Anomalous pseudo-parabolic Kirchhoff-type dynamical model

: In this paper, we study an anomalous pseudo-parabolic Kirchhoff-type dynamical model aiming to reveal the control problem of the initial data on the dynamical behavior of the solution in dynamic control system. Firstly, the local existence of solution is obtained by employing the Contraction Mapping Principle. Then, we get the global existence of solution, long time behavior of global solution and blowup solution for J ( u 0 ) ≤ d , respectively. In particular, the lower and upper bound estimates of the blowup time are given for J ( u 0 ) < d . Finally, we discuss the blowup of solution in finite time and also estimate an upper bound of the blowup time for high initial energy.


Introduction and main result
The paper is devoted to the study of an anomalous pseudo-parabolic Kirchho -type dynamical model as follows where s ∈ ( , ), N > s, Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂Ω. The Kirchho function M(t) = t λ− for t ∈ R + , here ≤ λ < N N− s , and q satisfy λ < q ≤ * s , where * s is the fractional critical exponent given by φ(x) − φ(y) |x − y| N+ s dy, for any φ ∈ C ∞ (R N ), where Bε(x) denotes the ball in R N with radius ε > centered at x ∈ R N . We can refer to [5,[23][24][25][35][36][37] for more details on nonlocal operators and nonlocal Sobolev spaces. Problem (1.1) is a class of nonlocal fractional di usion problem, which is related to the anomalous di usion theory. A usual model for anomalous di usion is the linear evolution equation involving the fractional Laplacian ∂ t u + (−∆) s u = , which derives asymptotically from basic random walks models, see [2,22,39] and references therein. We denote by u(x, t) the probability of nding the particle at the point x at time t. Through a series of calculations, we can obtain ∂ t u(x, t) = −cn,s(−∆) s u(x, t) for a suitable cn,s > , which shows that, for small time and space steps, the above probabilistic process approaches a fractional heat equation. Another nonlinear anomalous di usion equation is the fractional porous medium equation ∂ t u + (−∆) s (u m ) = with < s < and m > , which was rst proposed by De Pablo et al. in [32]. Many important results on these equations have been obtained, see an overview in [41] and references therein.
To the best of our knowledge, fractional Laplacian operator and related equations have a growing wide utilization in many important elds, as explained by Ca arelli in [3] and Vázquez in [40]. In particular, the steady state of problem (1.1) without strong damping term, rst proposed by Fiscella and Valdinoci in [12] by taking into account the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string, is a fractional version of the so-called stationary Kirchho model. Subsequently, the existence of weak solutions which solves the above stationary problem was obtained. Later, Fu and Pucci in [9] proved the existence of global solutions with exponential decay and showed the blow-up in nite time of solutions to the space-fractional di usion equation where Ω ⊂ R N is a bounded domain, p satis es < p ≤ * s − = (N + s)/(N − s) and N > s. In recent years, much interest has grown on Kirchho -type problems, see for example [12,28]. In these papers, to obtain the existence of weak solutions, the authors always assume that the Kirchho function M : R + → R + is a continuous and nondecreasing function and satis es the following condition: there exists m > such that M(t) ≥ m for all t ∈ R + . (1.2) A typical example is M(t) = m + bt m with m > , b ≥ for all t ∈ R + . Hence, we can divide the problem into degenerate and non-degenerate cases according to M( ) = and M( ) > respectively. For the nondegenerate Kirchho -type problem, we can refer to [10,30]. It is worthwhile pointing out that the degenerate case is rather interesting and is treated in well-known papers in Kirchho theory, see for example [4]. From a physical point of view, the fact that M( ) = means that the base tension of the string is zero. For some recent results in the degenerate case, see for instance [1,26,29,43,45]. In this regard, Pan et al. in [31] studied for the rst time the degenerate Kirchho -type di usion problem involving fractional p-Laplacian of following equation where Ω ⊂ R N , p < q < Np/(N − sp) with < p < N/s and ≤ λ < N/(N − sp). They obtained the existence of a global solution by combining the Galerkin method with potential well theory to discover the control mechanics of the initial data on the dynamical behavior of the solution. Yang et al. in [50] studied the same problem with pλ < q < Np/(N − sp), they obtained the blow up of solutions by applying the concave method. Moreover, the authors estimated an upper bound of blow-up time in the sub-critical initial energy case J(u ) < d and arbitrary positive initial energy case J(u ) > . Later, by implementing the same theory as shown in [31], for the initial boundary value problem of (1.3) with λ = and the more general nonlinearity f (u) instead of |u| q− u, authors in [19] studied the existence and nonexistence of global weak solutions in the cases of J(u ) < d, J(u ) = d and J(u ) > d, respectively. Moreover, the authors estimated an upper bound of blow-up time for low and high initial energies. Xiang et al. [44] considered the initial boundary value problem of the following Kirchho type equation In [48], Xu and Su used the family of the potential wells to prove the nonexistence of solutions with initial energy J(u ) ≤ d, and obtained nite time blowup with high initial energy J(u ) > d by comparison principle. Later, Xu et al. in [49] discussed the same problem, they established a new nite time blowup theorem for problem (1.5) and estimated the upper bound of blowup time for J(u ) > . Previously, Liu and Zhao in [21] considered the initial-boundary value problem u t − ∆u = f (u) with initial data J(u ) < d for I(u ) < and I(u ) ≥ , and initial data J(u ) = d for I(u ) ≥ . Xu in [46] studied the same problem with critical initial data J(u ) = d, I(u ) < . A powerful technique for treating the above problem is the so-called potential well method, which was established by Payne and Sattinger in [27]. Since then, the potential well method has been widely used to study the well-posedness of solution for evolution equations, such as [18,46,47]. Gazzola and Weth in [13] studied the initial-boundary value problem of u t − ∆u = |u| p− u, they proved nite time blow-up of solutions with high initial energy J(u ) > d by the comparison principle and variational methods. Recently, the threshold results of global existence and nite time blowup for several types of pseudo-parabolic equations were established in [33,42].
It is worthy pointing out that the Kirchho -type parabolic problem was studied by Han et al. [14] where the global existence and nite time blowup of solutions were proved in the sub-critical, critical and super-critical cases.
Here Ω ⊂ R N is a bounded domain, M(τ) = a + bτ with a, b > . In [15], by some di erential inequalities, the authors investigated the upper and lower bounds of blowup time for the weak solution to (1.6). Motivated by the above works, the main objective of this paper is to consider a more complicated case of the problem (1.4) studied in [6,7,44] by taking damping term (−∆) s u t in the fractional setting. More precisely, we focus on the local and global well-posedness of degenerate Kirchho 's model of parabolic type (1.1), by using potential well theory and concave function method.
The outline of this paper is as follows. In Section 2, we recall some necessary de nitions and properties of the fractional Sobolev spaces and introduce the family of potential wells. In Section 3, we prove the existence of local solutions for problem (1.1). In Section 4, we prove the global existence, the nite time blow-up, the asymptotic behavior for problem (1.1) and give the lifespan estimates of the blowup solution with J(u ) < d. In Section 5, we parallelly extend some conclusions for the sub-critical initial energy case to the critical initial energy. In Section 6, by constructing a new unstable set and using some di erential inequality techniques, we also study the nite time blowup solution for problem (1.1) and give the upper bound estimate of the blowup time at arbitrary positive initial energy level.

Preliminaries . Functional spaces
In this section, we rst recall some necessary de nitions and properties of the fractional Sobolev spaces, see also [5,11] for further details.
Throughout the paper, s ∈ ( , ), N > s and λ < q ≤ * s . We denote Q = R N \ G, where G = C(Ω) × C(Ω) ⊂ R N , and C(Ω) = R N \ Ω. W is a linear space of Lebesgue measurable functions from R N to R such that the restriction to Ω of any function u in W belongs to L (Ω) and The space W is equipped with the norm It is easy to get that · W is a norm on W. We shall work in the closed linear subspace By [35], we can get an equivalent norm on W de ned as We put where · X is an equivalent norm over W .
Then we de ne the potential energy functional of problem (1.1) as follows the Nehari functional Next we introduce the Nehari manifold Furthermore, we set N+ := {u ∈ X | I(u) > }, The potential well depth is de ned as Further we give some sets as follows

Lemma 2.5. Let δ > , then the properties of d(δ) can be summarized as follows:
λ and takes the maximum at δ = .
Proof. (i) When u ∈ N δ , then from lemma 2.4 (ii) it gives u X ≥ r(δ). Further by (2.8) and Therefore, the conclusion of (iii) is proved.

Existence and uniqueness of local solution
Inspired by [38], in which Taniguchi considered the existence of a local solution to a Kirchho -type wave equation with damping. In this section, we shall prove the local well-posedness of solution to the Kirchhotype pseudo-parabolic equation of the form (1.1).
For a given T > , we consider the space H = C([ , T], X ) endowed with the norm In the following, the existence and uniqueness of solution for the linear problem corresponding to (1.1) is proved.
which solves the linear problem

2)
Proof. The assertion follows from an application of the Galerkin method. By [36], for every h ≥ let W h = Span{ω , · · · , ω h }, where {ω j } is the orthogonal complete system of eigenfunctions of (−∆) s in W such that ω j W = and ω j = for all j. Then, {ω j } is orthogonal and complete in L (Ω) and W ; denote by {λ j } the related eigenvalues repeated according to their multiplicity. Let solving the problem for every η ∈ W h and t ≥ . For j = , · · · , h, taking η = ω j in (3.4) yields the following Kirchho fractional Laplacian problem for a linear ordinary di erential equation with unknown γ h For all j, the above Kirchho fractional Laplacian problem yields a unique global solution γ h j ∈ C [ , T]. In turn, this gives a unique v h de ned by (3.3) and satisfying (3.4). In particular, Next, the proof is divided into the following two cases.
for every h ≥ . Since u ∈ H, u X is bounded. We estimate the last term in the right-hand side thanks to Hölder and Young inequalities where c > and represent di erent constants between di erent lines. By combining (3.7), (3.8) and Hölder inequality, we obtain where C T > is independent of h. By this uniform estimate, the embedding W → L (Ω) and using (3 Sinceũ ∈ H, ũ X is bounded. We estimate the last term in the right-hand side thanks to Hölder and Young inequalities where A := u h X + cT is a constant. By using Gronwall's inequality again, we have Therefore, up to a subsequence, we may pass to the limit in (3.4) and obtain a weak solution v of (3.2) with the above regularity. Uniqueness follows arguing for contradiction: if v and ω are two solutions of (3.2) which share the same initial data, by subtracting the equations and testing with v t − ω t , instead of (3.7) we can get which immediately yields ω ≡ v. The proof of the lemma is now complete.
Next, we establish local existence and uniqueness of (1.1).
Next, we still divide the proof to two cases corresponding to Lemma 3.1.
Case 1: M( u W ) ≥ m > for any u ∈ W , where m is a constant. For the last term on the right-hand side of (3.15), we argue in the same spirit (although slightly di erently) as for (3.8) and we get  (3.17) where L := cTR (q− ) + M( u W ) u W is a constant. By using Gronwall's inequality, it gives Then taking the maximum over [ , T] gives Choosing T su ciently small, we get v H ≤ R .
Case 2: There is at least aũ ∈ X such that M( ũ W ) = .
In this regard, let R = u X and for any T > consider Similar to (3.11) in lemma 3.1, the corresponding solution v = Φ(u) satis es for all t ∈ ( , T] the energy identity where A := u X + cTR (q− ) is a constant. By Gronwall's inequality So v X ≤ u X + cTR (q− ) + A (e T − ). (3.22) Choosing T su ciently small, we get v H ≤ R . Combining Case 1 and Case 2, we show that Φ(M T ) ⊆ M T . Next we prove Φ is a contraction. Now take ω and ω in M T , subtracting the two equations (3.2) for v = Φ(ω ) and v = Φ(ω ), and setting v = v − v we obtain for all η ∈ W and a.e. t ∈ [ , T] where ς = ς(x, t) ≥ is given by Lagrange Theorem so that ς(t) ≤ (q − )(|ω (t)| + |ω (t)|) q− . Therefore, by taking η = v t in (3.23) and arguing as above, we obtain  Assume that u ∈ X , < e < d, δ < δ are the two roots of equation d(δ) = e for < δ < < δ < q λ , Tmax is the maximal existence time of u(t). Then (i) All weak solutions u of problem (1.1) with J(u ) = e belong to W δ for δ < δ < δ , ≤ t < Tmax , provided I(u ) > . (ii) All weak solutions u of problem (1.1) with J(u ) = e belong to V δ for δ < δ < δ , ≤ t < Tmax , provided I(u ) < .

. Global existence and nite time blowup of solution
In this section, we prove a threshold result of global existence and nonexistence of solutions for problem (1.1) with the sub-critical initial energy J(u ) < d. Next, we prove um(x, t) ∈ Wp for su ciently large m and ≤ t < ∞. If it is false, then there exists t such that um(x, t ) ∈ ∂Wp, then I(um(t )) = , um(t ) X ≠ or J(um(t )) = d.
By (4.4), it implies that J(um(t )) = d < J(um( )) is not true. On the other hand, If I(um(t )) = , um(t ) X ≠ , according to the de nition of d, we have J(um(t )) ≥ d, which is also contradictive with (4.4). Hence um(x, t) ∈ Wp for all ≤ t < ∞ and su ciently large m. Then by (4.4) and for su ciently large m, which yields t umτ X dτ < d, ≤ t < ∞.
Also, according to the embedding inequality um ≤ C um W , (4.5) implies So, we can get where C * is the embedding constant from W → L q (Ω). Therefore, there exist a u and a subsequence um, such that as m → ∞. So we can get and H ′′ (t) = (u, u t ) X . (4.9) Employing the Cauchy Schwartz inequality, we obtain As a consequence, we read the di erential inequality for almost every t ≥ . Next we de ne Setting ϕ = u(t) in (2.4) and using (2.7), it follows that Then we discuss the situation in two cases. It follows lemma 4.1 that I(u) < for t > . This implies θ * < , then we can get which implies Further by a simple computation, it gives (4.12) and q > λ, we get Obviously, we can also derive (4.14) in this case. Moreover, by J(u ) ≤ it implies that I(u) < for all t ≤ .
Then multiplying both sides of the rst equation in (1.1) by u, it follows that (u, u t ) X = −I(u), which together with (4.9) and the fact I(u) < gives H ′′ (t) > for all t ≥ . Then by H ′ ( ) = , it can be deduced that H ′ (t) > for all t > . Note that by H ′ (t) > and H(t) > for t > . From (4.15) and (4.16), it follows that there exists a nite time T > such that lim Then the proof is completed.

. Asymptotic behavior of solutions
Xu and Su in [48] studied the initial boundary value problem of semilinear pseudo-parabolic equation (1.5), obtained the asymptotic behavior of solutions with initial energy J(u ) ≤ d, which implies that the global solution to problem (1.5) decay exponentially. In this section, we shall consider the above decay behavior of the global solution in the fractional setting. when λ > , then and (u t , w) + u, w W + (u t , w) W = (|u| q− u, w). From Lemma 4.1 along with < J(u ) < d and I(u ) > , we get u(t) ∈ W δ for δ ∈ (δ , δ ) and t ∈ [ , ∞). Hence, it follows that I δ (u) ≥ and I δ (u) ≥ for δ ∈ (δ , δ ) and t ∈ [ , ∞). Then from (4.20) and the de nition of d(δ), we have From (4.21) we also have When λ = , by the Gronwall inequality, we have Therefore, there exists a constant β = ( − δ ) > such that When λ > , from (4.22) we have The proof is completed.

. Lower bound estimate of the blowup time
Luo [20] considered the semilinear pseudo-parabolic equation (1.5), obtained a lower bound for blow-up time at low initial energy. Inspiring by Luo's work. In this section, by the similar argument, we derive the lower bound estimate for blowup time of solution to problem (1.1) with J(u ) < d. Then by (4.25) and the embedding inequality u q ≤ u X , it implies So we see the following inequality Integrating the inequality (4.26) from to t, we have So letting t → T in (4.27), we can conclude that The proof is completed.

. Upper bound estimate of the blowup time
.
The proof is completed.
By the way, inspired by [20], we can also derive an upper bound for blow-up time when J(u ) < . and um ∈ Vp for ≤ t < ∞, satisfying Then we can get The rest of the proof is similar as that in theorem 4.1. Proof. Similar to the proof of Theorem 4.2, rst we assume that the critical initial energy solution exists globally. From I(u ) < and the continuity of I(u(t)) in t, it can be seen that there exists a su ciently smallt > such that I(u(t)) < for t ∈ [ ,t]. Moreover, by the fact that (u, u t ) X = −I(u(t)) > for t ∈ [ ,t], we have u t ≠ for t ∈ [ ,t]. Hence, by (2.7) and the continuity of J(u(t)) in t, it follows that J(u(t)) < d for t ∈ ( ,t]. Takingt ∈ ( ,t] as the new initial time, obviously there holds I(u(t)) < and J(u(t)) < d. The remainder of the proof is similar to that of Theorem 4.2.

. Asymptotic behavior of solutions
In this section, we consider the asymptotic behavior of solutions for problem (1.1) with the critical initial condition J(u ) = d. By the similar way of the proof of Theorem 4.3, we can give Theorem 5.3.
When λ > , then (i) Assume that I(u) > for ≤ t < ∞. Then from (u t , u) X = −I(u) < and u t X > , it follows that t u t X dτ is increasing on t ∈ [ , ∞). Picking any t > and setting d = d − t uτ X dτ, (5.5) by noticing (2.7), we have < J(u) ≤ d < d and u(t) ∈ W δ for δ ∈ (δ , δ ) and t ∈ [t , ∞), where δ < δ solve the equation d(δ) = d . Thus, I δ (u) ≥ for t ≥ t , which together with (4.21), gives that When λ = , making use of Gronwall's inequality, we can get When λ > , it follows that (ii) Let us suppose by contradiction that t > is the rst time such that I(u(t )) = . By (2.8), we get Meanwhile, (2.7) gives Hence we deduce J(u(t )) = d. Again from (5.6) we get t uτ X dτ = , that is u(t) ≡ for ≤ t ≤ t , which contradicts I(u ) > . Hence we have I(u) > and J(u) < d for < t < ∞. By the continuity of the functionals J(u) and I(u) in t, we reset the initial data to a small enough t > such that < J(u(t )) < d and I(u(t )) > . By (4.21) we get Making use of Gronwall's inequality, when λ = we can get when λ > , the result is same as the case (i). Therefore, when λ = , there exist constants E > , t > and γ > such that The proof is completed.

Blowup for arbitrary positive initial energy J(u ) >
In this section, we establish a nite time blowup theorem for the solution of problem (1.1) with arbitrary high initial energy. At the same time, we estimate the upper bound of the blowup time. Firstly, the invariance of the set N− is proved as follows.
Lemma 6.1 (The invariance of N− when J(u ) > ). Assume that λ < q < * s , u ∈ X , J(u ) > and the initial condition holds. Then u ∈ N− for all t ∈ [ , T], where C denotes the embedding constant for W → L (Ω), T is maximum existence time of u(t).
Proof. Let u(t) be any weak solution of problem (1.1). Multiplying (1.1) by u t (t) and integrating on Ω, then we have Further we could obtain Multiplying (1.1) by u and integrate on Ω × ( , t), we have which together with (6.1) indicates that I(u ) < . Next, we prove u(t) ∈ N− for all t ∈ [ , T). Arguing by contradiction, by the continuity of I(t) in t, we assume that there exists at ∈ ( , T) such that u(t) ∈ N− for ≤ t <t and u(t) ∈ N, then by (6.3) we have which implies that Then, we have By the de nition of J(u) and u(t) ∈ N, we derive to which together with (6.1) and (6.6), we can get i.e., u(t) λ X < u λ X , which contradicts (6.5).
such that where α, η( ) and η ′ ( ) will be determined in the later proof.
Proof. Arguing by contradiction, we assume the existence time of solution T = +∞. Integrating of (6.2) with respect to t, we have = − λ λ u λ W − q u q q + − λ q u q q = − λJ(u) + q − λ q u q q . (6.8) In the rest of the proof, we consider the following two cases.
As η(t) is a continuous function with respect to t, we can conclude that y(t) tends to ∞ at some t * which contradicts T = +∞.
Since J(u ) > , by the continuity of J(u(t)) in t, we can assume that there exists a rst time t > such that J(u(t )) = and J(u(ṫ)) < for someṫ > t . We take u(ṫ) as a new initial datum, then from Lemma 6.1, we have u(t) ∈ N− for t >ṫ. Then similar to the proof of Theorem 4.2, we can prove the nite time blowup of the solution.
Combining the above two cases, we conclude that u(t) blows up in nite time.

Conclusions and future works
Inspired by [35], it is natural to consider the following more general problem there exists K > , such that K(x) ≥ K |x| −(N+ s) for a.e. x ∈ R N \ { }.
A typical example for K is the singular kernel K(x) = |x| −(N+ s) . In this case, up to some normalization constant, L K φ(x) = (−∆) s φ(x). Using the arguments similar to Sects. 4-6 of this paper, we get the existence and nite time blow up of solutions, as well as the asymptotic behavior for problem (7.1). However, the global existence for super-critical initial energy, i.e., J(u ) > d can't be obtained because of the absence of the comparison principle. Thus, in order to prove the global well-posedness for problem (1.1) and (7.1) in the super-critical initial energy case, some new methods and strategies should be found, which will be the object of future work. At the same time, this work is helpful to analyze the observability and measurability of the control model in control system.