On global solution, energy decay and blow-up for 2-D Kirchhoff equation with exponential terms
Boundary Value Problems volume 2014, Article number: 230 (2014)
This paper is concerned with the study of damped wave equation of Kirchhoff type in , with initial and Dirichlet boundary condition, where Ω is a bounded domain of having a smooth boundary ∂ Ω. Under the assumption that g is a function with exponential growth at infinity, we prove global existence and the decay property as well as blow-up of solutions in finite time under suitable conditions.
MSC: 35L70, 35B40, 35B44.
Let Ω be a bounded domain with smooth boundary ∂ Ω, we are concerned with the initial-boundary value problem
where g is a source term with exponential growth at the infinity to be specified later, is a positive class function in . It is said that (1.1) is non-degenerate if there exists a constant such that for all . If there exists a point such that , then it is said that (1.1) is degenerate. In the case , (1.1) is usually a semilinear wave equation. In this paper, we only consider non-degenerate case.
It is known that Kirchhoff  first investigated the following nonlinear vibration of an elastic string for :
where is the lateral displacement at the space coordinate x and the time t, ρ the mass density, h the cross-section area, L the length, E the Young modulus, the initial axial tension, δ the resistance modulus, and f the external force.
Recently, Alves and Cavalcanti  studied the following problem with nonlinear damping term:
where g is a source term with exponential growth at infinity to be specified later, is a monotone continuous function with polynomial growth at infinity and with no restriction on the growth rate near the origin. There are few works in the literature dealing with the exponential source even for wave equation, the work  is a recent one in this direction. In  Ma and Soriano studied an evolution equation with exponential term of the following form:
with initial and Dirichlet boundary condition, where is a bounded domain with smooth boundary ∂ Ω, , grows like and satisfies the sign condition . More recently, Han and Wang  studied the following problem:
with initial and Dirichlet boundary condition, where is a bounded domain with smooth boundary ∂ Ω, is just the term considered in . In fact, when , the problem (1.3) was studied by Gazzola and Squassina in .
To the author’s knowledge, there are few works in the literature dealing with the exponential source for Kirchhoff equations. When the source term is a nonlinear function like for , the problem (1.1) has been discussed by many authors; see – and the references cited therein.
Motivated by there papers, in this study, we concentrate on studying the problem (1.1) with for constant . In what follows, we would like to introduce some well-known theory of elliptic problems. More precisely, defining the functional by
where . The critical points of the functional are the weak solutions of the elliptic problem
Defining the functional by
Related to the functional , we have the well-known Nehari manifold:
If g satisfies some suitable properties, it is possible to prove the functional satisfies the hypotheses of the mountain pass theorem due to Ambrosetti and Rabinowitz , and the level
called mountain pass level is a critical level for . By Theorem 4.2 in , the mountain pass level d can be characterized as
In order to study the problem (1.1), we define some additional functionals. Define
where . Then we can define
Now, as usual setting
The remainder of this paper is organized as follows. Section 2 is concerned with some notation, statement of assumptions and the main results. Sections 3 and 4 are devoted to the proofs of the main results.
2 Assumptions and main results
To state our results, we need the following assumptions.
(A1) Assume that is a function satisfying:
For each , there exists a positive constant such that
Near the origin we have
The function is increasing in .
(A2) There exists a positive constant such that
A typical example of functions satisfying (A1) is
where , , arbitrarily chosen. From (2.2), for each fixed, there exists such that
Moreover, from (2.1), for each and fixed, there exists such that
Hence, for each and fixed, there exists δ and satisfying
The assumptions (A1) and (A2) have been used in . The condition (2.3) is the well-known Ambrosetti-Rabinowitz condition, widely used in elliptic problem. Also as remarked in , , the mountain pass level d can be characterized by (1.6) provided (2.1), (2.2), and (2.3) hold.
Let Ω be a bounded domain in, . For all,
and there exist positive constants and such that
whereandis the-dimensional surface of unit sphere, specially, .
Now, we state our main results. First, we consider the problem (1.1) with for . We have the following global existence and decay result.
Assume that (A1) and (A2) hold, for. Then there exists an open set S in, which contains, ifand the initial energy, thenon. Furthermore, suppose that there exists a constantsuch that
then the problem (1.1) has a unique solutionsatisfying
Furthermore, we have the following energy decay estimate:
where κ is a positive constant.
Secondly, we consider the initial-boundary value problem (1.1) under the following general assumption.
(A3) Assume that the sign condition holds for all . is a positive function on , and
Then we can state the global existence and energy decay to the related problem (1.1).
Let (A1) and (A3) hold, then there exists an open set S in, which containssuch that if, then the conclusions of Theorem 2.1hold.
Our final result is concerned with the blow-up phenomenon. First of all we give the following assumption.
(A4) There exists a positive constant δ such that
Under the assumptions (A1) and (A4), and that either one of the following conditions is satisfied:
and (4.10) holds, where to be chosen later.
3 Global existence and energy decay
In this section, we will give the solvability in the class of and the energy decay of the problem (1.1). From now on we denote c or various positive constants.
3.1 Proof of Theorem 2.1
In this section we take for , and and . We employ the Galerkin method to construct a global solution. Let be a sequence of eigenvalues for in Ω and on ∂ Ω. Let be the corresponding eigenfunction to and take as a completely orthonormal system in . We construct approximate solutions in the form , where are determined by the following ordinary differential equations:
System (3.1) can easily be solved by Picard’s iteration method, hence it admits a local solution on some interval with . Note that is of class. We shall see that can be extended to , which needs some prior estimates for . But this procedure allows us to employ the energy method for an assumed smooth solution to the problem (1.1) (the results should be in fact applied to approximated solutions).
Now, it is easy to see the following energy identity:
as long as the approximated solutions exist. First we discuss the a priori estimate.
(a priori bounds)
Letbe a solution with the initial data, , and the initial energy. And the assumptions (A1) and (A2) hold. Thenon. Furthermore, there exists a constantsuch that
Since , it follows from the energy identity (3.2) and the initial energy that
which together with the Ambrosetti-Rabinowitz condition (2.3) implies
for all . At the same time, these estimates imply that the (approximated) solution can be extended to the whole interval . This concludes the proof of Lemma 3.1. □
Moreover, since , we have for all from Lemma 3.1. If , using (2.3), we have
for all . If , (3.5) is obvious.
Under the assumptions imposed on Lemma 3.1, and suppose that there exists a constantsuch that
Then we have the energy satisfies the decay estimates
on, where, and κ is a positive constant.
Hence, from Lemma 3.1 and (3.6), we deduce that
Multiplying (1.1) by and integrating over , we obtain
Thus, there exist two numbers and such that
Multiplying (1.1) by u and integrating over , we obtain
where . Since , we get
for some constant . Hence, there exists a constant such that
We are now in a position to obtain a priori bounds. Set
(a priori estimate)
Under the assumptions imposed on Lemma 3.2. Supposeis a local solution onsuch thatonfor some K and. Then we have the following estimate:
whereis a constant depending increasing onand ().
Multiplying (1.1) by and integrating over Ω, we obtain
On the other hand from , we deduce from the Moser-Trudinger inequality that
where is a positive constant, as long as we choose .
Hence, Sobolev’s inequality and the interpolation inequality imply
Integrating (3.18) over , noticing for some , we obtain
Thus, we complete the proof of Lemma 3.3. □
Let and put
By the same method as considered in , we can deduce that S is an open unbounded set, and if , the solution can be continued globally on and for all .
Uniqueness: Let and be two solutions; satisfies
with on and in Ω. Taking the inner product on both sides of (3.20) with , we can easily find that
Using assumption (A1), or more precisely (2.1), we estimate the last term as
Since , are two solutions, from (3.5), we obtain , . Repeating a similar procedure as estimating the term , after employing the Hölder and the Moser-Trudinger inequality, yields
On the hand the first and the second term on the right-hand side of (3.21) are bounded by
respectively. Thus, integrating (3.21) over , we obtain
which implies by Gronwall’s inequality. Thus, we complete the proof of Theorem 2.1.
As is well known, the difficult for Kirchhoff equations is proving the approximate solutions converge to the desired solution. Indeed, we prove the local existence solution for the problem (1.1) by Picard’s iteration method. To utilize the standard compactness argument for the limiting procedure, it suffices to derive some a priori estimates for (see Lemma 3.1 and Lemma 3.3). In this direction, we also mention  and .
3.2 Proof of Theorem 2.2
In this section, we will give the proof of Theorem 2.2, which is similar to the proof Theorem 2.1. We sketch it as follows.
Proof of Theorem 2.2
For brevity, we take the same notations , , , and as in the proof of Theorem 2.1, but since for all , we can deduce
Similar to the proof of Lemma 3.2, we have
From (3.22) and the Moser-Trudinger inequality, we have
where c is a positive constant, as long as we choose . Hence, by the Sobolev inequality, there exists a constant c such that
Then, by the same argument of Lemma 3.2 we can obtain the decay estimate
where κ is a positive constant. Hence, it suffices to show a priori bounds under the assumption , and on for some and . Set
By the same derivation as Lemma 3.2, using (A3), we deduce
Thus, we can prove Theorem 2.2 in the same way as Theorem 2.1. □
When , (1.1) is a wave equation, Alves and Cavalcanti  obtain the general energy decay result. Indeed, Lemma 3.3 (to be precise: (3.44)) in  plays an important role in the proof of energy decay, where the authors used the unique continuation property of wave equations; see  for details and , ,  for an application. But in our case, since is nonlinear, we cannot use the unique continuation property directly.
4 The blow-up in finite time
In this section, we shall discuss the blow-up properties for the problem (1.1). For this purpose, we use the following lemmas.
Let and be a nonnegative function satisfying
thenfor, whereis a constant, is the smaller root of the equation
Ifis a nonincreasing function on, , and satisfies the differential inequality
where, , then there exists a finite timesuch that
and the upper bound ofis estimated, respectively, by the following cases:
If and , then .
If , then .
If , then or , where .
A solution of (1.1) is called a blow-up solution if there exists a finite time such that
For the next lemma, we define
Assume that (A1) and (A4) hold, then we have
From (4.4), we obtain
From the above equation and the energy identity, we obtain
Therefore from the assumption (A4), we obtain (4.5). □
Now, we consider three different cases on the sign of initial energy .
If , then from (4.5), we have
Thus, we get for , where
If , then for . Furthermore, if , i.e. . Then we get for .
For the case that , we first note that(4.8)
By the Hölder inequality and the Young inequality, we have from (4.8)
Then, from the above inequality and (4.5), we obtain
Then satisfies (4.1). From Lemma 4.1, we see that if
then for all .
Consequently, we obtain the following lemma.
Assume that (A1) and (A4) hold and that either one of the following conditions is satisfied:
Next, we will estimate the lifespan of and prove Theorem 2.3.
where is some certain constant which will be chosen later. Then we get
For simplicity, we denote
By (4.9) and the Hölder inequality, we obtain
From (4.5), we have
where we have used Schwarz inequality in the last but one term. Therefore from (4.12), we have
Note that by Lemma 4.1, for . Multiplying (4.16) by and integrating it from to t, we have
We observe that
Then by Lemma 4.2, there exists a finite time such that and the upper bounds of are estimated, respectively, according to the sign of . This yields
The upper bounds of are estimated as follows by Lemma 4.2.
In case (i),
Furthermore, if , then we have
In case (ii),
In case (iii),
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The author would like to thank the Prof. Hua Chen in Wuhan University for pertinent discussions. The author would also like to thank the referee for the careful reading of this paper and for the valuable suggestions to improve its presentation and style. This work is supported by the Doctor Foundation of Henan University of Technology (2012BS058) and Plan of Nature Science Fundamental Research in Henan University of Technology (No. 2013JCYJ11).
The author declares to have no competing interests.
The author read and approved the final manuscript.
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Liu, G. On global solution, energy decay and blow-up for 2-D Kirchhoff equation with exponential terms. Bound Value Probl 2014, 230 (2014). https://doi.org/10.1186/s13661-014-0230-3