ=η−1(δ∥u∥X0p), that is, inverse function of η, then λ is strictly increasing on (0, +∞) and

   limδ→0+λ(δ)=0,limδ→+∞λ(δ)=+∞,

   Since λ u ∈ 𝓝_δ, d(δ) ⩽ J(λu), combining with Lemma 3.3(1), we get

   0⩽limδ→0+d(δ)⩽limδ→0+J(λu)=limλ→0+J(λu)=0,

   and

   limδ→+∞d(δ)⩽limδ→+∞J(λu)=limλ→+∞J(λu)=−∞.

   This completes the proof of (1).

   (2) Clearly, we only need to prove that any 0 < δ′ < δ″ < 1 or δ′ > δ″ > 1 and any u ∈ 𝓝_δ″, there exists a v ∈ 𝓝_δ′ and a constant ϵ(δ′,δ″) such that J(u) − J(v) ⩾ ϵ(δ′,δ″). Actually, for u ∈ 𝓝_δ″, we get I_δ″(u) = 0(This implies λ(δ″) = 1 from (3.2)) and ∥u∥_X0 ⩾ y(δ″) by Lemma 3.4(3). By the definition of λ in (3.3), then I_δ(λu) = 0. Therefore, choosing v = λ(δ′)u, then we have v ∈ 𝓝_δ′. Let g(λ) = J(λ(δ)u), then

   dg(λ)dλ=λp−1∥u∥X0p−∫Ωf(λu)udx=1λ(1−δ)∥λu∥X0p+Iδ(λu)=λp−1(1−δ)∥u∥X0p.

   If 0 < δ′ < δ″ < 1, since λ is strictly increasing (Claim 2) and λ(δ″) = 1, then

   J(u)−J(v)=g(1)−g(λ(δ′))=∫λ(δ′)1dg(λu)dλdλ=∫λ(δ′)1λp−1(1−δ)∥u∥X0pdλ⩾1p(1−δ″)(y(δ″))p(λ(δ′))p(1−λ(δ′))=ϵ(δ′,δ″)>0.

   if δ′ > δ″ > 1, then

   J(u)−J(v)=∫λ(δ′)1dg(λu)dλdλ=−∫1λ(δ′)λp−1(1−δ)∥u∥X0pdλ⩾1p(δ″−1)(y(δ″))p(λ(δ″))p(λ(δ′)−1)=ϵ(δ′,δ″)>0.

   Since d(δ) is continuous, it is obvious that it take its maximum d = d(1) at δ = 1.□

Lemma 3.6

Assume u ∈ X₀, 0 < J(u) < d and δ₁ < 1 < δ₂, δ₁, δ₂ satisfy the equation d(δ) = J(u). Then the sign of I_δ(u) does not change for δ₁ < δ < δ₂.

Proof

Clearly, J(u) > 0 implies ∥u∥_X0 ≠ 0. If the sign of I_δ(u) changes for δ₁ < δ < δ₂, then there exists a δ̄ ∈ (δ₁, δ₂) such that I_δ̄(u) = 0. Thus by the definition of d(δ) we get J(u) ⩾ d(δ̄), which contradicts J(u) = d(δ₁) = d(δ₂) < d(δ̄) by Lemma 3.5(2).□

For φ ∈ W^s,p(ℝ^N), due to the symmetry of the kernel, we have the following equality:

which leads to the following definition of weak solutions by using u(x) = 0 in 𝓒Ω.

Definition 3.1

A function u = u(x, t) ∈ L^∞(0, T; X₀) with u_t ∈ L²(0, T; L²(Ω)) is called a weak solution to problem (1.1) if u(x, 0) = u₀(x) and the following equality holds

here

for all φ ∈ X₀. Moreover,

Lemma 3.7

Assume that u is a weak solution of problem (1.1) with 0 < J(u₀) < d and δ₁ < 1 < δ₂, δ₁, δ₂ satisfy the equation d(δ) = J(u₀).

1. If I(u₀) > 0, then u(x, t) ∈ W_δ for δ₁ < δ < δ₂ and 0 < t < T.

2. If I(u₀) < 0, then u(x, t) ∈ V_δ for δ₁ < δ < δ₂ and 0 < t < T.

Proof

(1) For 0 < J(u₀) = d(δ₁) = d(δ₂) < d and I(u₀) > 0, from Lemma 3.6, we get u₀ ∈ W_δ for δ₁ < δ < δ₂. Next we prove u(x, t) ∈ W_δ for δ₁ < δ < δ₂ and 0 < t < T. Otherwise, there exists a t₀ ∈ (0, T) and a δ₀ ∈ (δ₁, δ₂) such that u(t₀) ∈ ∂ W_δ0, then

Clearly, J(u(t₀)) < d(δ₀) from (3.5). Thus I_δ0(u(t₀)) = 0, ∥u(t₀)∥_X0 ≠ 0, the definition of d(δ₀) implies J(u(t₀)) ⩾ d(δ₀), which contradicts (3.5).

(2) Similarly, u₀ ∈ V_δ for δ₁ < δ < δ₂. Next we show u(x, t) ∈ V_δ for δ₁ < δ < δ₂ and 0 < t < T. If not, there exists a t₀ ∈ (0, T) and a δ₀ ∈ (δ₁, δ₂) such that u(t₀) ∈ ∂ V_δ0, then

Clearly, I_δ0(u(t₀)) = 0. Assuming t₀ is the first time such that I_δ0(u(t₀)) = 0, the I_δ0(u(t)) < 0 for 0 ⩽ t < t₀. By Lemma 3.4(2), we get ∥u∥_X0 > y(δ) for 0 ⩽ t < t₀. Lemma 3.4(3) implies ∥u(t₀)∥_X0 ⩾ y(δ₀), combining with I_δ0(u(t₀)) = 0, we get u(t₀) ∈ 𝓝_{δ₀}. By the definition of d(δ₀), we get J(u(t₀)) ⩾ d(δ₀), which contradicts (3.5).□

Lemma 3.8

If conditions (a) − (c) hold, then

1. 0 is away from both 𝓝 and 𝓝₋, that is, dist(0, 𝓝) > 0 and dist(0, 𝓝₋) > 0.

2. For any ς > 0, the set J^ς ∩ 𝓝₊ is bounded in X₀.

Proof

(1) For u ∈ 𝓝, by the definition of d, Lemma 2.3(1) and the embedding from X₀ into L^r(Ω), we get

which implies that there exists a constant ρ > 0 such that dist(0, 𝓝) = infu∈N ∥u∥_X0 ⩾ ρ.

For u ∈ 𝓝₋, we get ∥u∥_X0 ≠ 0. Then the embedding from X₀ into L^r(Ω) implies that

which implies ∥u∥X0>(1ArC∗r)1r−p. Furthermore, dist(0, 𝓝₋) = infu∈N− ∥u∥_X0 > 0.

(2) For any u ∈ J^ς ∩ 𝓝₊, then J(u) < ς and I(u) > 0. Therefore, from condition (c), we get

which yields ∥u∥_X0 < p(q+1)ςq+1−p1p. Therefore, the set J^ς ∩ 𝓝₊ is bounded in X₀.□

Proof

Proof of Theorem 1.1. The proof is divided into three steps.

1. Noting that ps∗ > 2, we have X₀ ⊂ L². Choose sequence {e_i}_i∈ℕ ∈ C0∞ (Ω), which is an orthogonal basis of L²(Ω). Consider the following Galerkin approximations:

   un(x,t)=∑i=1ncin(t)ei(x),

   where functions ckn (t) : [0, T] → ℝ satisfy the following system of ordinary differential equations:

   (dundt,ej)L2(Ω)+(un,ej)X0=(f(un),ej)L2(Ω),j=1,2,⋯,nun(x,0)=∑j=1n(u(x,0),ej)L2(Ω)ej, (4.1)

   with

   (un,ej)X0=12∬Q|un(y,t)−un(x,t)|p−2(un(y,t)−un(x,t))|x−y|N+ps(ej(y)−ej(x))dydx,un(x,0)→u0inX0,as n→∞.

   It follows from (4.1) that

   dcjn(t)dt=−12∬Q|un(y,t)−un(x,t)|p−2(un(y,t)−un(x,t))|x−y|N+ps(ej(y)−ej(x))dydx+(f(un),ej)L2(Ω).

   Define

   Fjn(cn(t))=−12∬Q|∑i=1ncin(t)ei(y)−∑i=1ncin(t)ei(x)|p−2(∑i=1ncin(t)ei(y)−∑i=1ncin(t)ei(x))×(ej(y)−ej(x))1|x−y|N+psdydx+(f(∑i=1ncin(t)ei(x)),ej)L2(Ω),

   and set

   cn(⋅)=(cjn(⋅))j=1n,Fn(⋅)=(Fjn(⋅))j=1n,c0n=((u0,ej)L2(Ω))j=1n.

   Then (4.1) can be rewritten as

   dcn(t)dt=Fn(cn(t)),cn(0)=c0n. (4.2)

   Next, we aim to use the Peano’s theorem to prove the existence of (4.2). Multiplying the first equality of (4.2) by cⁿ(t), the we get

   12d|cn(t)|2dt=−12∬Q|∑i=1ncin(t)ei(y)−∑i=1ncin(t)ei(x)|p−2(∑i=1ncin(t)ei(y)−∑i=1ncin(t)ei(x))×(∑j=1ncjn(t)ej(y)−∑j=1ncjn(t)ej(x))1|x−y|N+psdydx+(f(∑i=1ncin(t)ei(x)),∑j=1ncjn(t)ej(x))L2(Ω).

   Clearly, the first term on the right side of the above equation is less than 0, then using Corollary 2.1(1), we get

   12d|cn(t)|2dt⩽(f(∑i=1ncin(t)ei(x)),∑j=1ncjn(t)ej(x))L2(Ω)⩽Ar∥∑i=1ncin(t)ei(x)∥Lr(Ω)r⩽Ar∑i=1n|cin(t)|r∑i=1n∥ei(x)∥Lr(Ω)r⩽Ar(C(n))r∑i=1n|cin(t)|r∑i=1n∥ei(x)∥X0r⩽Ar(C(n))r|cn(t)|r,

   were C(n) > 0 is a constant depended on n. Solving the ordinary differential inequality, we get

   |cn(t)|⩽1(|c0n|1−r/2+(2−r)Ar(C(n))rt)2r−2,t∈[0,T~),

   here, T~=|c0n|1−r/2(2−r)Ar(C(n))r. There exists a sufficiently small ε > 0 such that ∣cⁿ(t)∣ ⩽ C(T̃−ε) (C(T̃−ε) > 0 is a constant). Furthermore, F(cⁿ(t)) with t ∈ [0, T̃−ε] is bounded(denote by G₀ the boundedness). Denote

   T0=0,E:={(t,cn(t))∈R×Rn||t−T0|⩽T~,|cn(t)−c0n|⩽C(T~−ε)},∀t∈[0,T~).

   Peano’s theorem implies that there exists a solutions cⁿ(t) of (4.2) on [0,T₁], where

   T1=minT~−ε,c0nG0.

   For t ∈ [0, T₁], the following (4.6) still holds, then we have

   |cjn(t)|2⩽∑j=1n|cjn(t)|2=∑j=1n|cjn(t)|2∫Ω|ej|2dx=∥un∥L2(Ω)2⩽C∗2∥un∥X02<C∗2(dp(q+1)q+1−p)2p.

   Furthermore, F(cⁿ(t)) is bounded. Denote by T₁ the new initial point, Peano’s theorem implies that there exists a global solution cⁿ(t) of the ordinary differential equation (4.2) by repeating the similar argument.

   Multiplying (4.1) by dcjn(t)dt, and summing for j from 0 to n, then integrating with respect to t from 0 to t, we get

   ∫0t∥uτn∥L2(Ω)2dτ+J(un)=J(un(0)). (4.3)

   It follows from uⁿ(x, 0) → u₀(x) in X₀ that

   J(un(x,0))→J(u0(x))<dandI(un(x,0))→I(u0(x))>0,as n→∞.

   Therefore, for sufficiently large n, we get

   ∫0t∥uτn∥L2(Ω)2dτ+J(un)=J(un(0))<dandI(un(x,0))>0, (4.4)

   which implies that uⁿ(x, 0) ∈ W.

   Next, we prove uⁿ(x, t) ∈ W for sufficiently large n. Otherwise, there exists a t₀ ∈ (0, ∞) such that uⁿ(x,t₀) ∈ ∂ W, that is

   I(un(x,t0))=0,∥u∥X0≠0orJ(un(x,t0))=d.

   Clearly, J(uⁿ(x,t₀)) ≠ d from (4.4). If I(uⁿ(x, t₀)) = 0, ∥u∥_X0 ≠ 0, then J(uⁿ(x, t₀)) ⩾ d by the definition of d, which contradicts (4.4). Therefore, uⁿ(x, t) ∈ W and I(uⁿ) > 0. Combining with condition (c), we have

   J(un)⩾1p∥un∥X0p−1q+1∫Ωunf(un)dx=q+1−pp(q+1)∥un∥X0p+1q+1I(un).

   From (4.4), we get

   ∫0t∥uτn∥L2(Ω)2dτ+q+1−pp(q+1)∥un∥X0p+1q+1I(un)<d, (4.5)

   which implies

   ∥un∥X0p<dp(q+1)q+1−p, (4.6)

   ∫0t∥uτn∥L2(Ω)2dτ<d. (4.7)

   Furthermore, Corollary 2.1(1) and the embedding from X₀ into L^r(Ω) imply

   ∥f(un)∥Lrr−1(Ω)⩽Ar∥un∥Lr(Ω)r−1⩽Ar(C∗)r−1∥un∥X0r−1<Ar(C∗)r−1(dp(q+1)q+1−p)r−1p. (4.8)

   Therefore, there exists a u and a subsequence of {uⁿ}_n∈ℕ (still denoted by {uⁿ}_n∈ℕ) such that as n → ∞,

   un⇀∗uinL∞(0,∞;X0);tn⇀utinL2(0,∞;L2(Ω));f(un)⇀∗ξinL∞(0,∞;Lrr−1(Ω)).

   Since the embedding from X₀ into L^r(Ω) is compact and the embedding from L^r(Ω) into L²(Ω) is continuous, from Simon’s theorem [29] we get

   un→uinC([0,∞];Lr(Ω)),as n→∞. (4.9)

   Therefore, ξ = f(u).

   Next, we show that the function u is a weak solution of the problem (1.1). Choose v ⊂ C¹([0, ∞]; C0∞ (Ω)) with the following form

   v=∑j=1klj(t)ej,

   where l_j(t) ∈ C¹([0, ∞]) with j = 1, 2, ⋯, k(k ⩽ n). Multiplying the first equality of (4.1) by l_j(t) summing for j from 1 to n, integrating with respect to t from 0 to T, we get

   ∫0T(utn,v)L2(Ω)dt+∫0T(un,v)X0dt=∫0T(f(un),v)L2(Ω)dt.

   Letting n → ∞, combining with the theory of monotone operators as in [24], we get

   ∫0T(ut,v)L2(Ω)dt+∫0T(u,v)X0dt=∫0T(f(u),v)L2(Ω)dt. (4.10)

   Since C¹([0, ∞]; C0∞ (Ω)) is dense in L²(0, ∞; X₀), the identity in (4.10) holds for v ∈ L²(0, ∞; X₀). Moreover, by the arbitrariness of T > 0, we get

   (ut,φ)L2(Ω)+(u,φ)X0=(f(u),φ)L2(Ω),φ∈X0,a.e.t>0.

   By (4.9) and uⁿ(x, 0) → u₀(x) in X₀, then u(x, 0) = u₀(x). Assuming that u is sufficiently smooth such that u_t ∈ L²(0, ∞; X₀), taking v = u_t in (4.10), then (3.5) the holds. Since L²(0, ∞; X₀) is dense in L²(0, ∞; L²(Ω)), then (3.5) holds for weak solutions of problem (1.1).

2. Assuming both u, v are two bounded weak solutions for problem (1.1). Then by the definition of weak solution, for φ ∈ X₀, we get

   (ut,φ)L2(Ω)+(u,φ)X0=(f(u),φ)L2(Ω);(vt,φ)L2(Ω)+(v,φ)X0=(f(v),φ)L2(Ω).

   Subtracting the above two equalities, taking φ = u − v ∈ X₀, and then integrating for t from 0 to t, we get

   ∫0t∬Q|u(y,t)−u(x,t)|p−2(u(y,t)−u(x,t))−|v(y,t)−v(x,t)|p−2(v(y,t)−v(x,t)×[u(y,t)−u(x,t)−v(y,t)+v(x,t)]1|x−y|N+psdydxdt+∫0t∫Ωφtφdxdt=∫0t∫Ω(f(u)−f(v))φdxdt.

   Clearly, the fist term on the left side of the above equality is non-negative. By the continuity of f, boundedness of u, v and Cauchy-Schwarz inequality we get

   ∫0t∫Ωφtφdxdt⩽∫0t∫Ω(f(u)−f(v))φdxdt⩽C∫0t∫Ωφ2dxdt,

   here C > 0 depends on the bound of u, v. Furthermore,

   ∫Ωφ2dx⩽C∫0t∫Ωφ2dxdt

   by φ(x, 0) = 0. Gronwalľs inequality implies

   ∫Ωφ2dx=0.

   Thus φ = 0 a.e. in Ω × (0, ∞).

3. Taking φ = u in (3.4), we obtain

   12ddt∥u∥L2(Ω)2=−I(u). (4.11)

   From Lemma 3.2 and Lemma 3.7, we know that u(x, t) ∈ W_δ for δ₁ < δ < δ₂ and 0 < t < ∞ under the condition J(u₀) < d and I(u₀) > 0. Thus Lemma 3.6 implies I_δ1(u) ⩾ 0 for 0 < t < ∞. Therefore, the embedding from X₀ into L²(Ω) implies that

   12ddt∥u∥L2(Ω)2=−I(u)=(δ1−1)∥u∥X0p−Iδ1(u)⩽(δ1−1)C∗−p∥u∥L2(Ω)p.

   Thus for p < 2, we get

   ∥u∥L2(Ω)2⩽∥u0∥L2(Ω)2−p−(2−p)(1−δ1)C∗−pt+22−p,∀0⩽t<∞,

   where

   ∥u0∥L2(Ω)2−p−(2−p)(1−δ1)C∗−pt+=max∥u0∥L2(Ω)2−p−(2−p)(1−δ1)C∗−p,0.

   This means that the solution u vanishes at a finite time t∗=∥u0∥L2(Ω)2−p(2−p)(1−δ1)C∗−p.

   For p = 2, we get

   ∥u∥L2(Ω)2⩽∥u0∥L2(Ω)2e−2(1−δ1)C∗−2t,∀0⩽t<∞.

   For p > 2, we get

   ∥u∥L2(Ω)2⩽1(2−p)(δ1−1)C∗−pt+∥u0∥L2(Ω)2−p2p−2,∀0⩽t<∞.

   □

   Next, we are concerned with blow-up in finite time.