Global existence and nite time blowup for a nonlocal semilinear pseudo-parabolic equation

Abstract: In this paper, the initial boundary value problem for a nonlocal semilinear pseudo-parabolic equation is investigated, which was introduced to model phenomena in population dynamics and biological sciences where the total mass of a chemical or an organism is conserved. The existence, uniqueness and asymptotic behavior of the global solution and the blowup phenomena of solutionwith subcritical initial energy are established. Then these results are extended parallelly to the critical initial energy. Further the blowup phenomena of solution with supercritical initial energy is proved, but the existence, uniqueness and asymptotic behavior of the global solution with supercritical initial energy are still open.


Introduction
In this paper, we consider the initial boundary value problem of semilinear pseudo-parabolic equation with Neumann boundary condition in Ω, − Ω u dx = |Ω| Ω u dx = , in Ω, ∂u ∂ν (x, t) = , on ∂Ω × ( , T), (1.1) where u ∈ H (Ω), T ∈ ( , ∞] and p satis es Problem (1.1) can be used to model phenomena in population dynamics and biological sciences where the total mass of a chemical or an organism is conserved [1][2][3][4]. The nonlocal term acts to conserve the spatial integral of the unknown function as time evolves. Such equations give insight into biological and chemical problems where conservation properties predominate. So the model including this nonlocal term distinguishes the classical heat equation [5][6][7][8][9][10][11][12][13][14][15][16][17], the parabolic systems [18], the fractional Laplacian parabolic equation [19][20][21], variable exponent parabolic equation [22] and the pseudo-parabolic equation with singular potential [23]. Budd et al. in [1] studied the 1-dimensional nonlinear parabolic equation (1.2) with Neumann boundary condition and proved that the solution blows up for some special initial value. Fila and Levine [24] considered the n-dimensional case of (1.2) they showed that the global solution is bounded in some space with f (u) = |u| p− u. The initial boundary problem (1.3) with Neumann boundary and f (u) = |u| p was considered in [25], and the nite time blowup was proved for negative initial energy, i.e. J(u ) < and < p ≤ . Later this result was extended to p > in [26]. Then for f (u) = |u| p− u, the above blowup result was further extended to the positive initial energy case, i.e. J(u ) < d in [27] by the similar potential well method employed in [28]. In [29], the above restriction on the initial energy was relaxed to J(u ) > and the nite time blowup was derived for some large initial data. For problem u t − η∆u t − ∆u = f (u), (1.4) Showalter and Ting [30] investigated the initial boundary value problem with f (u) = . They proved global existence, uniqueness and regularity of solutions. Xu and Liu [31][32][33][34] studied the initial and boundary value problem of (1.4) for f (u) = u p . They proved the existence, asymptotic behavior of the global solutions and global nonexistence of solutions with subcritical and critical initial energy J(u ) ≤ d, and also obtained the global nonexistence of solutions with supercritical initial energy J(u ) > d by comparison principle, further estimated the upper bound of the blowup time for supercritical initial energy. Due to the role of the corresponding conservation properties in the real world and its connect to the biological and chemical model equation (1.4), it is interesting to consider (1.1) by investigating the e ect of the nonlocal term − Ω |u| p− udx. Another tough task is to "keep balance" between the bad side and the good side of the strongly dissipative term ∆u t , which helps the global existence by decaying the energy, meanwhile impedes blowup and delay the blowup time. As the goal of this paper is not to treat just one side of above, but to nd out for which initial data the solution exists globally, and for which the solution blows up in nite time, it is important but not easy to deal with the blowup case and keep such balance. As we know that the global well-posedness of solution to the evolution equation strongly relies on the initial data, especially the initial energy, we plan to conduct a comprehensive study in this paper on the global well-posedness of solution at subcritical and critical initial energy J(u ) ≤ d, and supercritical initial energy J(u ) > . For each initial energy level, we aim to obtain the global existence, asymptotic behavior of the solution as the time goes to in nity and nite time blow up of solution. But we do not succeed for all the cases. We summarize the main conclusions of this paper in the following Table 1, where the mark " √ " denotes the results we get in this paper and the question mark "?" indicates the problem unsolved still.
This article is structured as follows: (i) In Section 2, we introduce some functionals and the potential well. Also, we give several preliminary de nitions and lemmas. (ii) In Section 3, we prove the local existence of solution by the standard Galerkin method. (iii) In Section 4, we prove the existence, uniqueness, decay estimate of the global solution and global nonexistence of solution with J(u ) < d are inspired by [31]. (iv) In Section 5, we extend all the conclusions in Section 4 parallelly to the initial energy J(u ) = d are inspired by [15]. (v) In Section 6, we prove the global nonexistence of solution with J(u ) > by using the two di erent methods in [29,35,36], but the global existence and asymptotic behavior of solution at supercritical initial energy level are still open.

Preliminaries
In this section, we introduce some de nition, functionals and the potential well, and also establish some properties of them. Throughout the paper, we assume that Ω ⊂ R n is an open bounded smooth domain. We denote by · q the L q (Ω) norm for ≤ q ≤ ∞ and by ∇ · the Dirichlet norm in H (Ω). And for u, v ∈ H (Ω), denote by (u, v) * = Ω uvdx + Ω ∇u∇vdx the inner product in H (Ω). The inner product endow with the norm u H = u + ∇u for every u ∈ H (Ω), which is equivalent to ∇u .
Next, the energy functional J(u) and the Nehari functional I(u) are de ned as (2.5) and By I(u) and J(u), we introduce the stable set where d is the depth of potential well de ned as (the so-called mountain pass level in [37]) and the Nehari manifold is Next, we give the following de nition of weak solution.
Lemma 2 (Conservation Law). Let u be the weak solution of problem (1.1). If Ω u (x)dx = , then the integral of u is conserved, that is Ω u(x, t)dx = .
Proof. For the weak solution de ned by (2.7), we can take φ = as the test function, then by noticing De nition 3 (Maximal existence time). Let u(t) be a weak solution of problem (1.1). We de ne the maximal existence time T of u(t) as follows (i) If u(t) exists for ≤ t < ∞, then T = +∞.
(ii) If there exists a t ∈ ( , ∞) such that u(t) exists for ≤ t < t , but doesn't exist at t = t , then T = t .
Here, we give a estimate of nonlinear term |u| p− u by Gâteaux derivative.
By the property of Gâteaux derivative, we know for s ∈ ( , ), which together with the de nition of Gâteaux derivative For the Nehari functional I(u), we have the following properties. Proof.
(i) From < ∇u < r , we have we get ∇u ≥ r .
Lemma 7 (Depth d of potential well). Suppose that (H) holds. Then the potential well depth Proof. By the de nition of J(u), I(u) and the fact that I(u) = , we have which implies J(u) > . By Lemma 5 (iii), there holds

Local solution
In this section, by the standard Galerkin method we prove the local existence of solution for problem (1.1). From now on, the constants in the Sobolev, Poincaré and Young's inequalities, and other constants that may appear in calculation, are denoted uniformly by C > . Proof. The proof of this theorem is divided into three steps. The rst step is to show that the existence and uniqueness of solution to the linear problem corresponding to problem (1.1) by Galerkin method. The second step is to prove the existence and uniqueness of the local solution to problem (1.1) based on rst step by the Contraction Mapping Principle. And the third step is to get the assertion (3.11).
Step I: The existence and uniqueness of solution to the linear problem. Take u ∈ H (Ω) and let R = ∇u . For every T > consider the space H = C([ , T]; H (Ω)) endowed with the norm (3.12) and the set Next for every T > and u ∈ H T , we shall respectively prove the existence and uniqueness of solution to the following linear problem by Galerkin method Existence. Fix a positive integer m, assume that {ω j (x)} is an orthogonal complete basis in H (Ω) and L (Ω) such that Wm = Span{ω , · · · , ωm} and ω j = for all j. Denote by λ j the related eigenvalues of ω j (x). Let such that u m ∈ Wm and u m → u in H (Ω) as m → ∞, then u m ∈ H T . We seek m functions g m (t), . . . , gmm(t) ∈ C [ , T] to construct the approximate solutions to problem (3.13) vm(x, t) = m j= g jm (t)ω j (x), m = , , · · · (3.14) satisfying for any µ ∈ Wm and f (u) = |u| p− u − − Ω |u| p− udx. Taking µ = ω j in (3.15) we have the following Cauchy problem of system of linear ordinary di erential equations with the unknown g jm (t) by Ω vm( )dx = and Lemma 2, i.e. Ω vmdx = .
For u ∈ H T , i.e. ∇u ≤ R, we estimate the last term in the right hand side of (3.17) by Hölder, Sobolev and Young's inequalities as follows Then combining (3.17), (3.18) and u m ∈ H T , we obtain for every m ≥ , where C is independent of m. Therefore, we see that the sequence {vm} is bounded in In (3.15) letting m = k → ∞, we conclude the existence of a weak solution v of (3.13) with the above regularity.

Uniqueness.
Arguing by contradiction, if v and v are two weak solutions of (3.13) with the same initial datum, by subtracting the two equations corresponding to v and v respectively, and testing it Then the uniqueness is proved.
Step II: The existence and uniqueness of local solution to the nonlinear problem (1.1). By Step I, for any u ∈ H T and the unique solution v to problem (3.13) we can de ne v := Φ(u). We claim that Φ(H T ) ⊂ H T is a contractive map. First for given u ∈ H T , similar to ( .22) gives For su ciently small T, we see v H ≤ R, which implies that Φ(H T ) ⊂ H T . Next we prove such map is contractive. Taking u , u ∈ H T to be the known functions in the linear terms of (3.13) respectively, subtracting the two equations in form of (3.13) for v = Φ(u ) and v = Φ(u ) respectively, setting v = v − v and testing the both sides by v t , we can arrive at For the last term of (3.23), by Lemma 4, Hölder, Sobolev and Young's inequalities, we get for some δ T = CR p− T < as long as T is su ciently small. Therefore, the map v = Φ(u) is contractive. By the Contraction Mapping Principle, there exists a unique weak local solution to (1.1) de ned on [ , T].
Step III: Finite time blowup of local solution. By the arguments above, especially the requirements on δ T , the su cient conditions ensuring the contraction indicate that the existence time of the local solution only depends on the scale of the norm of the initial data by the connection between R and ∇u . Therefore, the local solution u is continued as long as u H remains bounded, see also Theorem 1 in [38] and Theorem 3.1 in [39] for similar argument. Hence, if the maximum existence time of local solution is Tmax < ∞, we have lim t→Tmax ∇u = ∞.
(3.25) By (2.5) and (2.8) we deduce which together with the following Gagliardo-Nirenberg interpolation inequality Combining Step I, Step II and Step III we complete the proof of Theorem 8.

Subcritical initial energy J(u ) < d
In this section, we shall prove the global existence, asymptotic behavior and nonexistence of solution for problem (1.1) with the subcritical initial energy J(u ) < d. First, we show that the invariance of some sets of problem (1.1).

Lemma 9 (Invariant sets for J(u ) < d).
Let p satisfy (H), u ∈ H (Ω), T be the maximal existence time of solution, then Proof.
(i) Arguing by contradiction, by the continuity of I(u) respect to t, we suppose that t ∈ ( , T) is the rst time such that u(t ) ∈ ∂W, that is I(u(t )) = , we know that J(u(t )) ≠ d. If I(u(t )) = , u(t ) H ≠ , then by the de nition of d we have J(u(t )) ≥ d, which contradicts (4.28).
(ii) The proof is similar to (i).
Next, we prove the global existence and uniqueness of solution with J(u ) < d, also show the asymptotic behavior of global solution. and by Lemma 9, for ≤ t < ∞ and su ciently large m we have um(t) ∈ W. Combining (4.32) and for su ciently large m, which yields a priori estimate Therefore, there exists a u and a subsequence {uν} of {um} such that where (p − )A = n n− and A = A = n n−(n− )p− < n n− by (H). As v(x, ) = , both (4.38) and (4.39) give in Ω × ( , ∞). Asymptotic behavior. Setting φ = u in (2.7) and noting that Note that (4.44), we can see that there exists δ > such that −δ := C C p+ By Gronwall's inequality, we have for δ > The proof of this theorem is complete.
Here, we show a relationship between ∇u and the depth of potential well d with the initial value condition u ∈ V, which is helpful for the proof of nite time blowup of solution.
Proof. Suppose that u(t) is a weak solution of (1.1) with J(u ) < d and I(u ) < , T be the maximal existence time. From Lemma 9, we can see that u(x, t) ∈ V, that is I(u) < for < t < T. By Lemma 5, we obtain Therefore, Lemma 7 implies that d < p − (p + ) ∇u .

Theorem 12 (Blow up for J(u ) < d). Assume that p satis es (H). If u ∈ H (Ω), then for u ∈ V the weak solution u(t) of (1.1) blows up in nite time.
Proof. First, Theorem 8 shows that the problem (1.1) admits a unique local solution u ∈ C [ , T]; H (Ω) . We will prove the existence time T of solution u(t) is nite with u ∈ V. Arguing by contradiction, we assume that the solution is global in time, i.e. T = +∞. By u ∈ V and Lemma 9, for t ∈ [ , +∞) we get u ∈ V. We introduce an auxiliary function as where < T < +∞. Obviously, F(t) > for any t ∈ [ , T ]. By the continuity of F(t) with respect to t, it follows that there exists a constant ς > , such that F(t) ≥ ς for t ∈ [ , T ]. Then From (4.45) and Cauchy-Schwarz inequality, we obtain By (4.47) and (4.46), we can get Hence we have Therefore, by [5] there exists a T > such that The proof is completed.

Critical initial energy J(u ) = d
In this section, we shall extend parallelly the global existence, asymptotic behavior and blowup of solution for problem (1.1) to the critical initial energy J(u ) = d. Proof. Global existence and uniqueness. First the condition J(u ) = d implies that u H ≠ . We de ne χs = − s and u s (x) = χs u (x) for s = , , · · · , which implies that < χs < and χs → as s → ∞. Next, we study the problem (1.1) with the condition The remainder of the proof is similar to Theorem 10.
Asymptotic behavior. From above we have proved that the solution u(t) of (1.1) is global, then we claim that u ∈ W for t > . Arguing by contradiction, we suppose that t > is the rst time that I(u(t )) = . By the de nition of d, we see that J(u(t )) ≥ d. By (2.8) which implies that for any t > . Hence we deduce J(u(t )) = d, which means (by (5.54)) t uτ H dτ = , that is u t ≡ for ≤ t ≤ t , which contradicts I(u ) > (by (4.46)). Hence we have u ∈ W for < t < ∞. By the continuity of J(u) and I(u) with respect to t, we take a t > as the initial time, then u(x, t) ∈ W for t > t . Hence from (4.42)-(4.44) we know that there exists a constant κ > as The proof of this theorem is complete.

Theorem 14 (Blow up for J(u ) = d). Assume that p satis es (H). If u ∈ H (Ω), J(u ) = d and I(u ) < , then the weak solution u(t) of (1.1) blows up in nite time.
Proof. From the continuity of I(u) and J(u) with respect to the time t, for I(u ) < and J(u ) = d > , which means there exists a su ciently small t > such that I(u(t )) < and J(u(t )) > . Combining (4.46), we have u t ≠ for < t ≤ t . Hence, by (2.8) there holds (5.55) By Lemma 9 and by taking t = t as the initial time, we get u(x, t) ∈ V for t > t . The remainder of the proof is the same as Theorem 12.

Supercritical initial energy blowup J(u ) >
In this section, we prove the blowup of solution to problem (1.1) with supercritical initial energy and suitable initial data by two di erent methods. We will use the following lemmas to estimate the blowup time and prove the blowup in nite time.
Since J(u) is continuous with respect to t, for J(u ) > and (6.57) there must exist a time t > such that J(u) < for t > t and J(u(t )) = . We choose u(t ) as a new initial datum of problem (1.1), then Lemma 16 gives u ∈ N− for t > t . Similar to the proof of Theorem 12, we obtain the blowup of solution in nite time.
Combining the Case (i) and Case (ii), we conclude that the blowup of solution in nite time.

Method II. (Inspired by [36])
Proof. First, Theorem 8 shows that the problem (1.1) admits a unique local solution u ∈ C [ , T); H (Ω) . We show that the solution u(t) blows up in nite time Tmax. If it is false, we suppose that the existence time T = +∞. By Lemma 16, we have u ∈ N−, i.e. I(u) < for all t ∈ [ , +∞). Next, we consider the following two cases. Case I. J(u) ≥ for all t > . In this case, we divide the proof into two steps.
Step 2. We estimate the upper bound of blow up time.
We introduce an auxiliary function as where the two positive constants η, ξ > will be determined later. By (6.59), we have H( ) = Tmax u H + ηξ > , (6.90) Combining (6.89) and (6.95), (6.91) gives By (6.56), we let which says that f (η, ξ ) is decreasing with η, then for any ξ ∈ , if we only focus on the initial condition about energy, we will nd that the conditions J(u ) < d and J(u ) > have overlapping parts, that is < J(u ) < d. Therefore, it is necessary to clarify their relationship. As the depth of potential well d, so-called the mountain pass level, plays a very important role in characterizing the initial data manifolds in terms of the signs of the Nehari functional i.e. I(u ) > , I(u ) < or the radium of the ball in H (Ω) space, i.e., ∇u < r = C p+ * p− and ∇u > r only under the condition J(u ) < d, we can get the results similar to the sharp condition, which is to show that the initial data u ∈ u ∈ H (Ω) | J(u) < d, I(u) > leading to global existence solution and u ∈ u ∈ H (Ω) | J(u) < d, I(u) < leading to nite time blowup solution. Although in Theorem 17 we extend the restriction J(u ) < d to J(u ) > , we only get a class of su cient conditions (di erent from those in Theorem 12) ensuring the nite time blowup of the solution instead of proving anything similar to the sharp condition that can be derived for the sub-critical case, i.e., J(u ) < d. Hence the conclusions for J(u ) < d and J(u ) > are di erent, and the conclusions for case J(u ) < d are better.
Indeed J(u ) < d and J(u ) > are only the conditions about the initial energy acting as part of the initial conditions, so when we compare the conclusions of these two conditions, we need to pay attention to other conditions also. On one hand, in the frame of J(u ) < d, we have an invariant set V in which I(u ) < , and a relation between I(u) < and ∇u > r = C p+ * p− , but we cannot get neither of them when we extend J(u ) < d to J(u ) > . On the other hand, in the frame of J(u ) > , (6.56) is an important condition on the initial data, and (6.56) seems interesting here. First we can derive u ∈ N− if u satis es (6.56), that means the initial datum given by (6.56) leads to I(u) < for any t > , which can be considered as another version of the invariant manifold V in the case J(u ) < d, but the di erence is that we replace u ∈ V by (6.56).