Nonexistence of global solutions to Klein - Gordon equations with variable coe ﬃ cients power - type nonlinearities

: In this article, we investigate the Cauchy problem for Klein - Gordon equations with combined power - type nonlinearities. Coe ﬃ cients in the nonlinearities depend on the space variable. They are sign preserving functions except one of the coe ﬃ cients, which may change its sign. We study completely the structure of the Nehari manifold. By using the potential well method, we give necessary and su ﬃ cient conditions for nonexistence of global solution for subcritical initial energy by means of the sign of the Nehari functional. When the energy is positive, we propose new su ﬃ cient conditions for ﬁ nite time blow up of the weak solutions. One of these conditions is independent of the sign of the scalar product of the initial data. We also prove uniqueness of the weak solutions under slightly more restrictive assumptions for the powers of the nonlinearities


Introduction
The aim of this article is to study the nonexistence of global solutions to Cauchy problem for Klein-Gordon equation:

(4)
The nonlinearities (2) and (3) differ by their source term of order p s ; to be precise, the nonlinearity (2) is not an odd function in u, while (3) is an odd function in u. This distinction makes the analysis of the initial problem (1) with nonlinearity (3) easier than the analysis of (1) with nonlinearity (2), as the reader can see in the article.
Moreover, all coefficients ( ) a x k in (2) and ( Alternative assumptions to hypotheses ( ) H 1 , ( ) H 2 , and ( ) H 3 for nonlinearities (2) and (3) include signpreserving coefficient ( ) a x s . In this case, we suppose that for every ∈ x n , one of the following hypotheses holds: Note that for = s 1, we have different signs of ( ) a x 1 in (H 4 ) and (H 6 ). In the previous assumption (4), we suppose that the powers p j , = … j m 1, 2, , are in a strictly increasing order. We also consider the case of two additional hypotheses ( * Note that nonlinearities (2) and (3) with hypotheses ( ) H i , = … i 1, 2, , 6, ( ) * H 2 and ( ) * H 3 cover most popular cases, "which frequently appear in the physical or mathematical models," as mentioned on page 4 in [1].
In the last 50 decades, the global behavior of the weak solutions to Klein-Gordon equation with nonlinear term ( ) f u has been intensively investigated. In the beginning, finite time blow up of all nontrivial solutions with nonpositive initial energy ( ) ≤ E 0 0 is proved in [2,3], see also [4][5][6] for other results. Later on, after the remarkable article of Payne and Sattinger [7], the case of positive subcritical initial energy is completely solved. Here, d is the depth of the potential well (or mountain pass level), see definition (25). For different types of nonlinearities, the weak solutions blow up for a finite time when ( ) < I u 0 0 and exist globally for ( ) > I u 0 0 , where ( ) I w is the Nehari functional defined in (7). For example, when ( ) | | = − f u a u u p 1 or ( ) | | = f u a u p , > p 1, = ≠ a const 0, the global behavior of the solutions is studied in [6][7][8][9][10]. More general cases of power-type nonlinearities with constant coefficients a k , = … k m 1, , and b j , = … j s 1, , ,  are investigated, e.g., in [11][12][13].
, then weak solutions either blow up for a finite time or are globally defined and tend to the so-called ground state solutions of the elliptic part of the Klein-Gordon equation [8].
For supercritical initial energy ( ( ) > E d 0 ), the problem is far from the final state. In fact, after the pioneering papers of Straughan [15] and Gazzola and Squasssina [16], only sufficient conditions for finite time blow up of the solutions have been found. These conditions have been extended for different types of nonlinearities ( ) f u , see, for example, [17][18][19][20][21], as well as [22] for the damped Klein-Gordon equation. As for hyperbolic problems with nonlinearities ( ) f x u , with variable coefficients, there are only few results. Existence of global solutions for one-dimensional case ( = n 1) is proved in [23] for small initial data and decaying coefficients ( ) a x and ( ) V x for the equation: Asymptotic stability, scattering, and decay for solutions to the one-dimensional Klein-Gordon equation with variable coefficients are studied in [24][25][26][27][28]. In [29], global existence of solutions to (1) with small data and singular nonlinearities ( ) is investigated in [30,31], when the initial data and the potential ( ) > V x 0 are radially symmetric functions. By means of the potential well method for positive subcritical energy, global existence and finite time blow up of the solutions is established, depending on the sign of the Nehari functional ( ) I u 0 . The damped Klein-Gordon equation with compactly supported initial data is studied in [32,33], where existence of global solutions is proved without any sign conditions of the nonlinear term.
Let us also mention paper [34] for the Klein-Gordon equation with nonlinearities, whose coefficients depend on a new variable, different from the space variable of the original problem.
Equation (1) with constant coefficients in the nonlinearity (2) or (3) models propagation of longitudinal strain waves in isotropic compressible elastic rods, see [38]. The quadratic-cubic nonlinearity appears in dislocation of crystals [11], while cubic-quintic nonlinearity appears in particle physics [39]. The study of asymptotic stability of kink-type solutions in the classical field theory leads to Klein-Gordon equation with quadratic and cubic nonlinearity with variable coefficients, see [23,26,27].
In the present article, we extend the results for nonexistence of solutions to problem (1) with constant coefficients in the nonlinearities (5) to the wider class of Klein-Gordon equations with variable coefficients power-type nonlinearities (2) or (3). The coefficients in the nonlinearitiy ( ) f x u , have a constant sign, except one of them, which may change its sign (see hypotheses H 1 , H 2 , and H 3 ). The case of constant sign of all variable coefficients is also treated in the article (see hypotheses H 4 , H 5 , and H 6 ), as well as the case of nonlinearity (2) with two coinciding powers (see hypotheses * H 2 and * H 3 ). We investigate in details the structure of the Nehari manifold and give necessary and sufficient conditions when the Nehari manifold is not empty. As a consequence, we obtain that the depth of the potential well is positive. By means of the potential well method, we propose necessary and sufficient conditions for nonexistence of global weak solutions with nonpositive or positive subcritical initial energy. For supercritical initial energy, we establish two sufficient conditions for nonexistence of global solutions. One of them does not depend on the sign of the scalar product of the initial data. We discuss relationships between two quantities of blowing up solutionsthe sign of the Nehari functional and the value of the initial energy.
For a single power nonlinearity, our results are valid without restriction on the sign of the nonlinearity coefficient. We also prove uniqueness of the weak solutions under slightly more restrictive assumptions for the powers in the nonlinearities and without any assumptions on the signs of coefficients.
There exist several results for nonexistence of global solutions to Klein-Gordon equations with constant coefficient nonlinearities, but our results are the first blow up results when the combined power-type nonlinearities have variable coefficients (including a sign-changing one).
This article is organized in the following way. In Section 2, some notations and definitions, as well as the idea of the improved concavity method of Levine, are given. Section 3 deals with the properties of the Nehari functional. The main results are formulated in Section 4 and proved in Section 6. Some discussion of the main results of the article is given in Section 5.

Preliminaries
In this section, we collect some preliminary results, which will be used in the rest of the article.

Notations and definitions
For functions depending on t and x, we use the following short notations: is a weak solution to (1), (2), The definition of a weak solution to (1) with nonlinearity (3) remains the same as the aforementioned definition for the nonlinearity (2).

Definition 2. The solution ( )
u t x , of problems (1), (2), and (4) (or to (1), (3), (4)) defined in the maximal The solution ( ) u t x , to Klein-Gordon equation conserves the energy E, i.e., Here, the energy E is defined as follows: We define two important functionals for the potential well method associated with the considered problem: the potential energy functional ( ) J w The Nehari manifold N , defined by contains all the nontrivial critical points of ( ) J u , because ( ) ( ) ′ = J u I u . Since the coefficient ( ) a x s may change its sign, we eliminate the integral that includes this coefficient, from the expressions ( ) J w and ( ) I w . We obtain the following important identity valid for both nonlinearities (2) and (3): In view of (4) and the constant sign of the remaining coefficients in (2) and (3), under each of hypotheses H 1 , H 2 , or H 3 (or H 4 , H 5 , or H 6 for the case of sign-constant coefficients), we conclude that When the functionals J , I , and B are evaluated on the solution ( ) ⋅ u t, to problem (1), then we use the short notations: By using (8), the conservation law (6) can be rewritten in the following way Klein-Gordon equations with power-type nonlinearities  5 Further on, we use the Sobolev imbedding theorem with constant C q :

Improved concavity method of Levine
In the proofs of our main theorems in the next sections, we use the concavity method of Levine [3].
, 2 satisfies a second-order ordinary differential inequality is a concave one, i.e., ( ) ″ < z t 0, which gives the name of the method. Later on, inequality (13) is generalized by Straughan [15], by Kalantarov and Ladyzhenskaya [40] and by Korpusov [41]. Let us mention that the concavity method is used in the proof of finite time blow up of the solutions to both nonlinear evolution equations and hyperbolic-parabolic systems (see [42] and the references therein).
For the application of the concavity method, we suppose by contradiction that there exists a global weak solution ( ) give us the following identities for the derivatives of ( ) t Ψ : The proofs of our main theorems follow from the corresponding results for ordinary differential equations (17) and (18). We recall our Theorem 2.3 in [43] and Theorem 3.2 in [14], which are extensions to the concavity method.
max max (17) or to the problem 3 Properties of the Nehari functional

Nonlinearity (2)
In this section, we study problem (1) with nonlinearity (2) and assume that one of hypotheses H 1 , H 2 , or H 3 holds. We begin with some basic properties of the Nehari manifold N .
. Suppose that (4) and one of hypotheses H 1 , H 2 , or H 3 hold for nonlinearity (2). Then there exists a number ≠ * λ 0 such that ∈ * λ w N iff: (a) for < s m, one of the following two possibilities is true, either Moreover, if either (19) or (20) with is true, then * λ is a unique negative number.
(b) for = s m, the following relation holds, then * λ is a unique positive number, while for Proof. Proof of sufficiency.
Let λ be any nonzero number. Then from (7), we obtain Hence, the properties of ( ) I λw are determined by the properties of the function ( ) g λ defined by Our aim is to find * λ such that ( ) = * g λ 0.
We study the behavior of ( ) If condition (19) holds, then at least one term in the sum (19) is positive (note that all terms in this sum are nonnegative), and hence, . Thus, from the continuity of ( ) g λ with respect to λ, there exists a positive number * λ such that ( ) = * g λ 0, i.e., ( ) = * I λ w 0. Let assumptions (20) hold, i.e., all terms in the first sum in (20) are zero, and hence, , and once again, there exists a positive number * λ such that ( ) = * g λ 0. In the other case, i.e., The proof of sufficiency in the case = s m coincides with the proof of the last part in case (20). The sufficiency is proved. Now we shall prove the uniqueness of positive (or negative) parameters * λ satisfying ( ) = * If, for example, > λ λ 2 1 , then the lhs of (23) is strictly positive, and from assumptions H 1 or H 2 , we obtain that the first sum in the rhs of (23) is nonpositive, while the second sum in the rhs is strictly negative due to (19), which is impossible. Hence, the positive multiple * λ of w, for which ( ) = * g λ 0, is unique.
Then, from H 1 or H 2 and the first equation in (20), we have that all terms in the sum ∑ = + k s m 1 of (20) are identically zero. By applying the same procedure as in the previous case, we conclude that equation (23) is reduced to the following one: The case (20) with is treated similarly to the previous one noting that the parameter * λ is a negative number. For = s m, the proof of uniqueness of positive/negative number * λ (depending on the sign of the integral in (21)) follows the same steps as the second part of case (20). The sufficiency is proved.

Proof of necessity.
Let   (21) is satisfied for the chosen function ( ) w x . Once again from Lemma 1, we conclude that a positive multiple of w lies in N.
Now we recall the definition of the critical energy constant (or the mountain pass level) d as follows: In the next lemma, we prove the boundedness from below of the critical energy constant d.
and the Nehari manifold N is bounded away from zero, i.e., Proof. Under the assumptions of this lemma and ( ) = I w 0, we obtain the chain of inequalities where C q is the Sobolev imbedding constant defined in (12).
From (28) Thus, for every function ∈ w N , its norm || || w 1 is bounded from below by a positive constant M, i.e., statement ( ) i is proved. By using (8), (10), and (29), we have for every ∈ w N Remark 2. Note that Corollary 1 and Lemma 2 guarantee that the depth of the potential well is positive and the potential well method is applicable in this case. Proof. First, we will show that one of conditions (19) or (20) is satisfied.
Let < s m. We follow the proof of necessity in Lemma 1. From H 1 or H 2 , we have that

Nonlinearity (3)
In this section, we investigate the properties of the Nehari functional for nonlinearity (3). The results are similar to the results for nonlinearity (2). For the reader's convenience, we formulate them below. The proof of Lemma 4 can be established following the ideas from Lemma 1, and we omit it.
Lemma 5. The conclusions in Lemmas 2 and 3, as well as in Corollary 1, remain true when we replace the nonlinearity (2) with the nonlinearity (3).
The proofs of Lemmas 2 and 3 for nonlinearity (3) are similar to the proofs for nonlinearity (2), and we omit them. (2) or (3) In this section, we introduce two important sets for the potential well method:

Sign-preserving properties of the Nehari functional for nonlinearity either
For weak solutions of (1) with nonlinearity either (2) or (3) and with initial energy ( ) < E d 0 , we formulate sign-preserving properties of the Nehari functional ( ) I w .
Theorem 2. Suppose (4) holds and one of hypotheses H 1 , H 2 , or H 3 is true. Assume ( ) u t x , is a weak solution of (1) with nonlinearity either (2) or (3), defined in the maximal existence time interval Let the initial energy be subcritical, i.e., ( ) < E d 0 . Then: .
. Suppose by contradiction that there exists some t 1 such that ( ) ⋅ ∉ u t W , 1 .
From the continuity of ( ) u t x , with respect to t, there exists some time t 2 , ≠ t t 2 1 , such that ( ) which contradicts our assumption ( ) < E d 0 . Hence, ( ) > I t 0 for every [ ) ∈ t T 0, max , i.e., ( ) ⋅ ∈ u t W , and the statement (i 1 ) is proved.
For the proof of (i 2 ), we suppose by contradiction that ( ) 1 , which contradicts (32). Statement (i 2 ) is proved. The proofs of ( ) ii and ( ) iii follow from the conservation law (11), written in the form (10)). In case (ii), ( ) < E 0 0; thus; ( ) < I t 0 holds from (33). In case (iii), ( ) = E 0 0, and by (33), we obtain ( ) ≤ I t 0. We proceed similarly to Lemma 2(i): from ( ) ≤ I t 0, we conclude  1 holds. We claim that if for some 4 1 , which contradicts the aforementioned conclusion that || ( )|| The proofs of all theorems are given in Section 6.  In the following theorem, we give two sufficient conditions for nonexistence of global solutions provided positive initial energy.

Case of hypotheses
For the particular case of nonpositive initial energy ( ) ≤ E 0 0, we have the following corollary. In all previous considerations, we assume that nonlinearities (2) or (3) include at least two power-type terms. In the following, we state a blow up result valid for a single power-type nonlinearity with arbitrary sign of the variable coefficient ( ) a x 1 . The result in this corollary is a consequence of Theorems 3 and 4 and Corollary 2.   Finally we state a result for the case of parameters satisfying either hypothesis * H 2 or hypothesis * H 3 , i.e., when two consecutive powers in (4) coincide.

Case of hypotheses H 4 , H 5 , or H 6
In this Section, we suppose that all variable coefficients in the nonlinearities (2) or (3) are sign-preserving.

Uniqueness of weak solutions
In the following theorem, we state uniqueness of the weak solutions to (1) under general power-type nonlinearity   (38) and conditions (39), has at most one weak solution ( ) ∈ u t x , .

Discussion
We discuss here the connection between the negative sign of the Nehari functional ( ) I t and the different values of the initial energies for solutions, which blow up for a finite time.  , we conclude that ( ) ≡ u t x , 0 0 . From Theorem 3 ( ) iii , problem (1) has no blowing up solution, which contradicts the assumption of the corollary. Hence, assertion ( ) i is true. To prove assertion ( ) ii , we rewrite (11) in the following way We apply (41) at the initial moment = t 0, and using (36), we obtain the desired estimate ( ) < I 0 0 as follows: By integrating (42) twice, we obtain i.e., ( ) t Ψ blows up at infinity. Moreover, the representation (16) of ( ) ″ t Ψ gives us the equalities Let us denote the rhs of the last equation by ( ) G t , i.e., Since from Theorem 2( ) We conclude from Theorem 1 that ( ) u t x , blows up for finite time < ∞ T max . This contradicts our assump- T 0, max . Using (11) and (10) where Since the requirement (35) is equivalent to from (45)   The proofs of Theorems 2, 5, and 6 follow the same arguments as the corresponding proofs of Theorems 2, 3, and 4 for a sign-changing coefficient, and we omit them.
Proof of Theorem 7. The proof of this theorem follows from the energy conservation law , e x p .

Conclusion
We study the Cauchy problem for Klein-Gordon equation with two combined power-type nonlinearities. All coefficients of the polynomials depend on the space variable and except one have a prescribed sign; the remaining coefficient may change its sign. One novelty in this article is the treatment of the nonlinearity which is not odd in u, including in this way the important quadratic-cubic nonlinearity. We investigate in details the properties of the Nehari manifold and apply the potential well method of Payne and Sattinger to investigate completely the blow up of ( ) L n 2 norm of weak solutions with subcritical initial energy. On the basis of the improved concavity method of Levine, for weak solutions with supercritical initial energy, we develop two new sufficient conditions, that guarantee nonexistence of global solutions. One of these sufficient conditions does not depend on the sign of the scalar product of initial data. For blowing up weak solutions, we examine carefully the relationship between the sign of the Nehari functional and the value of the initial energy. For Klein-Gordon equations with general variable coefficients in the nonlinearities, we prove uniqueness of the weak solutions.
Our results for nonexistence of global solutions to Klein-Gordon equations are the first blow up results where the combined power-type nonlinearities in the equations have variable coefficients (including a signchanging one).