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Criteria for finite time blow up for a system of Klein-Gordon equations

We give three conditions on initial data for the blowing up of the corresponding solutions to some system of Klein-Gordon equations on the three dimensional Euclidean space. We first use Levine's concavity argument to show that the negativeness of energy leads to the blowing up of local solutions in finite time. For the data of positive energy, we give a sufficient condition so that the corresponding solution blows up in finite time. This condition embodies datum with arbitrarily large energy. At last we use Payne-Sattinger's potential well argument to classify the datum with energy not so large (to be exact, below the ground states) into two parts: one part consists of datum leading to blowing-up solutions in finite time, while the other part consists of datum that leads to the global solutions.


INTRODUCTION
In this article, we are going to give several criteria for finite time blow up of solutions to the following system of Klein-Gordon equations (u(0), u t (0)) = (u 0 , u 1 ), in R 3 , where β ∈ R is a parameter to be specified in each criterion.
This system was introduced by Segal in [19] to model the motion of charged mesons in electromagnetic field. Similar systems of coupled wave and Klein-Gordon equations are also proposed to model some physical phenomena such as the interaction of mass and massless classical fields, and long longitudinal waves in elastic bi-layers, see [5,7,8] and references therein for more details.
For the mathematical study, we can apply a scaling argument to the corresponding homogeneous system, to see that the system (1.1) falls into the energy subcritical class. Hence, we can apply the Banach fixed point theorem in the energy space H 1 × H 1 to construct a local solution (u(t), v(t)) for t ∈ [0, T ) with T small (see Section 2 for details). As long as the solution (u(t), v(t)) exists, the energy functional remains invariant as t varies. However, as t varies, the kinetic energy (the first term) and the potential energy (the second term) might become arbitrarily large, so that we can not repeat the fixed point argument to extend the local solution to a global one. Even worse the solution might exhibit the blow up phenomenon. Here by the blow up of solutions, we mean that its associated 'mass' is going to become infinite in finite time. In this article, we will give some sufficient conditions on the initial datum for such a phenomenon to occur.
1.1. Negative energy regime. The first one we are going to give falls into the philosophy that 'negative energy leads to blow up in finite time', as is stated . Given the initial data ((u 0 , u 1 ), (v 0 , v 1 )) ∈ H 2 with H := H 1 × L 2 , let (u(t), v(t)) be the corresponding solution to (1.1). Assume either of the following holds: (i) the negative energy condition E(0) < 0; (ii) the zero energy conditon E(0) = 0, together with the angle condition that Then, (u(t), v(t)) blows up in finite time.
Remark 1.2. Comparing with the forthcoming Theorem 1.6, one can find the range of β is much larger in Theorem 1.1, and that for β in the common range [0, ∞), Theorem 1.6 implies Theorem 1.1. But for β ∈ [−1, 0), we would not expect a result as that in Theorem 1.6, see Remark 1.12 for the technical explanation. Therefore, we shall give the full proof of Theorem 1.1 in an independent section. One can also find that the proof, especially Lemma
We see that the negativeness of initial energy, allows us to conclude from (1.5) that y(t) and its derivative d dt y(t) will become strictly positive after some finite time, which leads to blowing up. However, if the energy is positive, we can not use this argument directly.
Fortunately, we can use the projection functional and Payne-Sattinger well argument to give two sufficient conditions for finite time blow up, which contains datum of positive energy. These will serve as our second and third criteria.
1.2. Criterion in terms of the projection functional. As is alluded to above, our second criterion will allow the energy to be positive and the angle condition to be non-negative, but at the cost that the initial energy is under control of y(t) and the projection functional P(t) defined for the solution (u(t), v(t)) to (1.1) by More precisely, this is stated as Remark 1.4. The projection functional P is indeed a two dimensional analogue of the scalar version in [9], where Kutev, Kolkovska and Dimova proved the same result for Klein-Gordon equation with more general nonlinearities. One can also find the motivation for introducing the scalar version of the functional P(t) in [9].
Remark 1.5. The advantage of Theorem 1.3 is that the initial energy can be arbitrarily large, in particular can be larger than that of ground states. Indeed, there have been results of blowing up for solutions with energy above the ground states, see [14,16]. What's more, in these two works, Nakanishi and Schlag gave the qualitative classification of solutions with energy slightly above the ground states.
In our case, datum with arbitrarily large energy, that leads to blowing up solution, is constructed in Section 6.
As mentioned at the end of Subsection 1.1, the key point in the proof of Theorem 1.3 is to establish a differential inequality as (1.5). We also argue by contradiction, assuming the solution does not blow up in finite time. In order to derive a contradiction, we carry out the following strategy.
(1) We first show that K(t) < 0 for all t ≥ 0 (see Lemma 4.2). This is proved by a contradiction argument. By the assumption we get K(0) < 0. Since K(t) is continuous in t (see Proposition 2.1), we can assume t 0 > 0 is the first time at which K vanishes. By Lemma 4.1, y(t) and d dt y(t) are strictly increasing on [0, t 0 ], and hence P(t) is well defined and is strictly increasing on [0, t 0 ]. We can then rewrite K(t 0 ) in terms of E(t 0 ) (see (4.7)) to obtain by using the energy conservation This is strictly negative and hence 0 = 1 4 K(t 0 ) < 0, a contradiction. (2) Using fundamental theorem of calculus and the monotonicity of d dt y(t), we can show (see Lemma 4.1). (3) By rewriting K(t) in terms of E(t) and using the conservation of energy once again (see (4.7)), we get It then follows from Cauchy-Schwarz inequality that We can use Levine's concavity argument (Lemma 2.2) to see that the solution blows up in finite time.
1.3. Criterion in the spirit of Payne and Sattinger. Our third criterion will be given with the help of the stationary solutions (φ, ψ) to (1.1), that is (φ, ψ) solves the equation If restricting β ≥ 0, then it follows from Proposition 3.5 in [20] that (1.12) has a non-trivial solution φ ,ψ , each component of which is non-negative, spherically symmetric and nonincreasing in radial direction. Consequently the lowest mountain pass level defined by is strictly positive. We use d to define the following two subsets of initial datum: We are now in a position to state our third criterion.
, v(t)) the corresponding local solution to (1.1). Then we have the following dichotomy is going to blow up in finite time.
Remark 1.7. It is easy to extend results in Theorem 1.6 to the following general system where µ 1 , µ 2 > 0 and β ≥ 0. Since the corresponding system of elliptic equations (see [20,Proposition 3.5]) has the lowest mountain pass level d being positive and possesses a constrained characterization of d, one has no difficulty in going through our proof for (1.15).
As far as the authors know, the more general systems of elliptic equations like (1.12) have been studied in [3,4,18,21,23] and [24], where nontrivial solutions are proven to exist. Thus as long as one can show d has a constrained characterization, we might expect to extend the results in Theorem 1.6 to the corresponding system of Klein-Gordon equations, by going through our argument in Section 5.
Remark 1.8. The result in Theorem 1.6 also holds for a modification of (1.1) by adding the global damping, see [25]. See also [27] for results about the scalar equation with damping term. Remark 1.9. By combining Payne-Sattinger's potential well argument with Kenig-Merle's compactness-rigidity argument, Xu [26] classified the solutions to some coupled Schrödinger system with energy below the ground states.
In order to prove Theorem 1.6, we shall give a new characterization of d. We define the functional is attained by some element in N ′ and it equals d. Remark 1.11. Theorem1.10 is indeed a generalization of Theorem 3.5 in [20], where N ′ is replaced by the Nehari manifold. It is as well a generalization of the scalar version, see [15]. Remark 1.12. As is mentioned in Remark 1.2, we can have results in Theorem 1.6 only for β ∈ [0, ∞). This is because we prove this theorem by using Theorem 1.10, which is stated only for β ≥ 0. In fact, we can not lower down the value of β strictly below zero by only using our argument. Therefore, it is curious for us whether or not the result in Theorem 1.10 holds for some β < 0.
An immediate corollary of Theorem 1.10, is the following minimal property of d.
Sending n to infinity and using the assumption K[u n , v n ] → n→∞ 0, we get Since (u n , v n ) H 1 ×H 1 0 and K[u n , v n ] < 0 for each n, we have (u n , v n ) ∈ N ′ . Consequently by Theorem 1.10, we have Combining the inequality above with (1.19), we get (1.17). This completes the proof.
We now sketch the proof of Theorem 1.6. As in [17], we argue by contradiction, assuming that the solution exists for all time. Using Corollary 1.13, we can show E is invariant under the flow of (1.1). It then follows that the differential expression is strictly positive and hence y(t) is convex on [0, ∞). By using Corollary 1.13 once again combining with Claim 5.9, we show that there exist a large number t * > 0 so that y(t) > 0 for t ≥ t * . This together with the strict convexity of y implies that y(t) → t→∞ +∞, in particular there exists a large number t * * > 0 so that for some δ > 0. By rewriting the expression of d 2 dt 2 y(t) in terms of E(t) and using the conservation of energy, we get Substituting (1.22) into (1.23), we get an inequality in analogue of (1.4). Consequently we can use Levine's concavity argument to get the contradiction, which will close our argument. We end this introductory part by describing the organization of this article. In Section 2, after giving some basic notations, we will establish the local Cauchy theory for (1.1) in the energy space. In Section 3, we will use Levine's concavity argument (see Lemma 2.2) to prove Theorem 1.1. In Section 4, we will prove Theorem 1.3. In Section 5, we will first show that (1.1) has nontrivial stationary solutions, which we will use to define the lowest mountain pass level d; next we will give a new characterization of d; using this new characterization, we will prove Theorem 1.6. In the last section, we will construct datum that satisfy each assumption in Theorem 1.1.

PRELIMINARIES
We devote this section to specifying some basic notations, with which we are going to work, and some basic results, which are basis for our forthcoming analysis.
Let L 2 be the space of square integrable real-valued functions defined on R 3 and H 1 the subspace of L 2 , consisting of elements whose gradients are also square integrable. Denote H := H 1 × L 2 . We equip the inner product on the product space H × H by setting Denote by · H×H the norm induced by this inner product. More generally, for an Hilbert space h, we will denote by ·, · h the inner product on it.

Moreover, the 'mass' functional y(t) is differentiable of the first and the second order for almost every t ∈ [0, T max ).
Proof. We rewrite (1.1) in the integral vectorial form Taking the H 1 × H 1 -norm on both sides of (2.1), we get for each T > 0 Applying Hölder inequality and then the Sobolev embedding H 1 ⊂ L 6 , we can bound Here C is a positive constant depending on the Sobolev embedding and β. If we take R = 2 (u 0 , u 1 ) Inserting the following algebraic relations , and using Hölder's inequality and Sobolev embedding Inserting the following algebraic relations v], and using the same argument, we obtain Substituting (2.6) and (2.7) into (2.5), and using Young's inequality, we get Up to now, we have constructed a unique solution (u(t), v(t)) in a small time interval. We next try to extend this solution onto a maximal time interval.
At time T , as long as ((u(T ), ∂ t u(T )), (v(T ), ∂ t v(T ))) H×H is finite, we can repeat the Banach fixed point argument, beginning at the time T , to extend the solution to live on a longer time interval, say [0, T +T 1 ] for some T 1 > 0 depending on ((u(T ), ∂ t u(T )), (v(T ), ∂ t v(T ))) H×H .
Repeating this process, we can extend the solution to live on the maximal time interval [0, T max ).
From the proof of the existence of local solutions, we know that the solution (u(t), v(t)) has its derivatives (∂ t u, ∂ t v) ∈ L ∞ ([0, T max ); L 2 × L 2 ). Hence y(t) is differentiable for almost every t ∈ [0, T max ). If we use the integration by parts, we can see that y(t) is also differentiable of second order, for almost every t ∈ [0, T max ).
We next recall two technical lemmas. The first one embodies the concavity argument, due to Levine (see [10]).
The second one is about Schwarz spherical rearrangement (see [2,Appendix III.3] and [11,Chapter 3] for the proof).
3. PROOF OF THEOREM 1.1 Recall the system we are going to consider is Here β ∈ R is a fixed parameter. Let (u 0 , u 1 ), (v 0 , v 1 ) ∈ H, then we can apply Proposition 2.1 to see that the equation (3.1) has a unique solution on the interval [0, T max ). Our goal here is to give some conditions on the initial datum so that T max is finite, i.e., the corresponding solution blows up in finite time.
Recall that the energy functional E(t) and the 'mass' functional y(t) are defined respectively in (1.2) and (1.3). By Proposition 2.1, we can differentiate y(t) in t-variable to obtain is the Nehari functional defined as in (1.14).

By definitions of the energy functional E(t) = E[u, v](t) and the Nehari functional K(t) := K[u, v](t), we can deduce by some algebraic calculations
Here γ is some parameter, which we are going to specify. But for γ ∈ (0, 1], if we assume β ≥ −1, the last term on the right hand side of this equality is non-negative so that we can control the kinetic energy by some proper combination of E and K. More precisely, we have the following result.
For each γ ∈ (0, 1], we then have With this lemma at hand, we are now ready to give Proof of Theorem 1.1. We argue by contradiction. Assume the solution does not blow up in finite time. Then at any time t, applying Cauchy-Schwarz inequality in (3.2), we get Recall that the differential expression (3.3) of dy dt reads . In order to treat the term −K(t), we invoke Lemma 3.1, to conclude that K(t) ≤ 2(1 + γ)E(t) for any γ ∈ (0, 1]. Together with the conservation of energy, we get Inserting this into the above differential equality (3.6), we get . It then follows that d dt y(t) tends to +∞ as t approaches infinity and hence so does y(t). In order to obtain the contradiction, we use (3.4) and the conservation of energy to conclude from (3.3) that . By (3.5), we bound this by Cauchy-Schwarz inequality from below We then apply Levine's lemma 2.2 to conclude that y(t) blows up before some finite positive time. But this contradicts the assumption that the solution exists globally. So the solution (u(t), v(t)) blows up in finite time. This finishes the proof of asserted result under the condition in (i).
Next we assume E(0) = 0 and (u 0 , v 0 ), (u 1 , v 1 ) > 0. As in the first item, we argue by contradiction. Assume the solution does not blow in finite time. By (3.7), we have Solving this differential inequality, complemented with the initial condition d dt This infers that there exists some t b > 0 so that y(t) > 0 for all t > t b . This allows us to rewrite (3.8) as Here c is some positive constant, for instance we can take We then apply Levine's lemma 2.2 to conclude that y(t) blows up before some finite positive time, which contradicts the initial assumption. Thus the solution blows up in finite time under the condition in (ii). This completes the proof of Theorem 1.1.

PROOF OF THEOREM 1.3
We devote this section to proving Theorem 1.3. Let (u 0 , u 1 ), (v 0 , v 1 ) ∈ H and (u(t), v(t)) , t ∈ [0, T max ) be a solution to (3.1). We define the projection functional P(t) as in (1.6) and the mass as in (1.3). As long as y(t) 0, we can rewrite the energy function E(t) as In order to prove Theorem 1.3, we first prove two lemmas. y(0) . Proof. We use (3.3) and the assumption K(t) < 0 to obtain which is just the asserted result in (iii). Consequently dy dt (t) is strictly increasing on [0, T ]. Solving this differential inequality together with the initial condition This is indeed the first one of the stated results in (i). What's more, as a consequence of the fundamental theorem of calculus, y(t) is increasing in t ∈ [0, T ]. Combining this with the non-negativeness of y(0), we also get y(t) > 0 for t ∈ (0, T ]. Thus the projection functional P in (1.6) is well defined for t > 0. Differentiating P in the time variable t and using (3.2), we obtain dP we use the assumption K(t) < 0 and apply Cauchy-Schwarz inequality to conclude that dP dt (t) > 0, which is just the second part of the stated results in (i). Consequently P(t) is increasing in t ∈ [0, T ]. It remains to show the last item (iv). On the one hand, by fundamental theorem of calculus, we have for any t ∈ [0, T ] Summing (4.5) and (4.6), we get From this, we see that for t ≥ t b := 1 2 dy dt (0) y(0) , the value of the term before the sign '≤' is negative, which is just the desired result in (iv). This finishes the proof of Lemma 4.1.
We next affirm that the assumption K(t) < 0 in Lemma 4.1 is indeed true.
We next show K(t) < 0 for t ∈ [0, T max ). It follows from Proposition 2.1 that K(t) is continuous in the time variable t. We argue by contradiction, assuming that there exists t 0 , so that K(t 0 ) = 0 and K(t) < 0 for t ∈ [0, t 0 ). Observe that t 0 > 0, thanks to the continuity of K(t) and K(0) < 0. This allows us to apply Lemma 4.1 on [0, t 0 ] to obtain Thus P(t) is well-defined for any t ∈ [0, t 0 ). By invoking to the defintions of K(t) and E(t), we have (4.7) This, together with the energy conservation law and the third assumption in Theorem 1.3, infers that (4.8) which is strictly negative since y(t) and P(t) are strictly increasing in t ∈ [0, t 0 ] by (ii) in Lemma 4.1. This contradicts the assumption that K(t 0 ) = 0. So we have that K(t) < 0 for any t ∈ [0, T max ). It remains to show (ii). It follows from (4.7) and the third assumption in Theorem 1.3 that for any t ∈ [t b , T max ) We are now ready to give Proof of Theorem 1.3. We argue by contradiction, assuming T max = ∞. By Lemma 4.2, we have K(t) < 0 for any t ∈ [0, ∞). This in turn allows us to use Lemma 4.1 to conclude that

Substituting this into the differential expression (3.3), it follows that for
By multiplying both sides by y(t) and using Cauchy-Schwarz inequality, we see that y(t) satisfies the differential inequality At time t = t b , it follows Lemma 4.1 that y(t b ) > 0 and d dt y(t b ) > 0. We can then apply Lemma 2.2 with to conclude that the solution u(t, x) blows up in finite time. 14
We shall give some sufficient conditions on the initial data (u 0 , u 1 ), (v 0 , v 1 ) so that the local solution can not be extended to a global one. The criterion we give here is in the spirit of Payne and Sattinger, with the help of stationary solutions to (1.1).

Remark 5.2. This result is indeed the analogue of the scalar version
Proof. We begin the proof by claiming Indeed, if this was not the case, we would have R 3 φ 4 dx = R 3 ψ 4 dx = 0, thanks to the assumption β > −1. We are first going to see what happens to φ. Since φ ∈ L 2 , for any δ > 0, there exists R > 0 such that φ L 2 (|·|≥R) < δ. On the bounded domain {x ∈ R 3 : |x| < R}, we use Hölder inequality to obtain which is 0, thanks to the assumption R 3 φ 4 dx = 0. By combining these two results, we see that φ L 2 (R 3 ) < δ. Since δ can be taken to be arbitrarily small, φ L 2 (R 3 ) = 0. To proceed, we shall show ∇φ L 2 = 0. For this we invoke the notion of Fourier transform. Letφ(ξ) be the Fourier transform of φ, then φ 2 L 2 = φ 2 L 2 = 0 and φ 2 H 1 = ξφ 2 L 2 + φ 2 L 2 , thanks to our assumption that φ ∈ H 1 . For M a parameter to be specified later, we decompose
By Lemma 5.1, j (φ,ψ) (λ) attains its maximum at some finite λ. This suggests us to renormalize (φ, ψ) so that at λ = λ * := 1 we have The left hand side is indeed the expression K[φ, ψ] defined by(1.14). Furthermore, the functional K enjoys as well the mountain-pass geometry (see [22] for detailed notions). This lemma can be proved by the same argument as that in Lemma 5.1. So we omit details. We emphasize here that Lemma 5.3 will play an important role in the proof of Lemma 5.5.
We are in a position to find the lowest mountain pass level for (5.1). If β ≥ 0, then we apply [20, Proposition 3.5] with λ 1 = µ 1 = µ 2 = 1 to conclude that (5.1) has a nontrivial solution φ ,ψ , each component of which is non-negative, spherically symmetric and decreasing.
Since φ ,ψ is nonzero and β is assumed to be non-negative, the quantity, known as the lowest mountain pass level, is strictly positive.
In order to use d to give some criterion of the blowing up for solution of (1.1), we need the characterization of d from the view point of the constrained minimization (see [15] and [14] for the scalar case). More precisely, we define the functional Define the constrained set to be We will consider the attainability of the quantity . Indeed, this is affirmed in the following result. Note that Theorem 1.10 follows directly from this lemma.
Proof. We follow the lines in the proof of Proposition 3.5 in [20]. Let {(v m,1 , v m,2 )} ⊂ N ′ be a minimizing sequence for d 1 , then we have In order to capture the compactness result from this boundedness, we invoke to the Schwarz symmetrization. For each m, let v * m,i be the Schwarz spherical rearrangement of v m,i , i = 1, 2. On the one hand, for each m and each i ∈ {1, 2}, v * m,i is a radial function. On the other hand, by Lemma 2.3, we have . Combining these two points with (5.9), we see that Here H 1 r is the subspace of H 1 , consisting of functions that are spherically symmetric. Thus, by [2, Theorem A.I ′ ], we conclude that there exists a couple (v * , r and strongly to (v * ,1 , v * ,2 ) in L 4 × L 4 (up to subsequence. Here we denoted this subsequence still by {(v * m,1 , v * m,2 )}). We shall show (v * ,1 , v * ,2 ) ∈ N ′ . Applying lower semi-continuity and then using Lemma 2.3, we first obtain Combining this with (5.11), we get We next bound the right hand side of (5.12). For each m, by applying Lemma 2.3 with p = 4 and using our assumption β ≥ 0, we have By the strong convergence of {(v * m,1 , v * m,2 )}, we see In order to get the convergence of the remaining part of the right hand side of (5.13), we do the following calculations Using Hölder's inequality, we can bound I as follows , which tends to 0 as m tends to infinity, thanks to the strong convergence of Applying a similar argument to II, we can see that By sending m to infinity on both sides of (5.15) and using (5.17) and (5.18), we obtain that v * m,1 converges strongly to v * ,1 2 v * ,2 2 in L 1 . Consequently we get By sending m to infinity on both sides of (5.13), and using (5.19) and (5.14), we see Substituting this equality into (5.12), we get From this, it follows that K[v * ,1 , v * ,2 ] ≤ 0.
It remains to prove (v * ,1 , v * ,2 ) (0, 0) in H 1 × H 1 . Since { v m,1 , v m,2 } ⊂ N ′ , we have for each m We bound further this inequality as follows where we used Hölder's inequality and the elementary inequality 2ab ≤ a 2 + b 2 in the first inequality, and the Sobolev embedding H 1 ⊂ L 4 in the second inequality. Here the positive constant C depends on the Sobolev embedding and β. Combining (5.22) and (5.23), we get for each m Thus, we can cancel out the common factor (v m,1 , v m,2 ) 2 H 1 ×H 1 on both sides of (5.24) to conclude .
We turn to the proof of second part. We note that if . Then by definitions of J and E 1 , we get To show the reversed inequality, we argue by contradiction by assuming K[v * ,1 , v * ,2 ] < 0. Since (v * ,1 , v * ,2 ) (0, 0) in H 1 × H 1 , it then follows from Lemma 5.3, that there exists a constant λ ∈ (0, 1), such that K[λv * ,1 , λv * ,2 ] = 0. Thus we have Let's continue the proof of Proposition 5.6. By Claim 5.7, we can take the sequence {t n } so that (u(t n ), v(t n )) (0, 0) in H 1 × H 1 . Then by Corollary 1.13, we get On the one hand, by the definition of E, the RHS of this inequality is strictly bigger that E(t 0 ). On the other hand, by definitions of J and E, the LHS is smaller than E(t 0 ). Combining these two points, we get d > d, which is impossible. This contradiction implies that ((u(t), ∂ t u(t)), (v(t), ∂ t v(t))) can not leave E, which completes the proof of Proposition 5.6.
Claim 5.8. As long as the solution (u(t), v(t)) exists at time t, we have K[u(t), v(t)] ≥ 0.

By Claim 5.8 and the definition of K[u(t), v(t)], we obtain
which in turn, together with the definition of the energy, implies Together with the energy conservation, this implies as long as the solution (u(t), v(t)) exists at the time t. Then by Proposition 2.1, we can extend the local solution onto a larger time interval. If we repeat the above argument on the extended time interval, we can extend the solution to be defined on the whole time line. Therefore, to finish the proof of (i), it remains to give Proof of Claim 5.8. Assume by contradiction, there was some time t 0 so that K 0 [u(t 0 ), v(t 0 )] < 0. We note that t 0 0, because we have by the assumption ((u 0 , u 1 ), is continuous in t, there exists a maximal interval (t − , t + ) containing t 0 , on which K[u(t), v(t)] < 0. It is clear that 0 is not in (t − , t + ), so that we can first assume t + is close to 0. From the ordered structure of the real line, we get from this that t + < 0 and hence t 0 < 0. If we restrict the solution (u(t), v(t)) of the equation (3.1) onto time interval [t 0 , 0], we see that ((u(t), ∂ t u(t)), (v(t), ∂ t v(t))) will enter W at time t = 0, before which it is in E. This contradicts the invariance of E under the flow of (3.1). Thus we are left with the case that t − is close to 0. In this case, by once again the order structure of the real line, we get from this that t − > 0 and hence t 0 > 0. If we restrict the solution (u(t), v(t)) of the equation (3.1) onto time interval [0, t 0 ], we see that ((u(t), ∂ t u(t)), (v(t), ∂ t v(t))) will enter W at time t = 0 in the reversed time direction, after which it is in E. Since (3.1) is invariant if the time is reversed, this also contradicts the invariance of E under the flow of (3.1). These two contradictions show that there is no such time t 0 at which K 0 [u(t), v(t)] < 0. Hence this completes the proof of Claim 5.8. 21 We next prove the second assertion (ii). Let ((u 0 , u 1 ), (v 0 , v 1 )) ∈ E and (u(t), v(t)) its corresponding solution to (1.1). We argue by contradiction. Assume the solution exists for all time.
By Proposition 5.6, we see that ((u(t), ∂ t u(t)), (v(t), ∂ t v(t))) remains in E for any t ≥ 0. Consequently we have K[u(t), v(t)] < 0. Substituting this strict inequality into (3.3), we get that for any t ≥ 0. From this we know that y(t) is a strictly convex function of t. What's more, we have the following strong result.
Claim 5.9. There exists a time t * , such that for all t ≥ t * we have Proof of Claim 5.9. We argue by contradiction. Assume d dt y(t) ≤ 0 for all t ≥ 0. Under this assumption, if y(0) = 0, then we have y(t) = 0 for all t ≥ 0. Consequently d 2 dt 2 y(t) = d dt y(t) = 0 for all t ≥ 0. But this contradicts (5.34). Thus in the following, we assume y(0) > 0.
Since y(t) is convex and decreasing, we may assume that y(t) converges to some nonnegative number A.
Admitting this claim, we can take a subsequence (t n ) with t n → ∞ as n tends to infinity, such that and (5.36) d 2 dt 2 y(t n ) → 0, as n tends to infinity. Using K[u(t), v(t)] < 0, we substitute (5.34) into this last limitation to obtain where in the last equality we have used the conservation of energy. On the other hand, since A > 0, we can take n large enough, say n ≥ L for some L > 0 so that y(t n ) 0. Then by Lemma 1.13, we have Recall that ((u 0 , u 1 ), (v 0 , v 1 )) ∈ E and hence d > E(u 0 , v 0 ). Using this strict inequality, we combine (5.38) and (5.39) to conclude 22 which is impossible. Thus the initial assumption is false. This finishes the proof of Claim 5.9, modulo the proof of Claim 5.10.
We now give This implies that E[u 0 , u 1 ; v 0 , v 1 ] ≥ 0. We are thus in the following two alternatives.
By using the interpolation and Sobolev inequality (or directly Gargliado-Nirenberg inequality), we get for each n (5.43) We first bound LHS of (5.43) from below. By (5.41), we can take N 1 > 0 so that for all n ≥ N 1 there holds We next bound RHS of (5.43) from above. By (5.42), we can take N 2 > 0 so that for all n ≥ N 2 we have . Since A = 0, for the same δ > 0, we can take N 3 > 0 so that for all n ≥ N 3 we have Since E[u 0 , u 1 ; v 0 , v 1 ] > 0, if we take δ > 0 small enough, we can get This is imppossible, since the data is in E.
Since the data is in E and E is invariant under the flow of (3.1) by Proposition 5.6, for each n we have the following strict inequality u 4 (t n ) + v 4 (t n ) + 2βu 2 (t n )v 2 (t n ) dx.
Combining this with (5.43), we obtain for each n Recalling that E[u 0 , u 1 ; v 0 , v 1 ] = 0 and the limitation (5.42), we take N > 0 so that for all n ≥ N there holds and hence We then see that (5.49) is impossible, since the product of two non-negative numbers can not be strictly smaller than zero. Combining Case 1 and Case 2, we see that A > 0. This finishes the proof of Claim 5.10.
Applying Cauchy-Schwarz inequality on the right hand side of (3.2), and substituting the resulted inequality into (5.53), we obtain At time t = t * * , it follows from the above discussion and Claim 5.9 that y(t * * ) > 0 and d dt y(t * * ) > 0. At this point, we can apply Lemma 2.2 with y(t) = ψ(t), t b = t * * , γ = 3 2 to conclude that y(t) becomes infinity before some finite time and hence (u(t), v(t)) blows up in finite time. This contradicts to the initial assumption that the solution exists for all the time. Thus the solution must blow up before some finite time. This completes the proof of (ii) in Theorem 1.6. 24 6. REMARKS ON THE CRITERIA From Theorems 1.1,1.3 and 1.6, we see that different conditions on initial data can lead its corresponding solution to blow up in finite time. Then it is natural to ask for the relations between these conditions. This is the first task of this section.
The second task is to give a comment on Theorem 1.3, indicating that there does exist initial data satisfying the assumptions there. The reason for us to do this is that there are too many (three, to be exact) restrictions on the datum, which forces us to make sure that they do not contradict with each other. 6.1. Comparing Theorem 1.1 with Theorem 1.6. In order to finish the first task, we first give a lemma.