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Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities

  • Natalia Kolkovska EMAIL logo , Milena Dimova and Nikolai Kutev
Published/Copyright: May 31, 2023
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Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

In this article, we investigate the Cauchy problem for Klein-Gordon equations with combined power-type nonlinearities. Coefficients in the nonlinearities depend on the space variable. They are sign preserving functions except one of the coefficients, which may change its sign. We study completely the structure of the Nehari manifold. By using the potential well method, we give necessary and sufficient 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 sufficient conditions for finite 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.

Keywords: Klein-Gordon equation; nonlinearities with variable coefficients; blow up; Nehari manifold; uniqueness
MSC 2010: 35B44; 35A24; 35L71

1 Introduction

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

(1) u t t − Δ u + u = f ( x , u ) , t > 0 , x ∈ R n , u ( 0 , x ) = u 0 ( x ) ∈ H 1 ( R n ) , u t ( 0 , x ) = u 1 ( x ) ∈ L 2 ( R n ) , x ∈ R n ,

with a nonlinear term f ( x , u ) . We consider the combined power-type nonlinearity

(2) f ( x , u ) = ∑ k = 1 , k ≠ s m a k ( x ) ∣ u ∣ p k − 1 u + a s ( x ) ∣ u ∣ p s , s ∈ [ 1 , m ] , m ≥ 2 ,

as well as the nonlinearity

(3) f ( x , u ) = ∑ k = 1 , k ≠ s m a k ( x ) ∣ u ∣ p k − 1 u + a s ( x ) ∣ u ∣ p s − 1 u , s ∈ [ 1 , m ] , m ≥ 2

under the assumptions

(4) 1 < p 1 < p 2 < ⋯ < p m ; p m < ∞ for n = 1 , 2 ; p m < n + 2 n − 2 for n ≥ 3 , a k ( x ) ∈ C ( R n ) , ∣ a k ( x ) ∣ ≤ A k for every k ∈ [ 1 , m ] and x ∈ R n .

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 k ( x ) in (2) and (3) are nontrivial and except a s ( x ) have a constant sign, by which we mean they are either nonpositive or nonnegative. For the case of a sign-changing a s ( x ) , we have the hypotheses ( H 1 ) , ( H 2 ) , and ( H 3 ) , and when a s ( x ) does not change its sign, we assume hypotheses ( H 4 ) , ( H 5 ) , and ( H 6 ) .

First, we suppose that for every x ∈ R n , one of the following hypotheses holds

  1. if s = 1 , then a 1 ( x ) is a sign-changing function and a j ( x ) ≥ 0 for j = 2 , 3 , … , m ;

  2. if s ∈ ( 1 , m ) , then a s ( x ) is a sign-changing function and a j ( x ) ≤ 0 for j = 1 , 2 , … , s − 1 ; a j ( x ) ≥ 0 for j = s + 1 , … , m ;

  3. if s = m , then a m ( x ) is a sign-changing function and a j ( x ) ≤ 0 for j = 1 , 2 , … , m − 1 .

Alternative assumptions to hypotheses ( H 1 ) , ( H 2 ) , and ( H 3 ) for nonlinearities (2) and (3) include sign-preserving coefficient a s ( x ) . In this case, we suppose that for every x ∈ R n , one of the following hypotheses holds:

  1. if s ∈ [ 1 , m ) , then a j ( x ) ≤ 0 for j = 1 , 2 , … , s ; a j ( x ) ≥ 0 for j = s + 1 , … , m ;

  2. if s = m , then a j ( x ) ≤ 0 for j = 1 , 2 , … , m ;

  3. if s = 1 , then a j ( x ) ≥ 0 for j = 1 , 2 , 3 , … , m .

Note that for s = 1 , we have different signs of a 1 ( x ) in ( H 4 ) and ( H 6 ).

In the previous assumption (4), we suppose that the powers p j , j = 1 , 2 , … , m are in a strictly increasing order. We also consider the case of two additional hypotheses ( H 2 ∗ ) and ( H 3 ∗ ) for nonlinearity (2) where two consecutive powers coincide, i.e., if p s − 1 = p s for 2 ≤ s ≤ m :

  • (H 2 ∗) For some 2 ≤ s ≤ m − 1 , we have

  • 1 < p 1 < ⋯ < p s − 1 = p s < p s + 1 < ⋯ < p m ; p m < ∞ for n = 1 , 2 ; p m < n + 2 n − 2 for n ≥ 3 ,

  • a k ( x ) ∈ C ( R n ) , ∣ a k ( x ) ∣ ≤ A k for k ∈ [ 1 , m ] , x ∈ R n ;

  • a s ( x ) is a sign-changing function and a j ( x ) ≤ 0 for j = 1 , 2 , … , s − 1 ; a j ( x ) ≥ 0 for j = s + 1 , … , m ;

  • (H 3 ∗) 1 < p 1 < ⋯ < p m − 1 = p m ; p m < ∞ for n = 1 , 2 ; p m < n + 2 n − 2 for n ≥ 3 ,

  • a k ( x ) ∈ C ( R n ) , ∣ a k ( x ) ∣ ≤ A k for k ∈ [ 1 , m ] , x ∈ R n ;

  • a m ( x ) is a sign-changing function and a j ( x ) ≤ 0 for j = 1 , 2 , … , m − 1 .

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 [4–6] for other results. Later on, after the remarkable article of Payne and Sattinger [7], the case of positive subcritical initial energy 0 < E ( 0 ) ≤ d 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 ∣ p − 1 u or f ( u ) = a ∣ u ∣ p , p > 1 , a = const ≠ 0 , the global behavior of the solutions is studied in [6–10]. More general cases of power-type nonlinearities with constant coefficients a k , k = 1 , … , m and b j , j = 1 , … , s ,

(5) f ( x , u ) = ∑ k = 1 m a k ∣ u ∣ p k − 1 u − ∑ j = 1 s b j ∣ u ∣ q j − 1 u , or f ( x , u ) = a 1 ∣ u ∣ p 1 + ∑ k = 2 m a k ∣ u ∣ p k − 1 u − ∑ j = 1 s b j ∣ u ∣ q j − 1 u , a 1 > 0 , a k ≥ 0 , k = 2 , … , m , b j ≥ 0 , j = 1 , … , s ; 1 < q s < q s − 1 < ⋯ < q 1 < p 1 < p 2 < ⋯ < p m ; p m < ∞ for n = 1 , 2 ; p m < n + 2 n − 2 for n ≥ 3 ,

are investigated, e.g., in [11–13].

When the initial energy is critical (i.e., E ( 0 ) = d ), then the solutions blow up if I ( u 0 ) < 0 , the scalar product ( u 0 , u 1 ) ≥ 0 and exist for every t ≥ 0 when I ( u 0 ) ≥ 0 , see [1,8,14]. If E ( 0 ) = d , I ( u 0 ) < 0 and ( u 0 , u 1 ) < 0 , 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 ( 0 ) > d ), 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, [17–21], 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:

u t t − u x x + ( m 2 + V ( x ) ) u = a ( x ) u 2 , m > 0 .

Asymptotic stability, scattering, and decay for solutions to the one-dimensional Klein-Gordon equation with variable coefficients are studied in [24–28]. In [29], global existence of solutions to (1) with small data and singular nonlinearities f ( x , u ) = γ ∣ u ∣ p − 1 u ∣ x ∣ s and f ( x , u ) = γ u { exp ( α u 2 ) − 1 } ∣ x ∣ s is proved in the two-dimensional case ( n = 2 ). Equation

u t t − Δ u + m 2 u = V ( x ) ∣ u ∣ p − 1 u , m ≠ 0 , n ≥ 3

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.

Polynomial nonlinearities with variable sign-changing coefficients like (2) and (3) appear for elliptic problems, see, e.g., [35–37], where existence of positive solutions or multiple sign-changing solutions is established.

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 solutions – the 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.

2 Preliminaries

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

2.1 Notations and definitions

For functions depending on t and x , we use the following short notations:

‖ u ‖ = ‖ u ( t , ⋅ ) ‖ L 2 ( R n ) , ‖ u ‖ 1 = ‖ u ( t , ⋅ ) ‖ H 1 ( R n ) , ( u , v ) = ( u ( t , ⋅ ) , v ( t , ⋅ ) ) = ∫ R n u ( t , x ) v ( t , x ) d x .

Definition 1

The function u ( t , x ) ∈ H ,

H ≔ C ( [ 0 , T max ) ; H 1 ( R n ) ) ∩ C 1 ( [ 0 , T max ) ; L 2 ( R n ) ) ∩ C 2 ( [ 0 , T max ) ; H − 1 ( R n ) ) ,

is a weak solution to (1), (2), and (4) in [ 0 , T max ) × R n if the identity

∫ R n u t ( t , x ) η ( x ) d x + ∫ 0 t ∫ R n ∇ u ( τ , x ) ∇ η ( x ) d x d τ + ∫ 0 t ∫ R n u ( τ , x ) η ( x ) d x d τ = ∫ 0 t ∫ R n f ( x , u ( τ , x ) ) η ( x ) d x d τ + ∫ R n u 1 ( x ) η ( x ) d x

holds for every η ( x ) ∈ H 1 ( R n ) and every t ∈ [ 0 , T max ) .

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 existence time interval [ 0 , T max ) , 0 < T max ≤ ∞ blows up at T max , if

limsup t → T max , t < T max ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 = ∞ .

The solution u ( t , x ) to Klein-Gordon equation conserves the energy E , i.e.,

(6) E ( t ) = E ( 0 ) , t ∈ [ 0 , T max ) .

Here, the energy E is defined as follows:

E ( t ) ≔ E ( u ( t , ⋅ ) ) ≔ 1 2 { ∣ ∣ u t ( t , ⋅ ) ∣ ∣ 2 + ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 2 } − ∫ R n ∫ 0 u ( t , x ) f ( x , z ) d z d x .

We define two important functionals for the potential well method associated with the considered problem: the potential energy functional J ( w )

J ( w ) ≔ 1 2 ∣ ∣ w ∣ ∣ 1 2 − ∫ R n ∫ 0 w ( x ) f ( x , z ) d z d x

and the Nehari functional I ( w )

(7) I ( w ) ≔ ∣ ∣ w ∣ ∣ 1 2 − ∫ R n f ( x , w ( x ) ) w ( x ) d x .

The Nehari manifold N , defined by

N ≔ { w ∈ H 1 ( R n ) : ∣ ∣ w ∣ ∣ 1 ≠ 0 , I ( w ) = 0 } ,

contains all the nontrivial critical points of J ( u ) , because J ′ ( u ) = I ( u ) .

Since the coefficient a s ( x ) 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):

(8) J ( w ) = 1 p s + 1 I ( w ) + p s − 1 2 ( p s + 1 ) ∣ ∣ w ∣ ∣ 1 2 + B ( w )

with functional B ( w ) defined by

(9) B ( w ) = ∑ k = 1 , k ≠ s m p k − p s ( p s + 1 ) ( p k + 1 ) ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x .

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

(10) B ( w ) ≥ 0 ∀ w ( x ) ∈ H 1 ( R n ) .

When the functionals J , I , and B are evaluated on the solution u ( t , ⋅ ) to problem (1), then we use the short notations:

J ( t ) ≔ J ( u ( t , ⋅ ) ) , I ( t ) ≔ I ( u ( t , ⋅ ) ) , B ( t ) ≔ B ( u ( t , ⋅ ) ) .

By using (8), the conservation law (6) can be rewritten in the following way

(11) E ( 0 ) = E ( t ) = 1 2 ∣ ∣ u t ∣ ∣ 2 + 1 p s + 1 I ( t ) + p s − 1 2 ( p s + 1 ) ∣ ∣ u ∣ ∣ 1 2 + B ( t ) .

Further on, we use the Sobolev imbedding theorem with constant C q :

(12) ∣ ∣ z ∣ ∣ L q ( R n ) ≤ C q ∣ ∣ z ∣ ∣ 1 , where 2 ≤ q ≤ ∞ for n = 1 ; 2 ≤ q < ∞ for n = 2 ; 2 ≤ q ≤ 2 n n − 2 for n ≥ 3 .

2.2 Improved concavity method of Levine

In the proofs of our main theorems in the next sections, we use the concavity method of Levine [3]. The main idea in this article is to replace the proof of blow up in H 1 ( R n ) norm of the weak solutions to (1) with blow up of their L 2 ( R n ) norm. The reason is that the L 2 ( R n ) norm of the solution Ψ ( t ) ≔ ∣ ∣ u ( t , ⋅ ) ∣ ∣ 2 satisfies a second-order ordinary differential inequality

(13) Ψ ″ ( t ) Ψ ( t ) − γ Ψ ′ 2 ( t ) ≥ 0 , γ > 1 , t ≥ 0

and the function z ( t ) = Ψ ( t ) 1 − γ 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 u ( t , x ) to (1). Then direct computations for Ψ ( t ) give us the following identities for the derivatives of Ψ ( t ) :

(14) Ψ ′ ( t ) = 2 ( u , u t ) , Ψ ″ ( t ) = 2 ∣ ∣ u t ∣ ∣ 2 + 2 ( u , u t t ) = 2 ∣ ∣ u t ∣ ∣ 2 − 2 ∣ ∣ u ∣ ∣ 1 2 + 2 ∫ R n u ( t , x ) f ( x , u ( t , x ) ) d x = 2 ∣ ∣ u t ∣ ∣ 2 − 2 I ( t ) ,

(15) Ψ ″ ( t ) = ( p s + 3 ) ∣ ∣ u t ∣ ∣ 2 − 2 ( p s + 1 ) E ( 0 ) + ( p s − 1 ) ∣ ∣ u ∣ ∣ 1 2 + 2 ( p s + 1 ) B ( t ) ,

(16) Ψ ″ ( t ) = ( p s + 3 ) ∣ ∣ u t ∣ ∣ 2 − 2 I ( t ) + 2 ( p s + 1 ) { J ( t ) − E ( 0 ) } .

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.

Theorem 1

(Theorem 2.3 in [43], Theorem 3.2 in [14]) Suppose γ > 1 and [ 0 , T max ) , 0 < T max ≤ ∞ is the maximal existence time interval of the nonnegative solution Ψ ( t ) ∈ C 2 ( [ 0 , T max ) ) to the problem

(17) Ψ ″ ( t ) Ψ ( t ) − γ Ψ ′ 2 ( t ) = α Ψ 2 ( t ) − β Ψ ( t ) + H ( t ) , t ∈ [ 0 , T max ) , α > 0 , β > 0 , H ( t ) ∈ C ( [ 0 , T max ) ) , H ( t ) ≥ 0 , for t ∈ [ 0 , T max )

or to the problem

(18) Ψ ″ ( t ) Ψ ( t ) − γ Ψ ′ 2 ( t ) = G ( t ) , t ∈ [ 0 , T max ) , G ( t ) ∈ C ( [ 0 , T max ) ) , G ( t ) ≥ 0 , for t ∈ [ 0 , T max ) .

If Ψ ( t ) blows up at T max , then T max < ∞ .

3 Properties of the Nehari functional

3.1 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 .

Lemma 1

Let w ∈ H 1 ( R n ) and ∣ ∣ w ∣ ∣ 1 ≠ 0 . 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:

  1. for s < m , one of the following two possibilities is true, either

    (19) ∑ k = s + 1 m ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x > 0

    or

    (20) ∑ k = s + 1 m ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x = 0 and ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x ≠ 0 .

    Moreover, if either (19) or (20) with ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x > 0 holds, then λ ∗ is a unique positive number. If (20) with ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x < 0 is true, then λ ∗ is a unique negative number.

  2. for s = m , the following relation

    (21) ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x ≠ 0

    is true. Moreover, when ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x > 0 holds, then λ ∗ is a unique positive number, while for ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x < 0 , λ ∗ is a unique negative number.

Proof

Proof of sufficiency.

Let λ be any nonzero number. Then from (7), we obtain

I ( λ w ) = λ 2 ∣ ∣ w ∣ ∣ 1 2 − ∑ k = 1 , k ≠ s m ∣ λ ∣ p k − 1 ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x − ∣ λ ∣ p s − 2 λ ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x .

Hence, the properties of I ( λ w ) are determined by the properties of the function g ( λ ) defined by

(22) g ( λ ) = ∣ ∣ w ∣ ∣ 1 2 − ∑ k = 1 , k ≠ s m ∣ λ ∣ p k − 1 ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x − ∣ λ ∣ p s − 2 λ ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x .

Our aim is to find λ ∗ such that g ( λ ∗ ) = 0 . We study the behavior of g ( λ ) for λ → 0 and λ → ± ∞ . It is obvious that lim λ → 0 g ( λ ) > 0 .

Suppose s < m . 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, lim λ → + ∞ g ( λ ) = − ∞ . 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,

∑ k = s + 1 m ∣ λ ∣ p k − 1 ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x = 0 .

If ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x > 0 , we have lim λ → + ∞ g ( λ ) = − ∞ , and once again, there exists a positive number λ ∗ such that g ( λ ∗ ) = 0 . In the other case, i.e., ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x < 0 , we have lim λ → − ∞ g ( λ ) = − ∞ and g ( λ ∗ ) = 0 for some negative λ ∗ .

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 g ( λ ∗ ) = 0 . Suppose (19) is true and g ( λ 1 ) = g ( λ 2 ) = 0 with λ i > 0 , i = 1 , 2 , λ 1 ≠ λ 2 . We multiply g ( λ 1 ) by λ 2 p s − 1 , g ( λ 2 ) by λ 1 p s − 1 and subtract both equalities to obtain

(23) ( λ 2 p s − 1 − λ 1 p s − 1 ) ∣ ∣ w ∣ ∣ 1 2 = ∑ k = 1 s − 1 λ 1 p k − 1 λ 2 p k − 1 ( λ 2 p s − p k − λ 1 p s − p k ) ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x + ∑ k = s + 1 m λ 1 p s − 1 λ 2 p s − 1 ( λ 1 p k − p s − λ 2 p k − p s ) ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x .

If, for example, λ 2 > λ 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.

Suppose (20) holds and ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x > 0 . Then, from H 1 or H 2 and the first equation in (20), we have that all terms in the sum ∑ k = s + 1 m 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:

( λ 2 p s − 1 − λ 1 p s − 1 ) ∣ ∣ w ∣ ∣ 1 2 = ∑ k = 1 s − 1 λ 1 p k − 1 λ 2 p k − 1 ( λ 2 p s − p k − λ 1 p s − p k ) ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x ,

and from arguments similar to previous ones, we obtain the uniqueness of positive λ ∗ for which g ( λ ∗ ) = 0 , i.e., I ( λ ∗ w ) = 0 .

The case (20) with ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x < 0 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 s < m . Suppose there exists a number λ ∗ ≠ 0 , such that λ ∗ w ∈ N , i.e., g ( λ ∗ ) = 0 .

From each of the assumptions H 1 , or H 2 on the signs of a k ( x ) , it follows that we have

∑ k = s + 1 m ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x ≥ 0 .

Thus, either one term in this sum is positive, and hence, ∑ k = s + 1 m ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x > 0 and (19) holds, or all terms are zero, and hence, ∑ k = s + 1 m ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x = 0 .

Consider the case ∑ k = s + 1 m ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x = 0 . Then the leading term with respect to λ in formula (22) for g ( λ ) is ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x . If ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x = 0 , then from H 1 or H 2 and ∣ ∣ w ∣ ∣ 1 ≠ 0 , we obtain g ( λ ) > 0 for all numbers λ , which contradicts our assumption for existence of λ ∗ ≠ 0 , such that g ( λ ∗ ) = 0 . Thus, ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x ≠ 0 and (20) holds. The necessity in case s < m is proved.

If s = m , then the necessity is proved following the arguments in the case s < m . The proof of Lemma 1 is completed.□

Corollary 1

Consider the nonlinearity (2). Suppose (4) and one of assumptions H 1 , H 2 , or H 3 hold. Then the Nehari manifold is not empty.

For example, if s < m (i.e., H 1 or H 2 holds), then for any function w ( x ) ∈ H 1 ( R n ) , ∣ ∣ w ∣ ∣ 1 ≠ 0 such that

(24) { x : w ( x ) ≠ 0 } ∩ { x : a m ( x ) > 0 } ≠ ∅ ,

the condition (19) is satisfied. Then from the sufficiency statements in Lemma 1, we conclude that a positive multiple of w lies in N.

Let H 3 hold. Then it follows that there exists a neighborhood R 1 of x 1 , such that a m ( x ) > 0 for x ∈ R 1 . Then one can choose a function w ( x ) ∈ H 1 ( R n ) with { x : w ( x ) ≠ 0 } ⊂ R 1 , w ( x ) > 0 for x ∈ R 1 and w ( x ) = 0 for x ∉ R 1 . Assumption (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:

(25) d = inf w ∈ N J ( w ) .

In the next lemma, we prove the boundedness from below of the critical energy constant d .

Lemma 2

Consider the nonlinearity (2). Suppose (4) and that one of assumptions H 1 , H 2 , or H 3 holds. Let w ∈ H 1 ( R n ) , ∣ ∣ w ∣ ∣ 1 ≠ 0 , and I ( w ) = 0 , i.e., w ∈ N . Then:

  1. the norm ∣ ∣ w ∣ ∣ 1 is bounded from below by a positive constant M ;

  2. the potential energy J ( w ) is strictly positive, i.e.,

    (26) J ( w ) ≥ p s − 1 2 ( p s + 1 ) M 2 > 0 for w ∈ N

    and the Nehari manifold N is bounded away from zero, i.e.,

    (27) d = inf w ∈ N J ( w ) > 0 .

Proof

Under the assumptions of this lemma and I ( w ) = 0 , we obtain the chain of inequalities

(28) ∣ ∣ w ∣ ∣ 1 2 = ∫ R n w ( x ) f ( x , w ( x ) ) d x ≤ ∑ k = s m A k ∫ R n ∣ w ( x ) ∣ p k + 1 d x ≤ ∑ k = s m A k C p k + 1 p k + 1 ∣ ∣ w ∣ ∣ 1 p k + 1 ,

where C q is the Sobolev imbedding constant defined in (12).

From (28), we obtain

1 ≤ ∑ k = s m A k C p k + 1 p k + 1 ∣ ∣ w ∣ ∣ 1 p k − 1 ,

and hence, there exists some k ¯ ∈ [ s , m ] such that A k ¯ C p k ¯ + 1 p k ¯ + 1 ∣ ∣ w ∣ ∣ 1 p k ¯ − 1 ≥ 1 m − s + 1 . Equivalently,

(29) ∣ ∣ w ∣ ∣ 1 ≥ { A k ¯ C p k ¯ + 1 p k ¯ + 1 ( m − s + 1 ) } − 1 p k ¯ − 1 ≔ M > 0 .

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

J ( w ) = p s − 1 2 ( p s + 1 ) ∣ ∣ w ∣ ∣ 1 2 + B ( w ) ≥ p s − 1 2 ( p s + 1 ) ∣ ∣ w ∣ ∣ 1 2 ≥ p s − 1 2 ( p s + 1 ) M 2 > 0 ,

and hence,

d = inf w ∈ N J ( w ) > 0 ,

which concludes the proof of Lemma 2.□

Remark 1

The functional J ( u ) is not bounded from below for functions w ∈ H 1 ( R n ) , but on the Nehari manifold, J ( u ) is bounded from below, as we proved in Lemma 2.

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.

Lemma 3

Consider the nonlinearity (2). Suppose that (4) holds and one of assumptions H 1 , H 2 , or H 3 is true. Let w ∈ H 1 ( R n ) , ∣ ∣ w ∣ ∣ 1 ≠ 0 , and I ( w ) < 0 . Then the inequality

I ( w ) < ( p s + 1 ) ( J ( w ) − d )

holds.

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

∑ k = s + 1 m ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x ≥ 0 .

We consider the case (20) since the remaining case ∑ k = s + 1 m ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x > 0 coincides with (19).

Suppose ∑ k = s + 1 m ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x = 0 but ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x = 0 . Then from H 1 or H 2 , we have for the function g ( λ ) defined in (22) that g ( λ ) > 0 for all λ , which contradicts assumption I ( λ w ) < 0 for λ = 1 . Therefore, condition (19) or (20) is fulfilled.

Arguments similar to the ones used for s < m are valid for the case s = m . Hence, the sufficiency assumptions in Lemma 1 hold.

From the sufficiency result in Lemma 1, there exists λ ∗ such that I ( λ ∗ w ) = 0 . Moreover, from (19), or from (20) or (21) with the additional condition ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x > 0 , we have that λ ∗ > 0 . Since I ( λ w ) > 0 for λ sufficiently close to 0 and I ( w ) < 0 , we conclude that the unique positive λ ∗ , for which I ( λ ∗ w ) = 0 , is in ( 0 , 1 ) , i.e., λ ∗ ∈ ( 0 , 1 ) .

Now we consider the remaining cases (20) and (21) with ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x < 0 , where the parameter λ ∗ in Lemma 1 is a negative number. In these cases, we obtain I ( − w ) = I ( w ) + 2 ∫ R n a s ( x ) ∣ w ( x ) ∣ p s w ( x ) d x , and hence, I ( − w ) < 0 . Since I ( λ w ) > 0 for ∣ λ ∣ sufficiently close to 0 and I ( − w ) < 0 , then the unique negative λ ∗ , for which I ( λ ∗ w ) = 0 , is in ( − 1 , 0 ) , i.e., ∣ λ ∗ ∣ < 1 .

From I ( λ ∗ w ) = 0 , ∣ λ ∗ ∣ < 1 , (8), (9), (10), and (25), under each of hypotheses H 1 , H 2 or H 3 , we obtain the following chain of inequalities:

d ≤ J ( λ ∗ w ) = p s − 1 2 ( p s + 1 ) ∣ λ ∗ ∣ 2 ∣ ∣ w ∣ ∣ 1 2 + ∑ k = 1 , k ≠ s m ∣ λ ∗ ∣ p k + 1 p k − p s ( p s + 1 ) ( p k + 1 ) ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x ≤ p s − 1 2 ( p s + 1 ) ∣ ∣ w ∣ ∣ 1 2 + ∑ k = 1 , k ≠ s m p k − p s ( p s + 1 ) ( p k + 1 ) ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x = J ( w ) − I ( w ) ( p s + 1 ) .

Lemma 3 is proved.□

3.2 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.

Lemma 4

Let w ∈ H 1 ( R n ) and ∣ ∣ w ∣ ∣ 1 ≠ 0 . Suppose (4) and that one of hypotheses H 1 , H 2 , or H 3 holds for nonlinearity (3). Then there exists a unique positive number λ ∗ > 0 such that λ ∗ w ∈ N iff:

  1. for s < m – one of the following two possibilities is true, either

    (30) ∑ k = s + 1 m ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x > 0

    or

    (31) ∑ k = s + 1 m ∫ R n a k ( x ) ∣ w ( x ) ∣ p k + 1 d x = 0 and ∫ R n a s ( x ) ∣ w ( x ) ∣ p s + 1 d x > 0 .

  2. for s = m , the inequality

    ∫ R n a s ( x ) ∣ w ( x ) ∣ p s + 1 d x > 0

    holds.

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.

3.3 Sign-preserving properties of the Nehari functional for nonlinearity either (2) or (3)

In this section, we introduce two important sets for the potential well method:

V = { w ∈ H 1 ( R n ) : I ( w ) < 0 } ,

W = { w ∈ H 1 ( R n ) : I ( w ) > 0 } ∪ { 0 } .

For weak solutions of (1) with nonlinearity either (2) or (3) and with initial energy E ( 0 ) < d , 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 [ 0 , T max ) , 0 < T max ≤ ∞ . Let the initial energy be subcritical, i.e., E ( 0 ) < d . Then:

  1. if 0 < E ( 0 ) < d , then:

    1. if there exists some t 0 ∈ [ 0 , T max ) such that u ( t 0 , ⋅ ) ∈ W , then u ( t , ⋅ ) ∈ W for every t ∈ [ 0 , T max ) ;

    2. if there exists some t 0 ∈ [ 0 , T max ) such that u ( t 0 , ⋅ ) ∈ V , then u ( t , ⋅ ) ∈ V for every t ∈ [ 0 , T max ) ;

  2. if E ( 0 ) < 0 , then I ( t ) < 0 , i.e., u ( t , ⋅ ) ∈ V for every t ∈ [ 0 , T max ) ;

  3. if E ( 0 ) = 0 , then for every nontrivial solution u ( t , x ) , we have I ( t ) < 0 , i.e., u ( t , ⋅ ) ∈ V .

Proof

Case ( i 1 ) . Let I ( t 0 ) > 0 . Suppose by contradiction that there exists some t 1 such that u ( t 1 , ⋅ ) ∉ W . From the continuity of u ( t , x ) with respect to t , there exists some time t 2 , t 2 ≠ t 1 , such that u ( t 2 , ⋅ ) ∈ ∂ W = { u ∈ H 1 : I ( u ) = 0 } .

From (22), we have I ( λ w ) > 0 for every w with ∣ ∣ w ∣ ∣ 1 ≠ 0 and for all sufficiently small λ , i.e., λ w ∈ W . Hence, 0 ∈ W and 0 ∉ ∂ W .

Therefore, ∣ ∣ u ( t 2 , ⋅ ) ∣ ∣ 1 ≠ 0 and I ( t 2 ) = 0 , i.e., u ( t 2 , ⋅ ) ∈ N . From the conservation law and the definition of the critical energy constant d , we obtain

(32) d = inf w ∈ N J ( w ) ≤ J ( t 2 ) = E ( 0 ) − 1 2 ∣ ∣ u t ( t 2 , ⋅ ) ∣ ∣ 2 ≤ E ( 0 ) ,

which contradicts our assumption E ( 0 ) < d . Hence, I ( t ) > 0 for every t ∈ [ 0 , T max ) , i.e., u ( t , ⋅ ) ∈ W and the statement ( i 1 ) is proved.

For the proof of ( i 2 ), we suppose by contradiction that u ( t 1 , ⋅ ) ∉ V for some t 1 ∈ [ 0 , T max ) , i.e., I ( t 1 ) = 0 . If u ( t 1 , ⋅ ) ≡ 0 , then u ( t 1 , ⋅ ) ∈ W and from ( i 1 ) it follows that u ( t 0 , ⋅ ) ∈ W , which contradicts u ( t 0 , ⋅ ) ∈ V . If u ( t 1 , ⋅ ) ≠ 0 , i.e., ∣ ∣ u ( t 1 , ⋅ ) ∣ ∣ 1 ≠ 0 , then u ( t 1 , ⋅ ) ∈ N , which contradicts (32). Statement ( i 2 ) is proved.

The proofs of ( i i ) and ( i i i ) follow from the conservation law (11), written in the form

(33) I ( t ) = ( p s + 1 ) E ( 0 ) − ( p s + 1 ) 2 ∣ ∣ u t ( t , ⋅ ) ∣ ∣ 2 − p s − 1 2 ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 2 − ( p s + 1 ) B ( t )

with B ( t ) ≥ 0 (see (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

(34) ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 2 ≤ ∫ R n u ( t , x ) f ( x , u ( t , x ) ) d x ≤ ∑ k = s m A k C p k + 1 p k + 1 ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 p k + 1

and for ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 ≠ 0 , we obtain from (34) that 1 ≤ ∑ k = s m A k C p k + 1 p k + 1 ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 p k − 1 . Therefore, ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 ≥ M > 0 , where M is defined in (29). To sum up, when I ( t ) ≤ 0 , either ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 ≡ 0 or ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 ≥ M holds.

We claim that if for some t 3 ∈ [ 0 , T max ) , we have ∣ ∣ u ( t 3 , ⋅ ) ∣ ∣ 1 = 0 , then ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 ≡ 0 for all t ∈ [ 0 , T max ) , i.e., u ( t , ⋅ ) is a trivial solution. Otherwise, there exists t 4 ∈ [ 0 , T max ) , such that 0 < ∣ ∣ u ( t 4 , ⋅ ) ∣ ∣ 1 < M , which contradicts the aforementioned conclusion that ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 ≥ M whenever ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 ≠ 0 and I ( t ) ≤ 0 hold. Our claim follows. In ( i i i ) , we consider nontrivial solutions, so ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 ≠ 0 for all t ∈ [ 0 , T max ) , and by (33), we obtain I ( t ) < 0 for t ∈ [ 0 , T max ) . This completes the proof of Theorem 2.□

4 Main results

In this section, we state our main results for nonexistence of weak solutions to problem (1) with nonlinearities either (2) or (3) and powers satisfying (4). The case of variable coefficients { a j ( x ) } , j = 1 , 2 , … , m satisfying one of hypotheses H 1 , H 2 , or H 3 is treated in Section 4.1, while results under one of hypotheses H 4 , H 5 , or H 6 on the variable coefficients { a j ( x ) } are given in Section 4.2. The uniqueness of the weak solutions under slightly more restrictive assumptions for the powers of the nonlinearities is formulated in Section 4.3.

The proofs of all theorems are given in Section 6.

4.1 Case of hypotheses H 1 , H 2 , or H 3

We start with nonexistence of weak solutions under one of hypotheses H 1 , H 2 , or H 3 , i.e., when the coefficient a s ( x ) , s ∈ [ 1 , m ] , may change its sign.

Theorem 3

Assume that (4) holds together with one of the hypotheses H 1 , H 2 , or H 3 . Suppose u 0 and u 1 are initial data to (1) with nonlinearity either (2) or (3) and with positive subcritical energy, i.e., 0 < E ( 0 ) < d . Then:

  1. for I ( 0 ) < 0 , problem (1) has no global in time weak solution;

  2. for I ( 0 ) > 0 and ∣ ∣ u 0 ∣ ∣ 1 ≠ 0 , problem (1) has no blowing up weak solution;

  3. for ∣ ∣ u 0 ∣ ∣ 1 = 0 (hence I ( 0 ) = 0 ), problem (1) has no blowing up weak solution.

In the following theorem, we give two sufficient conditions for nonexistence of global solutions provided positive initial energy.

Theorem 4

Assume that (4) holds together with one of hypotheses H 1 , H 2 , or H 3 . Then problem (1) with nonlinearity either (2) or (3) has no global in time weak solution if at least one of the following assumptions (35) or (36) holds:

  1. (35) 0 < E ( 0 ) < p s − 1 2 ( p s + 1 ) ∣ ∣ u 0 ∣ ∣ 2 + p s − 1 p s + 1 ( u 0 , u 1 ) ,

  2. (36) ∣ ∣ u 0 ∣ ∣ 2 ≠ 0 , ( u 0 , u 1 ) > 0 , 0 < E ( 0 ) < p s − 1 2 ( p s + 1 ) ∣ ∣ u 0 ∣ ∣ 2 + 1 2 ( u 0 , u 1 ) 2 ∣ ∣ u 0 ∣ ∣ 2 .

For the particular case of nonpositive initial energy E ( 0 ) ≤ 0 , we have the following corollary.

Corollary 2

Assume that (4) holds together with one of the hypotheses H 1 , H 2 , or H 3 . Suppose u 0 and u 1 are initial data to (1) with nonlinearity either (2) or (3). Then:

  1. for E ( 0 ) < 0 , problem (1) has no global in time weak solution;

  2. for E ( 0 ) = 0 , except the trivial solution u ( t , x ) ≡ 0 , problem (1) has no global in time weak solution.

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 1 ( x ) . The result in this corollary is a consequence of Theorems 3 and 4 and Corollary 2.

Corollary 3

Suppose

(37) f ( x , u ) = a 1 ( x ) ∣ u ∣ p 1 − 1 u or f ( x , u ) = a 1 ( x ) ∣ u ∣ p 1 , a 1 ( x ) ∈ C ( R n ) , ∣ a 1 ( x ) ∣ ≤ A ; 1 < p 1 < ∞ for n = 1 , 2 ; 1 < p 1 < n + 2 n − 2 for n ≥ 3 .

  1. If E ( 0 ) ≤ 0 , then problem (1) with (37) has no nontrivial global in time weak solution.

  2. If 0 < E ( 0 ) < d , then

    1. for I ( 0 ) < 0 , problem (1) with (37) has no global in time weak solution;

    2. for I ( 0 ) > 0 , problem (1) with (37) has no blowing up weak solution.

  3. If initial data u 0 , u 1 satisfy the inequality

    0 < E ( 0 ) < p 1 − 1 2 ( p 1 + 1 ) ∣ ∣ u 0 ∣ ∣ 2 + p 1 − 1 p 1 + 1 ( u 0 , u 1 ) ,

    or the inequalities

    ∣ ∣ u 0 ∣ ∣ 2 ≠ 0 , ( u 0 , u 1 ) > 0 , 0 < E ( 0 ) < p 1 − 1 2 ( p 1 + 1 ) ∣ ∣ u 0 ∣ ∣ 2 + 1 2 ( u 0 , u 1 ) 2 ∣ ∣ u 0 ∣ ∣ 2 ,

    then problem (1) with (37) has no global in time weak solution.

  4. Moreover, problem (1) with (37) has at most one weak solution provided additionally

    1 < p 1 < ∞ for n = 1 ; 1 + 1 n ≤ p 1 < ∞ for n = 2 ; 1 + 1 n ≤ p 1 ≤ n − 1 n − 2 for n ≥ 3 .

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.

Corollary 4

For nonlinearity (2), the conclusions in Theorems 3 and 4, and Corollary 2  remain true under the hypotheses H 2 ∗ or H 3 ∗ on the place of hypotheses H 1 , H 2 , and H 3 .

For example, the blow up results in Corollary 4 hold for f ( x , u ) = a 1 ( x ) ∣ u ∣ p − 1 u + a 2 ( x ) ∣ u ∣ p with a nonpositive function a 1 ( x ) and a sign-changing function a 2 ( x ) , where p satisfies the growth conditions 1 < p < ∞ for n = 1 , 2 and 1 < p < n + 2 n − 2 for n ≥ 3 .

4.2 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.

Theorem 5

For nonlinearity (2), the conclusions in Theorems 3 and 4 and Corollary  2 remain true when we replace the hypotheses H 1 , H 2 , or H 3 with hypotheses H 4 , H 5 , or H 6 .

Theorem 6

For nonlinearity (3), the conclusions in Theorems 3 and 4, and Corollary 2  remain true when we replace the hypotheses H 1 or H 3 with hypotheses H 4 or H 6 .

Theorem 7

Suppose that the hypothesis H 5 for the coefficients of the nonlinearity (3) is valid. Assume (4) holds. Then we have ∣ ∣ u ∣ ∣ 1 2 < 2 E ( 0 ) and the solution u ( t , x ) to (1) can not blow up.

4.3 Uniqueness of weak solutions

In the following theorem, we state uniqueness of the weak solutions to (1) under general power-type nonlinearity

(38) f ( x , u ) = ∑ j = 1 k b j ( x ) ∣ u ∣ p j − 1 u + ∑ j = k + 1 m b j ( x ) ∣ u ∣ p j

without any sign conditions of the variable coefficients b j ( x ) , j = 1 , 2 , … , m . We suppose that

(39) b j ( x ) ∈ C ( R n ) , ∣ b j ( x ) ∣ ≤ B for j ∈ [ 1 , m ] and x ∈ R n ; 1 < p j < ∞ for n = 1 , j ∈ [ 1 , m ] ; 1 + 1 n ≤ p j < ∞ for n = 2 , j ∈ [ 1 , m ] ; 1 + 1 n ≤ p j ≤ n − 1 n − 2 for n ≥ 3 , j ∈ [ 1 , m ] .

Theorem 8

(Uniqueness of the weak solutions) Equation (1) with nonlinearity (38) and conditions (39), has at most one weak solution u ( t , x ) ∈ H .

5 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.

Corollary 5

Suppose (4) holds together with one of hypotheses H 1 , H 2 , or H 3 . Let a weak solution to (1) with nonlinearity either (2) or (3) blows up for a finite time T max . Then:

  1. if E ( 0 ) < d , then I ( t ) < 0 for every t ∈ [ 0 , T max ) ;

  2. if d ≤ E ( 0 ) and (36) holds, then I ( 0 ) < 0 ;

  3. if d ≤ E ( 0 ) , ( u 0 , u 1 ) ≤ 0 , and (35) hold, then I ( 0 ) < 0 ;

  4. if d ≤ E ( 0 ) and additionally the L 2 ( R n ) norm of u ( t , ⋅ ) blows up for a finite time, then there exists t 0 ∈ [ 0 , T max ) such that

    (40) I ( t 0 ) < 0 .

Proof

Case ( i ) . When E ( 0 ) ≤ 0 , then I ( t ) < 0 for t ∈ [ 0 , T max ) from Theorems 2 ( i i ) and 2 ( i i i ) .

If 0 < E ( 0 ) < d , we suppose by contradiction that for some t 0 ∈ [ 0 , T max ) assertion ( i ) does not hold, i.e., there exists t 0 ∈ [ 0 , T max ) such that I ( t 0 ) ≥ 0 . First, let I ( t 0 ) > 0 . Then from Theorem 2 ( i 1 ) we have u ( t , ⋅ ) ∈ W , i.e., I ( t ) > 0 for every t ∈ [ 0 , T max ) . Hence, I ( 0 ) > 0 and from Theorem 3, problem (1) has no blowing up solution, which contradicts our assumptions.

Now let I ( t 0 ) = 0 and ∣ ∣ u ( t 0 , ⋅ ) ∣ ∣ 1 ≠ 0 . Then u ( t 0 , ⋅ ) ∈ N , and we obtain a contradiction as in (32).

In the remaining case I ( t 0 ) = 0 and ∣ ∣ u ( t 0 , ⋅ ) ∣ ∣ 1 = 0 , we conclude that u ( t 0 , x ) ≡ 0 . From Theorem 3 ( i i i ) , problem (1) has no blowing up solution, which contradicts the assumption of the corollary. Hence, assertion ( i ) is true.

To prove assertion ( i i ) , we rewrite (11) in the following way

(41) 1 p s + 1 I ( t ) = E ( 0 ) − 1 2 ( u , u t ) 2 ∣ ∣ u ∣ ∣ 2 − 1 2 u t − ( u , u t ) ∣ ∣ u ∣ ∣ 2 ⋅ u 2 − p s − 1 2 ( p s + 1 ) ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 2 − B ( t ) .

We apply (41) at the initial moment t = 0 , and using (36), we obtain the desired estimate I ( 0 ) < 0 as follows:

1 p s + 1 I ( 0 ) < − 1 2 u 1 − ( u 0 , u 1 ) ∣ ∣ u 0 ∣ ∣ 2 ⋅ u 0 2 − p s − 1 2 ( p s + 1 ) ∣ ∣ ∇ u 0 ∣ ∣ 2 − B ( 0 ) < 0 ,

hence, ( i i ) is true.

Let assumptions in ( i i i ) hold. Then from (35), we obtain

0 < E ( 0 ) < p s − 1 2 ( p s + 1 ) ∣ ∣ u 0 ∣ ∣ 2 + p s − 1 p s + 1 ( u 0 , u 1 ) ≤ p s − 1 2 ( p s + 1 ) ∣ ∣ u 0 ∣ ∣ 2 ,

and from inequality (41) for t = 0 , we obtain I ( 0 ) < 0 .

Let assumptions of ( i v ) be valid, i.e., the function Ψ ( t ) ≔ ∣ ∣ u ( t , ⋅ ) ∣ ∣ 2 blows up at T max < ∞ . Then, from Lemma 2.2 in [43], it follows that there exists some t 0 ∈ [ 0 , T max ) such that Ψ ( t 0 ) > 2 ( p s + 1 ) E ( 0 ) p s − 1 and Ψ ′ ( t 0 ) > 0 , i.e., E ( 0 ) < p s − 1 2 ( p s + 1 ) ∣ ∣ u ( t 0 , ⋅ ) ∣ ∣ 2 and ( u ( t 0 , ⋅ ) , u t ( t 0 , ⋅ ) ) > 0 . Thus, both conditions (35) and (36) are true at the moment t 0 < T m . We argue as in case ( i i ) , and from (41) at the point t 0 , we conclude that I ( t 0 ) < 0 . Corollary 5 is proved.□

Remark 3

The inequality d ≤ E ( 0 ) in assertions ( i i ) , ( i i i ) , and ( i v ) is meaningful since if E ( 0 ) < d , then ( i ) is valid and the Nehari functional I ( t ) is negative for every t ∈ [ 0 , T max ) .

In case ( i v ) , the blow up of L 2 ( R n ) norm of u ( t , ⋅ ) is needed. For the case of constant signs of the nonlinearity coefficients and for n = 1 , the blow up of u ( t , ⋅ ) in H 1 ( R 1 ) norm is equivalent to the blow up in L 2 ( R n ) norm, see Lemma 4.3 of [43]. For n = 2 , the equivalence of H 1 ( R 2 ) norm with L 2 ( R 2 ) norm of u ( t , ⋅ ) follows from the proof of Lemma 3.1 of [44] for 1 < p m < ∞ . For n ≥ 3 , the equivalence of blow up in H 1 ( R n ) and L 2 ( R n ) norms of u ( t , ⋅ ) is proved in Lemma 3.1 of [44] under additional requirement 1 < p m < n + 4 n . Careful analysis of the proofs of aforementioned results for equivalence of blow up in H 1 ( R n ) and L 2 ( R n ) norm shows, that conclusions are valid for the case of variable coefficients, satisfying one of hypotheses H 1 , H 2 , or H 3 and the corresponding restrictions on p m .

Remark 4

In Example 4.4 from [44], it was shown that there exists a single-power nonlinearity with constant coefficient and initial data u 0 , u 1 with ( u 0 , u 1 ) > 0 , for which assumption (35) holds (hence, the solution blows up), and at the same time, we have I ( 0 ) > 0 . This example does not contradict the result ( i i i ) of this corollary.

6 Proof of the main results

6.1 Proof of the main results for hypotheses H 1 , H 2 , and H 3

Proof of Theorem 3

Case ( i ) . We suppose by contradiction that problem (1) has a weak solution u ( t , x ) defined for every t ≥ 0 , i.e., T max = ∞ . Let Ψ ( t ) = ∣ ∣ u ( t , ⋅ ) ∣ ∣ 2 . From Theorem 2 ( i 2 ) , when u ( t , x ) is a nontrivial solution, we have I ( t ) < 0 and ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 ≠ 0 for every t ≥ 0 . From Lemma 3, (16), and (27), we obtain the inequality

(42) Ψ ″ ( t ) = ( p s + 3 ) ∣ ∣ u t ∣ ∣ 2 − 2 I ( t ) + 2 ( p s + 1 ) { J ( t ) − E ( 0 ) } > ( p s + 3 ) ∣ ∣ u t ∣ ∣ 2 + 2 ( p s + 1 ) ( d − E ( 0 ) ) ≥ 2 ( p s + 1 ) ( d − E ( 0 ) ) > 0 .

By integrating (42) twice, we obtain

Ψ ( t ) ≥ ( p s + 1 ) ( d − E ( 0 ) ) t 2 + Ψ ′ ( 0 ) t + Ψ ( 0 ) ,

and hence, lim t → ∞ Ψ ( t ) = ∞ , i.e., Ψ ( t ) blows up at infinity.

Moreover, the representation (16) of Ψ ″ ( t ) gives us the equalities

Ψ ″ ( t ) Ψ ( t ) − p s + 3 4 Ψ ′ 2 ( t ) = ( p s + 3 ) { ∣ ∣ u t ∣ ∣ 2 ∣ ∣ u ∣ ∣ 2 − ( u , u t ) 2 } + 2 ∣ ∣ u ∣ ∣ 2 { ( p s + 1 ) ( J ( t ) − E ( 0 ) ) − I ( t ) } = ( p s + 3 ) { ∣ ∣ u t ∣ ∣ 2 ∣ ∣ u ∣ ∣ 2 − ( u , u t ) 2 } + 2 ∣ ∣ u ∣ ∣ 2 { ( p s + 1 ) ( J ( t ) − d ) − I ( t ) + ( p s + 1 ) ( d − E ( 0 ) ) } .

Let us denote the rhs of the last equation by G ( t ) , i.e.,

G ( t ) = ( p s + 3 ) { ∣ ∣ u t ∣ ∣ 2 ∣ ∣ u ∣ ∣ 2 − ( u , u t ) 2 } + 2 ∣ ∣ u ∣ ∣ 2 { ( p s + 1 ) ( J ( t ) − d ) − I ( t ) + ( p s + 1 ) ( d − E ( 0 ) ) } .

Since from Theorem 2 ( i 2 ) , I ( t ) < 0 for t ≥ 0 , we obtain from Lemma 3 that I ( t ) < ( p s + 1 ) ( J ( t ) − d ) . Hence, G ( t ) > 0 . Now all assumptions in Theorem 1 for problem (18) are satisfied since γ = p s + 3 4 > 1 and G ( t ) > 0 . We conclude from Theorem 1 that u ( t , x ) blows up for finite time T max < ∞ . This contradicts our assumptions that u ( t , x ) is globally defined. Case ( i ) in Theorem 3 is proved.

Case ( i i ) . Suppose by contradiction that u ( t , x ) , defined in [ 0 , T max ) , 0 ≤ T max ≤ ∞ , blows up at T max . From Theorem 2 ( i 1 ) we conclude, that I ( t ) > 0 for t ∈ [ 0 , T max ) . Using (11) and (10), we obtain for t ∈ [ 0 , T max ) the estimate

(43) ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 2 ≤ 2 ( p s + 1 ) p s − 1 E ( 0 ) .

The above estimate contradicts Definition 2 for blow up, and thus, u ( t , x ) does not blow up at T max .

Case ( i i i ) . Since ∣ ∣ u 0 ∣ ∣ 1 = 0 , then u 0 ( x ) ≡ 0 and u ( 0 , x ) = u 0 ( x ) ∈ W . From Theorem 2 ( i 1 ) it follows that u ( t , x ) ∈ W for t ∈ [ 0 , T max ) . Repeating the proof of the case ( i i ) , we obtain estimate (43), which completes the proof of Theorem 3.□

Proof of Theorem 4

Case ( i ) . We suppose that (35) holds and problem (1) has a weak solution u ( t , x ) defined for every t ≥ 0 , i.e., T max = ∞ . Let Ψ ( t ) = ∣ ∣ u ( ⋅ , t ) ∣ ∣ 2 .

We rewrite (15) in the following way

(44) Ψ ″ ( t ) = ( p s − 1 ) Ψ ( t ) − 2 ( p s + 1 ) E ( 0 ) + G 1 ( t ) ,

where

G 1 ( t ) = ( p s + 3 ) ∣ ∣ u t ∣ ∣ 2 + ( p s − 1 ) ∣ ∣ ∇ u ∣ ∣ 2 + 2 ( p s + 1 ) B ( t ) ≥ 0 .

Let α = p s − 1 > 0 and β = 2 ( p s + 1 ) E ( 0 ) > 0 . Then the solution Ψ ( t ) to (44) is

(45) Ψ ( t ) = 1 2 Ψ ( 0 ) + 1 α Ψ ′ ( 0 ) − β α exp ( α t ) + 1 2 Ψ ( 0 ) − 1 α Ψ ′ ( 0 ) − β α exp ( − α t ) + β α + 1 α ∫ 0 t G 1 ( τ ) sinh ( α ( t − τ ) ) d τ .

Since the requirement (35) is equivalent to

(46) Ψ ( 0 ) + 1 α Ψ ′ ( 0 ) − β α > 0 ,

from (45) and (46) it follows that

lim t → ∞ Ψ ( t ) = ∞ .

Moreover, from (15) we obtain the following equality

(47) Ψ ″ ( t ) Ψ ( t ) − p s + 3 4 Ψ ′ 2 ( t ) = ( p s + 3 ) { ∣ ∣ u t ∣ ∣ 2 ∣ ∣ u ∣ ∣ 2 − ( u , u t ) 2 } − 2 ( p s + 1 ) E ( 0 ) ∣ ∣ u ∣ ∣ 2 + ( p s − 1 ) ∣ ∣ u ∣ ∣ 1 2 ∣ ∣ u ∣ ∣ 2 + 2 ( p s + 1 ) ∣ ∣ u ∣ ∣ 2 B ( t ) = α Ψ 2 ( t ) − β Ψ ( t ) + H ( t ) ,

where

H ( t ) = ( p s + 3 ) { ∣ ∣ u t ∣ ∣ 2 ∣ ∣ u ∣ ∣ 2 − ( u , u t ) 2 } + ( p s − 1 ) ∣ ∣ ∇ u ∣ ∣ 2 ∣ ∣ u ∣ ∣ 2 + 2 ( p s + 1 ) ∣ ∣ u ∣ ∣ 2 B ( t ) ≥ 0 .

Thus Ψ ( t ) is a solution to (17) with γ = p s + 3 4 > 1 , α = p s − 1 > 0 , β = 2 ( p s + 1 ) E ( 0 ) > 0 . According to Theorem 1, it follows that u ( t , x ) blows up for a finite time T max < ∞ , which contradicts our assumption T max = ∞ .

Case ( i i ) . The proof follows the same ideas of the proof of Theorem 4.2 in [13], because the inequality B ( t ) ≥ 0 holds for nonlinearity with constant coefficients as well as for nonlinearities (2) and (3) with variable coefficients, satisfying one of H 1 , H 2 , or H 3 .

The proof of Theorem 4 is completed.□

Proof of Corollary 2

We suppose by contradiction that problem (1) has a weak solution u ( t , x ) defined for every t ≥ 0 . Let Ψ ( t ) = ∣ ∣ u ( ⋅ , t ) ∣ ∣ 2 . In case ( i ) , when E ( 0 ) < 0 , we obtain from (15) that Ψ ″ ( t ) > − 2 ( p s + 1 ) E ( 0 ) > 0 . By integrating the last inequality twice, we obtain, similarly to the proof of Theorem 4 ( i ) , that lim t → ∞ Ψ ( t ) = ∞ . From (15), we obtain equality (18) with γ = p s + 3 4 and the rhs function G 2 ( t ) , defined by

G 2 ( t ) ≔ ( p s + 3 ) { ∣ ∣ u t ∣ ∣ 2 ∣ ∣ u ∣ ∣ 2 − ( u , u t ) 2 } + ∣ ∣ u ∣ ∣ 2 { ( p s − 1 ) ∣ ∣ u ∣ ∣ 1 2 + 2 ( p s + 1 ) B ( t ) − 2 ( p s + 1 ) E ( 0 ) } > 0 .

From Theorem 1, we conclude that the solution u ( t , x ) blows up for a finite time, which contradicts our assumption.

In case ( i i ) , we obtain from Theorem 2 ( i i i ) that I ( t ) < 0 for t ≥ 0 . We argue similarly to Lemma 2 ( i ) and conclude that ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 ≥ M > 0 for every t ≥ 0 . Here, constant M is defined in (29). From (15), we have Ψ ″ ( t ) > ( p s − 1 ) M 2 . Further on, we proceed as in the proof of case ( i ) and obtain a contradiction with the assumption that a weak solution u ( t , x ) is globally defined. The proof of Corollary 2 is completed.□

The proof of Corollary 4 follows the arguments used in the previous theorems, and we omit it.

6.2 Proof of the main theorems for hypotheses H 4 , H 5 , and H 6

In this section, we consider nonlinearities with sign-preserving coefficients satisfying one of hypotheses H 4 , H 5 , or H 6 . A careful analysis shows that for nonlinearity (2), the proofs of Lemmas 1, 2, and 3 under one of H 4 , H 5 , or H 6 follow the same lines as the corresponding proofs under the assumptions H 1 , H 2 , and H 3 . There are only small differences, e.g., in the proof of the new Corollary 1, under assumption H 5 , the requirement (24) should be replaced by the following one:

{ x : w ( x ) ≠ 0 } ∩ { x : a m ( x ) ≠ 0 } ≠ ∅ .

Therefore, we state the following.

Lemma 6

For nonlinearity (2), the conclusions in Lemmas 1, 2, and 3, as well as in Corollary 1, remain true when we replace the hypotheses H 1 , H 2 , or H 3 with the corresponding hypotheses H 4 , H 5 , or H 6 .

Similarly, we have for nonlinearity (3) the following.

Lemma 7

For nonlinearity (3), the conclusions in Lemmas 4, 2, and 3, as well as in Corollary 1, remain true when we replace the hypotheses H 1 or H 3 with the hypotheses H 4 or H 6 .

Note that under assumption H 4 , the result in Lemma 4 holds only under the restriction (30) (condition (31) is never true). Moreover, Lemma 2 is true when the sum ∑ k = s m in the inequalities (28) is taken over ∑ k = s + 1 m .

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 ( t ) = E ( 0 ) = 1 2 { ∣ ∣ u t ∣ ∣ 2 + ∣ ∣ u ∣ ∣ 1 2 } − ∑ k = 1 m ∫ R n a k ( x ) p k + 1 ∣ u ( t , x ) ∣ p k + 1 d x .

Since all terms in the aforementioned equality are positive, then ∣ ∣ u ∣ ∣ 1 2 < 2 E ( 0 ) , and the solution u ( t , x ) cannot blow up, which concludes the proof of Theorem 7.□

6.3 Proof of the uniqueness result in Theorem 8

Next we give the proof of Theorem 8.

Proof of Theorem 8

Suppose that u ( t , x ) and v ( t , x ) are weak solutions to (1), (38), and (39) in the interval [ 0 , T max ) , 0 < T max ≤ ∞ and u ( t , x ) ≠ v ( t , x ) for some t ∈ [ 0 , T max ) and x ∈ R n . We define

t 0 = inf t ∈ [ 0 , T max ) { t : u ( t , x ) ≠ v ( t , x ) for some x ∈ R n } ,

and function w ( t , x ) = u ( t , x ) − v ( t , x ) , which satisfies the problem

(48) w t t − Δ w + w = f ( x , u ( t , x ) ) − f ( x , v ( t , x ) ) , ( t , x ) ∈ [ 0 , T max ) × R n , w ( 0 , x ) = w t ( 0 , x ) = 0 , for x ∈ R n .

Multiplying equation (48) with w t ( t , x ) and integrating on [ 0 , t ) × R n , t ∈ ( t 0 , T ) , T < T max , we obtain

(49) ∣ ∣ w t ( t , ⋅ ) ∣ ∣ 2 + ∣ ∣ w ( t , ⋅ ) ∣ ∣ 1 2 = 2 ∫ 0 t ∫ R n ( f ( x , u ( τ , x ) ) − f ( x , v ( τ , x ) ) ) w t ( τ , x ) d x d τ ≤ 2 ∫ 0 t ∫ R n ∑ j = 1 m p j ∣ b j ( x ) ∣ ∫ 0 1 ∣ ( 1 − θ ) u ( τ , x ) + θ v ( τ , x ) ∣ p j − 1 d θ ∣ w ( τ , x ) ∣ ∣ w t ( τ , x ) ∣ d x d τ ≤ 2 B max j ∈ [ 1 , m ] { p j } ∫ 0 t ∫ R n ∣ w ( τ , x ) ∣ ∣ w t ( τ , x ) ∣ ∑ j = 1 m ( ∣ u ( τ , x ) ∣ + ∣ v ( τ , x ) ∣ ) p j − 1 d x d τ .

For n ≥ 2 , we apply the generalized Hölder inequality and obtain for p = max j ∈ [ 1 , m ] { p j }

(50) ∣ ∣ w t ( t , ⋅ ) ∣ ∣ 2 + ∣ ∣ w ( t , ⋅ ) ∣ ∣ 1 2 ≤ 2 p B ∫ 0 t ∑ j = 1 m ∣ ∣ ( ∣ u ( τ , ⋅ ) ∣ + ∣ v ( τ , ⋅ ) ∣ ) p j − 1 ∣ ∣ L 2 n ( R n ) ∣ ∣ w ( τ , ⋅ ) ∣ ∣ L 2 n n − 1 ( R n ) ∣ ∣ w t ( τ , ⋅ ) ∣ ∣ d τ .

From the inequality ( a + b ) q ≤ 2 q ( a q + b q ) , valid for q > 0 , a ≥ 0 , b ≥ 0 , we have

∫ R n ( ∣ u ( τ , x ) ∣ + ∣ v ( τ , x ) ∣ ) 2 n ( p j − 1 ) d x 1 2 n ≤ 2 2 n ( p j − 1 ) 2 n ∫ R n ( ∣ u ( τ , x ) ∣ 2 n ( p j − 1 ) + ∣ v ( τ , x ) ∣ 2 n ( p j − 1 ) ) d x 1 2 n ≤ 2 p j − 1 + 1 2 n { ∣ ∣ u ( τ , ⋅ ) ∣ ∣ L 2 n ( p j − 1 ) ( R n ) ( p j − 1 ) + ∣ ∣ v ( τ , ⋅ ) ∣ ∣ L 2 n ( p j − 1 ) ( R n ) ( p j − 1 ) } .

By applying the Sobolev inequality (12), we obtain for the previous chain of inequalities

∫ R n ( ∣ u ( τ , x ) ∣ + ∣ v ( τ , x ) ∣ ) 2 n ( p j − 1 ) d x 1 2 n ≤ 2 p − 1 + 1 2 n C 2 n ( p j − 1 ) ( p j − 1 ) { ∣ ∣ u ( τ , ⋅ ) ∣ ∣ 1 ( p j − 1 ) + ∣ ∣ v ( τ , ⋅ ) ∣ ∣ 1 ( p j − 1 ) } .

Thus, (50) becomes

∣ ∣ w t ( t , ⋅ ) ∣ ∣ 2 + ∣ ∣ w ( t , ⋅ ) ∣ ∣ 1 2 ≤ 2 p + 1 p B ∑ j = 1 m C 2 n ( p j − 1 ) ( p j − 1 ) C 2 n n − 1 ∫ 0 t ∣ ∣ w ( τ , ⋅ ) ∣ ∣ 1 ∣ ∣ w t ( τ , ⋅ ) ∣ ∣ { ∣ ∣ u ( τ , ⋅ ) ∣ ∣ 1 ( p j − 1 ) + ∣ ∣ v ( τ , ⋅ ) ∣ ∣ 1 ( p j − 1 ) } d τ ≤ K n ∫ 0 t { ∣ ∣ w ( τ , ⋅ ) ∣ ∣ 1 2 + ∣ ∣ w t ( τ , ⋅ ) ∣ ∣ 2 } d τ ,

where

(51) K n = 2 p + 1 p B ∑ j = 1 m C 2 n ( p j − 1 ) ( p j − 1 ) C 2 n n − 1 K 0 ( p j − 1 ) ,

(52) K 0 = max { sup t ∈ [ 0 , T ] ∣ ∣ u ( t , ⋅ ) ∣ ∣ 1 , sup t ∈ [ 0 , T ] ∣ ∣ v ( t , ⋅ ) ∣ ∣ 1 } .

For n = 1 , by the Sobolev inequality ∣ ∣ z ∣ ∣ ∞ ≤ 2 2 ∣ ∣ z ∣ ∣ 1 , we obtain from (49) the estimate

∣ ∣ w t ( t , ⋅ ) ∣ ∣ 2 + ∣ ∣ w ( t , ⋅ ) ∣ ∣ 1 2 ≤ 2 B p ∫ 0 t ∣ w ( τ , x ) ∣ ∣ w t ( τ , x ) ∣ ∑ j = 1 m { ∣ ∣ u ( τ , ⋅ ) ∣ ∣ ∞ + ∣ ∣ v ( τ , ⋅ ) ∣ ∣ ∞ } p j − 1 d τ ≤ 2 B p ∫ 0 t ∣ w ( τ , x ) ∣ ∣ w t ( τ , x ) ∣ ∑ j = 1 m 2 2 p j − 1 { ∣ ∣ u ( τ , ⋅ ) ∣ ∣ 1 + ∣ ∣ v ( τ , ⋅ ) ∣ ∣ 1 } p j − 1 d τ ≤ K 1 ∫ 0 t { ∣ ∣ w ( τ , ⋅ ) ∣ ∣ 1 2 + ∣ ∣ w t ( τ , ⋅ ) ∣ ∣ 2 } d τ ,

where

(53) K 1 = B p 2 p − 1 2 ∑ j = 1 m K 0 p j − 1 .

Thus, for every n ≥ 1 , the following inequality holds

(54) ∣ ∣ w t ( t , ⋅ ) ∣ ∣ 2 + ∣ ∣ w ( t , ⋅ ) ∣ ∣ 1 2 ≤ K n ∫ 0 t { ∣ ∣ w ( τ , ⋅ ) ∣ ∣ 1 2 + ∣ ∣ w t ( τ , ⋅ ) ∣ ∣ 2 } d τ

with constant K n given in (51) for n ≥ 2 and in (53) for n = 1 .

From (54), we obtain for every ε > 0 the inequality

∣ ∣ w t ( t , ⋅ ) ∣ ∣ 2 + ∣ ∣ w ( t , ⋅ ) ∣ ∣ 1 2 ≤ ε + K n ∫ 0 t { ∣ ∣ w ( τ , ⋅ ) ∣ ∣ 1 2 + ∣ ∣ w t ( τ , ⋅ ) ∣ ∣ 2 } d τ .

By using the Grönwall inequality, we obtain the estimate

(55) ∣ ∣ w t ( t , ⋅ ) ∣ ∣ 2 + ∣ ∣ w ( t , ⋅ ) ∣ ∣ 1 2 ≤ ε exp ( K n t ) .

After the limit ε → 0 in (55), it follows that w ( t , x ) = 0 , w t ( t , x ) = 0 for t ∈ [ 0 , T max ] and x ∈ R n , which contradicts our assumption. Theorem 8 is proved.□

7 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 2 ( R n ) 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 sign-changing one).

Acknowledgments

We thank the reviewers for the valuable suggestions that helped us improve the paper.

  1. Funding information: The authors were partially supported by Grant No BG05M2OP001-1.001-0003, financed by the Science and Education for Smart Growth Operational Program (2014-2020) and co-financed by the European Union through the European structural and Investment funds. Moreover, the research of the first author was partially supported by the Bulgarian Science Fund under Grant KΠ-06-H22/2.

  2. Author contributions: All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.

  3. Conflict of interest: The authors state no conflicts of interest.

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Received: 2022-12-14
Revised: 2023-03-24
Accepted: 2023-04-04
Published Online: 2023-05-31

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2022-0584
Keywords for this article
Klein-Gordon equation; nonlinearities with variable coefficients; blow up; Nehari manifold; uniqueness
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Articles in the same Issue

  1. Special Issue on Future Directions of Further Developments in Mathematics
  2. What will the mathematics of tomorrow look like?
  3. On H 2-solutions for a Camassa-Holm type equation
  4. Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
  5. Control of multi-agent systems: Results, open problems, and applications
  6. Logical perspectives on the foundations of probability
  7. Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
  8. A non-smooth Brezis-Oswald uniqueness result
  9. Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
  10. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
  11. Positive solution for a nonlocal problem with strong singular nonlinearity
  12. Analysis of solutions for the fractional differential equation with Hadamard-type
  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
  17. On the use of L-functionals in regression models
  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
  20. Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
  25. On the uniqueness of limit cycles for generalized Liénard systems
  26. A derivative-Hilbert operator acting on Dirichlet spaces
  27. Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
  30. Some results on the value distribution of differential polynomials
  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
  32. Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
  33. Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
  34. Homotopy cartesian squares in extriangulated categories
  35. A unified perspective on some autocorrelation measures in different fields: A note
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  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
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  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
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  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
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