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


In this paper, we study the fractional p-Laplacian evolution equation with arbitrary initial energy,

$$\begin{array}{}
\displaystyle u_t(x,t) + (-{\it\Delta})_p^s u(x,t) = f(u(x,t)), \quad x\in {\it\Omega}, \,t \gt 0,
\end{array} $$
where 
$\begin{array}{}
(-{\it\Delta})_p^s
\end{array} $ is the fractional p-Laplacian with 
$\begin{array}{}
p \gt \max\{\frac{2N}{N+2s},1\}
\end{array} $ and s ∈ (0, 1). Specifically, by the modified potential well method, we obtain the global existence, uniqueness, and blow-up in finite time of the weak solution for the low, critical and high initial energy cases respectively.


Introduction
Let Ω ⊂ R N (N ) be a bounded domain with smooth boundary, T ∈ ( , ∞], p > max{ N N+ s , } and s ∈ ( , ). In this paper, we study the following fractional p-Laplacian evolution equation with the zero Dirichlet boundary value condition: Here, the function f is a given function satisfying the following conditions: (a) f ∈ C and f ( ) = f ( ) = .
(b) f (u) is monotone, convex for u > and concave for u < .
For instance, we could take f (u) = λ |u| r− u + λ |u| q− u with λ , λ . The fractional Laplacian as a generalization of the integer order Laplace operator has been studied in classical monographs such as [14,30] and so on. Recently, more attention has been focused on the study of the di erential equation involved with non-local fractional operators. These equations could be used in many elds, such as digit image processing [10], obstacle problem [28], phase transitions [2] and so on. We also refer the reader to the monographs [3,6,20] and the references therein for a fairly large description of rigorous mathematics and diverse applications. Usually, non-local generalizations of the Laplacian (both linear and nonlinear) are examined with smooth kernels and the fractional Laplacian is represented as an integral operator over the whole R N . In this work, we focus on non-local operators acting on bounded domains which correspond to the regional fractional Laplacian and can be interpreted as a nonlocal version of the Laplacian equipped with Dirichlet boundary conditions (see [24]). The non-local generalization of the p-Laplacian we are going to study is hence the nonlinear pendant to the fractional regional Laplacian mentioned above and appears, for instance, as a type of nonlinear di usion [31]. Here, we mention that though it is also possible to de ne the fractional Laplacian via Fourier transform, this approach is restricted to the case of p = .
In 2016, Vázquez [32] considered the existence, uniqueness and a number of quantitative properties of strong nonnegative solutions of a Dirichlet problem with fractional p-Laplacian evolution equation where < s < and p > . When equation (1.3) is coupled with the Neumann boundary condition and the Cauchy initial condition, the existence, uniqueness and asymptotic behavior of strong solutions are obtained by the semigroup methods in [19]. Very recently, authors in [1] established the existence and suitable regularity of the weak solution and the entropy solution with general data. They also gave some properties of the solution, such as extinction, non nite speed of propagation, according to the values of p. Some regularities are also established in [12].
When p = and f is nonlinear, Musso et al. [21] consider in nite time blow-up for positive solutions of the fractional heat equation with critical exponent where N > s and s ∈ ( , ). Later, the nite time blow-up problem for (1.4) with s < N < s was investigated in [4]. When p ≠ , Gal and Warm [8] studied the existence of the strong solution and the blow-up for the initial data with nite energy and a positive low bound. For the bounded initial data, the existence, uniqueness and global behavior of solutions are proved in [9]. Recently, Pan, Zhang and Cao [22] studied the following parabolic equation involving the fractional p-Laplacian: where [u]s,p is the Gagliardo p-seminorm of u, p < q < Np N−sp with < p < N s and λ < N N−sp . Under some appropriate assumptions, the authors obtained the existence of a global solution to the problem (1.5) by the Galerkin method and potential well theory under J(u ) = d or < J(u ) < d, where J(u ) is the initial energy. However, authors did not study the problem for J(u ) > d and blow-up in nite time for arbitrary initial energy. In this paper, we consider the global existence and blow-up in nite time for the problem (1.1) for the more general f (u) when the initial energy J(u ) is arbitrary.
Potential well method was rstly proposed by Sattinger [25] to study non-linear hyperbolic boundaryinitial value problem. Since then, many authors have studied the existence of solutions for evolution equations by potential well theory [17,18,23,34]. Especially, authors [17,18,34] improved the results of Sattinger's by introducing a family of potential wells not only obtained some new results on global existence and invariant sets of solutions, but also discovered the vacuum isolating of solutions. Recently, the properties of a class of fourth-order parabolic equation were obtained with J(u ) > d in [7,11,35].
In this paper, motivated by the works mentioned above, we consider the global existence, uniqueness and blow-up property of problem (1.1) with arbitrary initial energy J(u ). The main results regarding the low initial energy (J(u ) < d) are as following. The notations used below are de ned in Section 2 and 3. Theorem 1.1. Let conditions (a) − (d) hold and u ∈ X . If J(u ) < d and I(u ) > , then problem (1.1) admits a global weak solution u ∈ L ∞ ( , ∞; X ) with u t ∈ L ( , ∞; L (Ω)) and u(t) ∈ W for t < ∞. The weak solution is unique if it is bounded. Moreover, there exists δ ∈ ( , ) such that the following estimates hold: where (x)+ := max{x, }, which implies u vanishes in nite time; , then problem (1.1) admits a global weak solution u ∈ L ∞ ( , ∞; X ) with u t ∈ L ( , ∞; L (Ω)) and u(t) ∈ W ∪ ∂W for t < ∞. The weak solution is unique if it is bounded. If I(u) > for < t < ∞, then there exists t > and δ ∈ ( , ) such that the following estimates hold: which implies u vanishes in nite time; (iii) if p > , then If there exists t * > such that I(u) > for < t < t * and I(u(t * )) = , then there exists a weak solution u(x, t) which vanishes in nite time t * . Theorem 1.4. Let conditions (a) − (d) hold, u ∈ X . If J(u ) = d and I(u ) < , then there exists a nite 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(u ) > d), we introduce the following sets: B = {u ∈ X |the solution u of problem ( . ) blows up in nite time}; The following theorem indicates that there exists blow-up solutions to problem (1.1) for any high initial energy. This paper is organized as follows. Some necessary de nitions and properties of the functional spaces used in this paper are given in Section 2. In Section 3, we de ne some notations and give some results about potential well theory. Section 4 and Section 5 are devoted to the cases J(u ) < d and J(u ) = d, respectively. In Section 6, we consider the case J(u ) > d.

Functional Space and Preliminaries
In this section, let's recall some necessary de nitions and properties of the functional spaces introduced by R. Servadei and E. Valdinoci in [26,27]. Let < s < , < p < ∞ be real numbers and the fractional critical exponent p * s be de ned as p * s = Np N−sp if sp < N and p * s = ∞, otherwise. De ne the Banach space Let Ω an open set in R N and Q = ( The space X is a normed linear subspace of W s,p (R N ) and also endowed with the norm It is known that (X , · X ) is a uniformly convex re exive Banach space, and its norm is equivalent to We list two useful lemmas used in the sequel. For more properties of the space X (Ω), readers could refer to [5,26,27].
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].
, the equality holds only for u = .

Preliminaries of potential well
Throughout this paper, C * is the optimal embedding constant from X into L r (Ω) with r ∈ ( , p * s ). To state our results, we need to introduce some notations and de nitions of some functionals and sets.
For u ∈ X (Ω), set and de ne the Nehari manifold If conditions (a)-(c) hold, then functionals J and I are well-de ned and continuous on X (Ω), which can be proved by the similar argument for Lemma 3.1 and Lemma 3.2 in [33]. The potential well and its corresponding set are de ned respectively by is the depth of the potential well W.
The following lemmas follow the line of [18], but there exist some di erences because of the generality of f (u) and existence of (−∆) s p u. Hence, we will give the speci c process of proof. Proof. Fix u ∈ N, from Corollary 2.1(1) and the embedding X into L r (Ω), we get Proof. Assuming that J(u) , by a direct computation, we get According to condition (c), we get It is obvious from p < q+ that there does not exist u p X satisfying (3.1). Therefore, this contradiction implies the assumption J(u) does not hold.
For any δ > , de ne the modi ed functional and Nehari manifold as follows: The corresponding modi ed potential well and its corresponding set are de ned respectively by De ne and the (open) sublevels of J J ς = {u ∈ X |J(u) < ς}.
Obviously, by the de nition of J(u), N, J ς and d, we get Obviously, λς is non-increasing and Λς is non-decreasing.
Proof. ( ) It follows from the de nition of J(u), condition (c) and Corollary 2.1(1) that Obviously, assertion ( ) follows from r > p. . By a direct computation, Therefore, there exists a λ * > such that Moreover, combining with condition (d) and u X ≠ , we get Next, we prove that λ * = λ * (u) is uniquely determined. Assume that there exists two roots λ , λ of which is a contradiction. Therefore, we conclude that J(λu) gets its minimum in [λ , λ ] at either λ or λ . If the rst case happens, then there exists a σ > such that J(λu) at [λ , λ + σ] is not only maximum, but also minimum. That is, J(λu) is a constant. Therefore, we obtain which contradicts (3.2). We conclude that J(λ u) is not the minimum. By the same argument, J(λ u) also is not the minimum. So we complete the proof of ( ). Next, we will prove ( ). By the de nition of I(u), we get It is obvious that ( ) holds from ( ).

Lemma 3.5. The function d(δ) satis es the following properties:
is strictly increasing on < δ , strictly decreasing on δ and take its maximum d = d( ) at δ = .
Proof. ( ) For any u ∈ X with u X ≠ and δ > , there exists a unique λ = λ(δ) such that I δ (λu) = from Lemma 3.3(3). Furthermore, we get It is obvious that We have the following two claims: Claim 1: η(λ) is strictly increasing on ( , +∞). Indeed, under the condition (d), we get which yields Claim 1.

The low initial energy J(u ) < d
In this section, we consider the global existence and blow-up of weak solution under the condition that J(u ) < d. Speci cally, we prove that if I(u ) > , then problem (1.1) admits a global weak solution and if I(u ) < , all solutions to problem (1.1) blow up in nite time.
Proof of Theorem 1.1. The proof is divided into three steps.
Then (4.1) can be rewritten as dc n (t) dt = F n (c n (t)), c n ( ) = c n . (4.2) Next, we aim to use the Peano's theorem to prove the existence of (4.2). Multiplying the rst equality of (4.2) by c n (t), the we get were C(n) > is a constant depended on n. Solving the ordinary di erential inequality, we get here,T = |c n | −r/ ( −r)Ar(C(n)) r . There exists a su ciently small ε > such that |c Peano's theorem implies that there exists a solutions c n (t) of (4.2) on [ , T ], where For t ∈ [ , T ], the following (4.6) still holds, then we have It follows from u n (x, ) → u (x) in X that J(u n (x, )) → J(u (x)) < d and I(u n (x, )) → I(u (x)) > , as n → ∞.
Therefore, for su ciently large n, we get t u n τ L (Ω) dτ + J(u n ) = J(u n ( )) < d and I(u n (x, )) > , (4.4) which implies that u n (x, ) ∈ W. Next, we prove u n (x, t) ∈ W for su ciently large n. Otherwise, there exists a t ∈ ( , ∞) such that u n (x, t ) ∈ ∂W, that is I(u n (x, t )) = , u X ≠ or J(u n (x, t )) = d.
Clearly, J(u n (x, t )) ≠ d from ( . ). If I(u n (x, t )) = , u X ≠ , then J(u n (x, t )) d by the de nition of d, which contradicts ( . ). Therefore, u n (x, t) ∈ W and I(u n ) > . Combining with condition (c), we have From ( . ), we get t u n τ L (Ω) dτ + q + − p p(q + ) u n p X + q + I(u n ) < d, (4.5) which implies Furthermore, Corollary 2.1(1) and the embedding from X into L r (Ω) imply Therefore, there exists a u and a subsequence of {u n } n∈N (still denoted by {u n } n∈N ) such that as n → ∞, Since the embedding from X into L r (Ω) is compact and the embedding from L r (Ω) into L (Ω) is continuous, from Simon's theorem [29] we get u n → u in C([ , ∞]; L r (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 where l j (t) ∈ C ([ , ∞]) with j = , , · · · , k(k n). Multiplying the rst equality of (4.1) by l j (t) summing for j from to n, integrating with respect to t from to T, we get Letting n → ∞, combining with the theory of monotone operators as in [24], we get Since C ([ , ∞]; C ∞ (Ω)) is dense in L ( , ∞; X ), the identity in (4.10) holds for v ∈ L ( , ∞; X ). Moreover, by the arbitrariness of T > , we get By (4.9) and u n (x, ) → u (x) in X , then u(x, ) = u (x). Assuming that u is su ciently smooth such that u t ∈ L ( , ∞; X ), taking v = u t in (4.10), then (3.5) the holds. Since L ( , ∞; X ) is dense in L ( , ∞; L (Ω)), then (3.5) holds for weak solutions of problem (1.1).
Step 2: Uniqueness of bounded weak solution Assuming both u, v are two bounded weak solutions for problem (1.1). Then by the de nition of weak solution, for φ ∈ X , we get Subtracting the above two equalities, taking φ = u − v ∈ X , and then integrating for t from to t, we get Clearly, the st 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 here C > depends on the bound of u, v. Furthermore, Gronwall's inequality implies Ω φ dx = .
Step From Lemma 3.2 and Lemma 3.7, we know that u(x, t) ∈ W δ for δ < δ < δ and < t < ∞ under the condition J(u ) < d and I(u ) > . Thus Lemma 3.6 implies I δ (u) for < t < ∞. Therefore, the embedding from X into L (Ω) implies that Thus for p < , we get This means that the solution u vanishes at a nite time t * = For p > , we get Next, we are concerned with blow-up in nite time.
From the above discussion, there exists t * > such that Following the concavity method introduced by Levine [ where θ = q− . Since a concave function must always lie below any tangent line, we can obtain Then from the fact that M(t * ) > and M (t * ) > , we get that and M(t) → ∞ as t → T * , which contradicts our assumption that u is a global weak solution of problem (1.1). The proof is completed. Proof of Theorem 1.3. Let λ k = − k , k = , , · · · . Consider the following initial value problem:

The critical initial energy J(u ) = d
(5.1) Noticing that I(u ) , by Lemma 3.3(3) we can deduce that there exists a unique λ * = λ * (u ) such that I(λ * u ) = . Then from λ k < λ * and Lemma 3.3(2)(3), we get I(u k ) = I(λ k u ) > and J(u k ) = J(λ k u ) < J(u ) = d. In view of Theorem 1.1, for each k, then problem (5.1) admits a global weak solution u k ∈ L ∞ ( , ∞; X ) with u k t ∈ L ( , ∞; L (Ω)) and u k ∈ W satisfying Applying the similar argument in Theorem 1.1, there exists a subsequence {u k } k∈N which converges to a function u and u is a weak solution of problem (1.1) with I(u) and J(u) d for t < ∞. The proof of uniqueness for bounded weak solution is the same as that in Theorem 1.1.
Let us consider u L (Ω) . Firstly, assume that I(u) > for < t < ∞. Then (4.11) implies u t ≠ . Therefore, by Lemma 3.2 and (3.5) for any t > we have Taking t = t as the initial time, from Lemma 3.7, we know that u(x, t) ∈ W δ for δ < δ < δ and t < t < ∞ under the condition J(u(t )) < d and I(u(t )) > , where δ < < δ are the two roots of d(δ) = J(u(t )). Thus Lemma 3.6 implies I δ (u) for t < t < ∞. Therefore, the embedding from X into L (Ω) implies Thus for p < , we get This means that the solution u vanishes at a nite time T * = For p > , we get Secondly, assume that I(u) > for < t < t * and I(u(t * )) = . Then (4.11) implies u t ≠ for < t < t * . Therefore, by (3.5) we have By the de nition of d, we easily know u(t * ) X = , which implies u(t * ) = . De ne u(t) ≡ for t t * . Then, such a weak solution u(x, t) vanishes in nite time t * .
Next, we are concerned with blow-up in nite time.
Proof of Theorem 1.4. By the same argument in (4.15) of Theorem 1.2, we get Since J(u ) = d, I(u ) < , by the continuity of J(u) and I(u) with respect to t, there exists a t such that J(u(x, t)) > and I(u(x, t)) < for < t t . We have u t ≠ from (4.11) for < t t , moreover t uτ L (Ω) dτ > . Then from (3.5) Taking t = t as the initial time and by Lemma 3.7(2), we get u(x, t) ∈ V δ for δ < δ < δ and t < t, where δ < < δ , δ , δ satisfy the equation d(δ) = d . Therefore, I δ (u) < and u X > γ(δ) for δ < δ < δ and t < t by Lemma 3.4 (2). Furthermore, combining with Lemma 3.6, we get I δ (u) and u X γ(δ ) for t < t. The rest of this proof is the same as that one of Case 2 in Theorem 1.2., here, we omit the process.

The high initial energy J(u ) > d
In this section, we give some su cient conditions for global existence of weak solutions and blow-up in nite time regarding the high initial energy. Before proving our results, we need the following lemma. Ar u αr where α ∈ ( , ) since r < p * s . (6.2) can be written as Clearly, the right term of (6.3) is bounded and away from by Lemma 3.8(1) and the de nition of N ς . Therefore, we get λς > by the de nition of λς. Embedding X into L (Ω), we get u L (Ω) C * u X . Combining with the de nition of N ς , it is obvious that Λς < +∞.
Next, we are concerned with blow-up in nite time for any high initial energy.