Nikolay KUTEV　Natalia KOLKOVSKAMilena DIMOVAInstitute of Mathematics and Informatics，Bulgarian Academy of Sciences，1113 Sofia，BulgariaE-mail:kutev@math.bas.bg;natali@math.bas.bg;mkoleva@math.bas.bg

FINITE TIME BLOW UP OF THE SOLUTIONS TO BOUSSINESQ EQUATION WITH LINEAR RESTORING FORCE AND ARBITRARY POSITIVE ENERGY?

Nikolay KUTEVNatalia KOLKOVSKA?Milena DIMOVA
Institute of Mathematics and Informatics，Bulgarian Academy of Sciences，1113 Sofia，Bulgaria
E-mail:kutev@math.bas.bg;natali@math.bas.bg;mkoleva@math.bas.bg

Finite time blow up of the solutions to Boussinesq equation with linear restoring force and combined power nonlinearities is studied.Sufficient conditions on the initial data for nonexistence of global solutions are derived.The results are valid for initial data with arbitrary high positive energy.The proofs are based on the concave method and new sign preserving functionals.

Boussinesq equation with linear restoring force;finite time blow up;arbitrary high positive energy;combined power nonlinearities;sign preserving functionals

2010 MR Subject Classification35L30;35L75;35B44

1　Introduction

In this paper we consider the Cauchy problem for nonlinear Boussinesq equation with linear restoring force

here β1≥0，β2＞0 and m＞0 are constants.Our purpose is to study the blow up of the weak solution to（1.1）under the following assumptions on the initial data:

The nonlinear term f（u）satisfies either condition（H1）or（H2）:

For simplicity we assume that a1＞0 in（H2）.By the change v=-u in（1.1）a negative constant，a1＜0，can be also considered in（H2）.Throughout this paper for functions，depending on t and x，we use the following short notations:

Equation（1.1）is a universal model for nonlinear wave dynamics in weakly dispersive media. For example，（1.1）describes the transverse deflections of an elastic rod on elastic foundation，see［1-4］.

The combined power type nonlinearities（H1）and（H2）arise in a number of mathematical models of physical processes.The quadratic-cubic nonlinearityappears in some models of propagation of longitudinal strain waves in an isotropic cylindrical compressible elastic rod［3］，while the cubic-quintic nonlinearityin the theory of atomic chains［5］and in shape memory alloys［6］.

According to［7］，the nonlinearities（H1）and（H2）make the problem under consideration more complicated than the nonlinearity

The reason for that is the lack of invariance under rescaling of the equation and the initial data.

Here we are focusing on the finite time blow up results for the weak solutions to（1.1）. The methodology of proving nonexistence of the solutions substantially depends on the initial energy level.There are three initial energy levels，treated in different ways:subcritical energy（0＜E（0）＜d）;critical energy（E（0）=d）;supercritical energy（E（0）＞d）.Here the initial energy E（0）is given by（2.2），while d is the depth of the potential well，defined in（2.4）.

In the case of subcritical initial energy（0＜E（0）＜d）the global behavior of the weak solution to（1.1）is fully characterized by means of the potential well method，introduced in［8］for the nonlinear wave equation in bounded domains.In the framework of the potential well method the finite time blow up of the solution is proved provided the condition I（0）＜0，where I（0）is the Nehari functional（see（2.3））.For problem（1.1）with nonlinearity（1.3）the potential well method is developed in［9］.Note，that the potential well method can be extendedto problems with more general type nonlinearities as（H1）and（H2）（see［10］forand nonlinearity（H1））.For the critical initial energy casethe existence time of the weak solutions to（1.1）is finite if I（0）＜0 and the additional conditionholds（see［11］for m=0 and nonlinearity（1.3））.

For supercritical initial energy（E（0）＞d）the theory for global existence or nonexistence of solutions to（1.1）is far from its final state.For m=0 there were few results concerning finite time blow up of the solutions with arbitrary positive initial energy，see［12-14］.In these papers the nonexistence of global solutions was proved by the well-known concave method of Levine（see［17］，Theorem 4）and some modifications of the method developed in［15］.For example，for nonlinearity（1.3）the assumptions on the initial data in［13］can be formulated in the following form

According to our best knowledge，in the case m＞0 the blow up result from［9］is the only result for blow up of the solutions with arbitrary positive energy given in the literature.In［9］for problem（1.1）with nonlinearity（1.3）we propose a new approach for proving the blow up which is based on sign invariant functionals.

The aim of this research is twofold:first，to generalize the approach from［9］for nonlinearities（H1）and（H2）and second，to prove nonexistence of global solution under following more general conditions on the initial data:

Under conditions（1.4）and（1.5）we prove that the Nehari functional I（t）is strictly negative for t≥0（see Theorem 2（i），（ii））.Moreover，we give explicitly the value tbof the time so that for t≥tbthe Nehari functional is sufficiently negative and the concave method of Levine can be applied.These new properties of the Nehari functional are based on the sign invariance of the new functionalWe construct explicitly initial data with arbitrary high positive energy，for which the existence time of the corresponding solution to（1.1）is finite.

The plan of the paper is as follows.In Section 2 some preliminary tools and definitions are formulated.In Section 3 sign preserving properties of some functionals under the flow of（1.1）are proved.By means of these sign preserving functionals，the main result of the paper，i.e.，the nonexistence of global solution to（1.1），is obtained in Section 4.Moreover，initial data with arbitrary high positive energy，satisfying all assumptions of the main theorem，are given in Section 4.

2　Preliminaries

First we formulate the local existence and the uniqueness of the solutions to problem（1.1）:

Theorem 2.1Suppose（1.2）holds and f（u）satisfies either（H1）or（H2）.Then there exists a maximal existence time Tm≤∞and a unique local weak solution to（1.1）

The proof of Theorem 2.1 is similar to the proof in［16，Theorem 2.1 and Lemma 3.1］and we omit it.

Now we introduce some important functionals and constants，crucial for our investigations and coming from the potential well method:the potential energy functional J（u），the Nehari functional I（u），the Nehari manifold N and the depth d of the potential well（the mountain pass level of J）:

When the argument of the functional I is a function u of t and x，i.e.，we use the short notation

Our nonexistence theorem is based on the following modification of the concave method of Levine［17］（see also［18］）:

for every t≥b and for some γ＞1.If either

then

From monotonicity of Ψ it follows thatThus（i）is proved.

Remark 2.3In Lemma 2.2 we prove that Ψ（t）is positive in contrast to［17］and［18］，where the positiveness of Ψ（t）is an additional assumption.Moreover，the result of Lemma 2.2 is true not only under condition（2.6）（as in［17，18］），but also under condition（2.7）.In fact the additional assumptionin（2.7）is always satisfied in the blow up theorems.However，，or equivalently，in Theorem 4.1，gives a wider class of initial data，for which the blow up result holds.

3　Sign Preserving Functionals

Now we introduce the functionals

In order to reveal the idea of the definition oflet us consider forthe decomposition

The energy E（t）can be estimated from below in the following way:

Our conjecture is that the influence of the termsin（3.4）is negligible for the finite time blow up of the solutions to（1.1），because of the orthogonality conditions

The success of our study is due to the sign invariance of the functionalsunder the flow of（1.1）.Note that the sign invariance of these functionals is interesting in itself. We need first the following auxiliary result.

Lemma 3.1Suppose（1.2）and（1.4）hold andsatisfies either（H1）or（H2）.Letbe the weak solution of problem（1.1）and

Then the following assertions for functionalsdefined in（3.1）and（3.2），respectively，hold:

（iii）

Proof（i），（ii）Using equation（1.1）we get

Straightforward computations lead to the following formulas for

（iii） Inequality（3.5）follows from the convexity ofand the monotonicity ofIndeed，

and from（3.3），（3.6）and the assumption for the negative sign ofwe get for s≥τ the inequalities

which proves（3.5）and Lemma 3.1.

The next theorem is concerned with the invariance of the functional I（t）for the flow governed by（1.1）and the additional assumption（1.5）.For this purpose we define the constant tas

Theorem 3.2Suppose（1.2），（1.4），（1.5）hold andsatisfies either（H1）or（H2）.

Then the following assertions are valid for the weak solution

Proof（i）From the conservation law（2.1）we have for the solution u withthe identity

for both cases（H1）and（H2）.Since，it follows that for every

（i）is proved at the initial moment t=0.

To establish（i）for t＞0 we use the proof by contradiction.Suppose that there existsFrom Lemma 3.1 we haveFrom（1.5），（3.8）and the monotonicity ofthe following chain of inequalities holds

（ii）From（i），（1.5），（3.7），（3.8），（3.9）and Lemma 3.1 we have for everythe estimate

4　Main Results

Our main result is formulated in the following theorem.

Theorem 4.1Suppose（1.2），（1.4），（1.5）hold and f（u）satisfies either（H1）or（H2）. Then

（i）every weak solution of（1.1）blows up for a finite time t?＜∞.Moreover，there exists a sequence

Thus all conditions of Lemma 2.2 are fulfilled with dataHence the functionblows up in a finite time，which contradicts our assumption.Moreover， there exists a sequencesuch thatThis means that eitherIn the second case from the conservation law（2.1）it follows that at least one of the normstends to infinity-From the imbedding theorem，we obtain that

Now we give initial data with arbitrarily high energy，for which the solution of（1.1）blows up in a finite time.

Theorem 4.2For every positive constant K there exist infinitely many initial datasuch thatand the existing time for the corresponding solutionis finite，i.e.

and define the initial data as

where the constants r＞0 andμ＞0 will be chosen below.For example，a possible choice of w，v is when w is an even function and v is an odd one.We fix functions w and v.

Straightforward computations give us the following expressions for the norms and energy:

Let K be an arbitrary positive number.From（4.1）it follows，thatandOne has to choose r andμsuch that both relations for initial data，inequality（1.5）andare satisfied.These relations are equivalent to

First we note that for all sufficiently large r the inequalityholds，because the leading term of R with respect to r has a negative sign.Thus we can choose r sufficiently large so that

which guarantees that the lhs of（4.2）is less or equal to the rhs of（4.2）.Now we chooseμso that

Thus for initial data（4.1）with already chosen parameters r andμall conditions of Theorem 4.1 are satisfied and the solution uKblows up in a finite time t?，where eitherIn this way Theorem 4.2 is proved.

Remark 4.3The result of Theorem 4.1 does not contradict the potential well method for subcritical and critical energies（0＜E（0）≤d）.Since assumptions（1.4）and（1.5）imply the inequality I（0）＜0，the potential well method is valid and gives nonexistence of global solution.The advantage of Theorem 4.1 is that the blow up result holds for initial data with supercritical energy E（0）＞d.

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?February 5，2015;revised November 18，2015.The authors are partially supported by Grant No. DFNI I-02/9 of the Bulgarian Science Fund.