On global solution, energy decay and blow-up for 2-D Kirchhoff equation with exponential terms
© Liu; licensee Springer. 2014
Received: 28 June 2014
Accepted: 3 October 2014
Published: 23 October 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.
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.
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.
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.
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:
The function is increasing in .
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.
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.
where κ is a positive constant.
Secondly, we consider the initial-boundary value problem (1.1) under the following general assumption.
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.
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
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).
as long as the approximated solutions exist. First we discuss the a priori estimate.
(a priori bounds)
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. □
for all . If , (3.5) is obvious.
on, where, and κ is a positive constant.
(a priori estimate)
whereis a constant depending increasing onand ().
where is a positive constant, as long as we choose .
Thus, we complete the proof of Lemma 3.3. □
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 .
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
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.
If and , then .
If , then .
If , then or , where .
Therefore from the assumption (A4), we obtain (4.5). □
- (1)If , then from (4.5), we have
If , then for . Furthermore, if , i.e. . Then we get for .
- (3)For the case that , we first note that(4.8)
then for all .
Consequently, we obtain the following lemma.
Next, we will estimate the lifespan of and prove Theorem 2.3.
The upper bounds of are estimated as follows by Lemma 4.2.
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).
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