Abstract
In recent years, the small initial boundary value problem of the Kirchhoff-type wave system attracts many scholars’ attention. However, the big initial boundary value problem is also a topic of theoretical significance. In this paper, we devote oneself to the well-posedness of the Kirchhoff-type wave system under the big initial boundary conditions. Combining the potential well method with an improved convex method, we establish a criterion for the well-posedness of the system with nonlinear source and dissipative and viscoelastic terms. Based on the criteria, the energy of the system is divided into different levels. For the subcritical case, we prove that there exist the global solutions when the initial value belongs to the stable set, while the finite time blow-up occurs when the initial value belongs to the unstable set. For the supercritical case, we show that the corresponding solution blows up in a finite time if the initial value satisfies some given conditions.
1. Introduction
This paper studies the initial boundary value problem for the following Kirchhoff-type wave system:where is a bounded domain in () with a smooth boundary , , , is a nonnegative function like , with , and : : , are given functions which will be specified later.
1.1. Historical Research
Kirchhoff-type wave system with nonlinear source and dissipative and viscoelastic terms have various applications in the field of physics and mechanics, which is the model to describe the motion of deformable solids. A single Kirchhoff-type wave equation is proposed:and (6) has its roots for the small amplitude vibrations of a string when and , but the tension of the string can not be ignored (see, e.g., Carrier et al. [1]). While (6) is used to describe the dynamics of an elastic string with fading memory when , this equation shows that the dynamic equilibrium of the object depends on both the present state of deformation and the history of the deformation gradient. Pohozaev and Tesei [2] proved that the solution exists in time if the datum satisfy an analytic-type condition for the case . This result of the case was extended by Torrejon and Yong [3]; they obtained the existence of weakly asymptotic stable solution. Later, Munoz Rivera [4] showed the existence of global solutions for small initial value and the exponential decay of the total energy. Then Wu and Tsai [5] established the global existence and energy decay under the assumption . Recently, this decay estimate was improved for a weaker condition on in [6].
Problem (6) is simplified to the following format without viscoelastic term:and some results of (7) concerning global well-posedness have been established in [7–11] for the case of . The above problem without source and dissipative terms is called Kirchhoff-type equation when is not a constant function, which was first introduced by Kirchhoff [12]; it describes the nonlinear vibrations of an elastic string. up to now, there are numerous results related to global well-posedness, including global existence, decay result, and blow-up properties; we refer the reader to [13–17].
Most recently Xu and Yang considered the initial boundary value problem of the following equation in [18]; they gave a blow-up result under supercritical energy:
Wave system such as (1) and (2) goes back to Reed [19] who proposed a similar system, but it does not contain and the viscoelastic terms . Subsequently, concerning blow-up and nonexistence, results in wave systems were discussed. Agre and Rammaha [20] studied the following concrete system:in with initial and boundary conditions of Dirichlet type, where and satisfy (A1) and (A2). They obtained several results concerning global well-posedness of a weak solution and showed that any weak solution blows up in finite time at negative initial energy. Afterwards, Alives et al. made further efforts as regards (9) in [21]. They obtained the global existence, uniform decay rates, and blow-up of solutions in finite time by involving the Nehari manifold when the initial energy is nonnegative and less than the mountain pass level value. And this blow-up result was improved by Said-Houari [22] when the initial data are large enough. In [23], Rammaha and Sakuntasathien studied a more general case of (9) by degenerating damping terms. Several results on the existence of local and global solutions as well as uniqueness are obtained by considering the constraint on the parameters of the system. Furthermore, they proved that the weak solutions blow up in finite time whenever the initial energy is negative and the exponent of the source term is more dominant than the exponents of both damping terms. Moreover, many studies of the global well-posedness for wave systems with dissipative terms have been researched in [24–28].
Wave systems with viscoelastic terms and dissipative terms have not been fully studied. In [29] the following coupled nonlinear wave equations with dispersive terms, viscoelastic dissipative terms, and nonlinear weak damping terms are considered:A global nonexistence theorem for certain solutions with positive initial energy is proved. Reference [30] further considered a fourth-order wave system similar to (10). In that case, the energy increases exponentially when time goes to infinity and the initial data are large enough.
Recently [31] considered a system of two coupled wave equations with dispersive and strong dissipative terms under Dirichlet boundary conditions:where The global existence of weak solutions and uniform decay rates (exponential one) of the solution energy were established.
Many researches considered the initial boundary value problem with global existence and blow-up of solutions for the nonlinear wave equations as follows:where is a bounded domain in () with a smooth boundary . When the viscoelastic terms are absent in (13), [20] showed local and global existences of a weak solution that any weak solution blows up in finite time with negative initial energy as the same way used in [22]. Later, Said-Houari extended this blow-up result to positive initial energy. At the same time, Liu [32] studied the following Cauchy problem for the coupled system of nonlinear Klein-Gordon equations with damping terms:and the existence of standing wave with ground state was stated and a sharp criterion for global existence and blow-up of solutions when was established. The author introduced a family of potential wells and the invariant sets and vacuum isolating behavior of solutions for and , respectively. Furthermore, he proved the global existence and asymptotic behavior of solutions when the condition is . Finally, a blow-up result with arbitrarily positive initial energy was obtained.
Conversely, in the presence of the memory term (), some results related to the asymptotic behavior and blow-up of solutions of viscoelastic systems were discussed. For example, [33] studied problem (13) with and and obtained the fact that the decay rate of the energy function is exponential under suitable conditions on the functions , certain initial data in the stable set. On the contrary, when the initial data is in the unstable set, the solutions blow up in finite time under positive initial energy. For and , [26] established local and global existence as well as finite time blow-up (the initial energy ). The latter blow-up result has been improved by [29] by considering a class of positive initial data. On the other hand, Messaoudi and Tatar [34] considered the following similar wave equations without considering coupling coefficient of and :where the functions and satisfy the following assumptions: and for some constant and , the solution goes to zero with an exponential or polynomial rate which depending on the decay rate of the relaxation functions was obtained. This result was improved in [35] to weaker conditions on the relaxation functions and more general coupling functions.
Additionally, Liu and Wang [36] considered the following nonlinear hyperbolic systems with damping and source termsThe author defined potential well and the outer manifold of the potential well associated with system (17) and got the global existence in the case of and discussed the global nonexistence of solutions for problem (1)–(5) in the case of and on page 88 of [36]. In [37], (1)–(5) was considered with and without imposing the memory terms (). The rate of decay of the exponential or polynomial energy of the damping terms was obtained.
Recently, Wu [38] considered the initial boundary value problem for system (1)–(5) with . The assumptions for problem (1)–(5) are as follows:(A1)The nonlinear source terms , satisfy where with .(A2) is a nonnegative function for satisfying (A3)The nonlinearity of satisfies(A4)For the relaxation functions and are of class and satisfy The solutions are global in time when the functions , , and , satisfy suitable conditions and certain initial data is in the stable set. The author established the rate of decay of solutions by a difference inequality given by Nakao [39] and intended to study the blow-up phenomena of problem (1)–(5). The blow-up of solutions when the energy is negative or subcritical case was proved by adopting and modifying the methods used in [29]. In this way, the above results in [38] allowed a bigger region for the blow-up results and improved the results of Messaoudi and Said-Houari [29]. More specifically, the decay result in [38] extends the one in [16, 37] to problem (1)–(5), where is not a constant function and the equations considered in [38] have more dissipations.
1.2. Unsolved Problems
It is well known that in the absence of the nonlinear source term the damping term ensures global existence. In addition, without the dissipative term, the nonlinear source term causes finite time blow-up of solution. Moreover, the viscoelastic materials possess a capacity of storage and dissipation of mechanical energy; therefore, it is interesting to investigate the well-posedness of solution for the viscoelastic equation with dissipative term and nonlinear source term.
We can see that problem (1)–(5) contains system (9) (), systems (10), (13), and (15) (), system (11) (), and system (17) () as special cases. In fact, the viscoelastic terms change the frame of the equations comparing without those viscoelastic conditions, which makes the structure of the equations and solutions more complex and which also makes energy decay faster. So the classical methods can not be applied to investigate properties of solutions. Therefore, much less is known for (1)–(5) with viscoelastic terms. We can note that in all of the above studies except [38] only the global well-posedness of solutions was proved in a relatively rough variational framework, but the global existence and finite time blow-up for problem (1)–(5) at the other initial energy levels have not been discussed yet. The global existence and blow-up results in [38] are under the assumption of negative initial energy or . In other words, to our knowledge there are no results on global well-posedness of solutions to the initial boundary value problem for a couple of nonlinear wave equations with coupling coefficient of and , viscoelastic terms , and nonlinear weak dissipative terms , . It is natural to ask a question of how the solution behaves for problem (1)–(5), which is what we want to deal with in this paper. Moreover, regarding the initial energy level, the present paper is also a comprehensive study for low energy case and high energy situation. To our knowledge this is the first try to consider this problem. The most attractive one is that we attain a blow-up result with arbitrary positive initial energy for a wave system of Kirchhoff type.
By reviewing above known results and also [19–38], we will face the fact that the following unsolved problems arise naturally. Firstly, from [38] we know the global existence for the definitely positive energy, but we know less for the initial energy which may be negative. Secondly, for general initial energy, which means the initial energy is not necessarily definitely positive, what will happen for the solution when the initial energy or ?
We restricts our attention to considering the global existence and blow-up at two different initial energy levels. Since the initial energy level plays a crucial role in dealing with the well-posedness of problem (1)–(5), the two cases are, respectively, tackled with different tools. For the subcritical case , there have been many tools tackling the hyperbolic problem without viscoelastic terms in [16, 17]. We may refer the tools to deal with (1)–(5) with viscoelastic terms. By the well-known works [40–44], we see that the supercritical case is not easy to deal with. Filippo and Marco [45] made the initial attempt to consider the global well-posedness of hyperbolic problem at high initial energy level . However, they focused on particular source term , and our work is on the viscoelastic terms condition and the complex source term . Based on the general comparison principle in [9], we try to resolve the above open problems with variational methods. In this paper, we consider the initial boundary value problem for system (10) with and the nonlinear source terms, coupling coefficient , the nonlinearity of , and the relaxation functions and satisfying the assumptions (A1)–(A4), respectively. In addition, we consider nonlinear damping terms of the form and as in the first equation and in the second equation of (10), respectively.
1.3. The Main Results and Organization of the Paper
In this paper, we mainly discuss the following problems.(1)Case : different from the method applied in [38], we introduce a family of potential wells to obtain the results: invariant sets, global existence, and finite time blow-up.(2)Case : we obtain the finite time blow-up of solutions for problem (1)–(5) whose initial data have arbitrarily high initial energy.
We can summarize our main conclusions in Table 1 and use the question mark “?” to indicate the open problem.
The organization of the paper is as follows.
In Section 2, we introduce some notations, assumptions, and preliminaries.
From Sections 3–5, we prove the main results.
2. Notations and Primary Lemmas
In this section, we shall give some lemmas and some notations which will be used throughout this work. We use the standard Lebesgue space and Sobolev space with their usual norms and products as follows:
We will use the embedding for , if or , if . In this case, the embedding constant is denoted by ; that is,
From assumption (A1) one can easily verify that Moreover we have the following result. Note that the following conclusion (Lemma 1) was assumed throughout many papers (see [16–31, 33, 34]); however in our opinion we think this conclusion is a deduction of assumption (A1). Thus we present this conclusion as follows and similar proof can be found in [29].
Lemma 1. There exist two positive constants and such that
Proof. We can see that taking then the right-hand side of inequality (26) is trivial. For the left-hand side, the result is also trivial if . If, without loss of generality, , then either or .
For , we have Consider the continuous function So . If then, for some , we have This implies that , which is impossible. Thus . Therefore Consequently, and then If , similarly we have This leads to the desired result and completes the proof of Lemma 1.
As in [29], we still have the following results.
Lemma 2 (see [29]). Suppose that (20) holds. Then there exists a positive constant such that, for any , one has
We also need the following technical lemma in the course of the investigation.
Lemma 3 (see [29]). For any and , one haswhere .
Now, we are in a position to state the local existence result to problem (1)–(5), which can be established by combining arguments of [15, 17, 20, 26].
Theorem 4 (local existence). Let and be given. Assume that (A2)–(A4) are satisfied. Then there exists a couple solution of problem (1)–(5) such that for some .
Remark 5 (see [46]). Condition (A1) is necessary to guarantee the hyperbolicity of the equation in (1) and (2) and condition (21) is needed to establish the local existence result.
Next for problem (1)–(5) we introduce potential energy functional:Potential energy functional:Nehari functional:For the definition of please see assumption (A1) in the beginning of this paper. Moreover we introduce the potential well (stable set)and the outer space of potential well (unstable set)Moreover we define or equivalently where .
Lemma 6 (depth of potential well). The depth of potential well , where is defined in (26) and is the best imbedding constant from into .
Proof. From the definition of , we have ; that is, . Then on the one hand from Lemma 1 we get Notice that, from assumption (A4) and the definitions and , we have that isOn the other hand notice ; moreover by virtue of (38), (39), and (78), we get and hence we have .
Lemma 7 (nonincreasing energy). Let be a solution of problem (1)–(5); then is a nonincreasing function for ; that is,
Proof. Multiplying (1) by and (2) by , integrating them over , and then adding the results together and integrating by parts, it follows that Exploiting Lemma 3 on the third term and the fourth term on the right side of the above equality, we can obtain (48) for any regular solution.
3. Global Existence under the Case
Now we give the following definition of weak solution for problem (1)–(5).
Definition 8 (weak solution). A function is called a weak solution of problem (1)–(5) on , if it satisfies with andwith
Lemma 9 (invariant set ). Let , , and (A1)–(A4) hold. Then all solutions of problem (1)–(5) with belong to , provided .
Proof. Let be any local weak solution of problem (1)–(5) with and and be the existence time of . Then it follows from Lemma 7 that . Thus if suffices to show that for . Suppose that there exists such that . From the continuity of the solution in time, there exists such that . Then from the definition of we have which is a contradiction.
Then we give the global existence of solutions for problem (1)–(5) with low initial energy level .
Theorem 10 (global existence when ). Let , , and (A1)–(A4) hold. Assume that and . Then problem (1)–(5) admits a global weak solution , , and for .
Proof. Let be a basis in given by the eigenfunction of the operator and it constructs a complete orthogonal system such that for all . Then is orthogonal and complete in and in . Let be the space generated by , . Construct the approximate solutions of problem (1)–(5): satisfyingMultiplying (54) and (55) by , , respectively, and summing for and adding these two equations, we can deduce Integrating the above equation with respect to , we havewhereFrom , (56), and (57) we get that as Therefore we have as . Then for sufficiently large we haveNote thatHence, from (62) and (63), we getBy andtaking into account (56) and (57), we can get for sufficiently large . From (62) and an argument similar to the proof of Lemma 9 we can prove that for and sufficiently large . Thus (64) givesfor sufficiently large and . Inequality (66) givesFurthermore, according to (68), the following results hold:Hence integrating (54) and (55) with respect to , for every and , we haveTherefore, up to a subsequence, by (67)–(71), we may pass to the limit in (72) and obtain a weak solution of problem (1)–(5) with the above regularity (67)–(71) and (50). On the other hand, from (56) and (57) we have in and in .
4. Finite Time Blow-Up When
Let us turn to discuss blow-up properties of solutions for system (1)–(5) when , . We firstly give the following definition of finite time blow-up of weak solution for problem (1)–(5).
Definition 11 (finite time blow-up). A solution of problem (1)–(5) is called a blow-up solution if there exists a finite time such that
By the same argument as Lemma 9, we can get the following lemma.
Lemma 12 (invariant set ). Let , , and (A1)–(A4) hold. Then all solutions of problem (1)–(5) with belong to , provided .
In order to prove Theorem 14 we state some relations of the depth of potential well , norm , and function as follows.
Lemma 13. Under the assumptions of Lemma 15, one has
Proof. From Lemma 6 for the depth of potential well , we have where is defined in (26) and is the best imbedding constant from into . By Lemma 15, we get ; that is, . Moreover by Lemma 1 and Sobolev embedding inequality, implies that Notice that, from assumption (A4) and the definitions and , we have that isHence we obtain
A finite time blow-up result of solutions for problem (1)–(5) is showed as follows.
Theorem 14 (finite time blow-up when ). Let , , and (A1)–(A4) hold; , hold. Assume that , , and satisfyThen the existence time of solution of problem (1)–(5) is finite.
Proof. Let be any solution of problem (1)–(5) with and . Next, we prove the solution of problem (1)–(5) blows up in finite time. Suppose by contradiction that the solution is global. Then, for any , we define a auxiliary function byClearly for all . From the continuity of in , it is easy to see that there exists (independent of the choice of ) such thatThen for we haveApplying Young’s inequality to estimate the fourth term on the right side of (84), we have for any . Similarly we have for any . Then (84) arrives atOn the other hand from (83), we have Using the Schwarz inequality, (88) takes on the form and therefore (88) becomesThen by (81), (87), and (90), we have Now we defineFrom the definition of , (92) becomesThen from Lemma 2 with , (93) arrives atFrom assumption (A4) on and we can derivewhere We next estimate the terms , , and one by one as follows. For the term from we haveHere by taking and and by (80) we have From Lemma 16 we can derive With the fact we have So from (95) we have . Therefore we can derive Setting , this inequality becomes This proves that reaches in finite time, say . Since is independent of the initial choice of , we may assume that . This tells us that which completes the proof.
5. A Finite Time Blow-Up When
We first present the following lemmas in order to prove Theorem 18.
Lemma 15. Let condition (A4) hold and the nonlinear viscoelastic terms and satisfyIf is a twice continuously differentiable function and satisfies the inequalityand the initial conditionfor every , where is the corresponding solutions of problem (1)–(5) with and , then the function is strictly increasing on .
Proof. Consider the following auxiliary ordinary differential equation:with the initial conditionfor every .
Clearly we can find the following function:as a solution of the ODE (109) and (110) for every .
Now in order to show that we need to prove thatFrom (105) a direct computation on (111) yieldsfor every , which says that Moreover from (108) and (114) it implies that Suppose by contradiction that the first inequality of (113) is invalid; then there exists such that From the continuity of the solution in time, there exists such thatOn the other hand we have the following ordinary differential inequality for every . This ordinary differential inequality can be solved as which contradicts (118). Thus we prove the first inequality of (113), which together with (114) states (113). So we complete this proof.
Lemma 16. Let , , , and be the solution of problem (1)–(5) with the initial data and . Assume the initial data satisfythen the map is strictly increasing as long as .
Proof. Letthen we haveAdding (124) and (125) we havewhich, from the fact that , implies thatTherefore, applying Lemma 15 with the fact that we can obtain that the map is strictly increasing.
In the following, we show the invariance of the unstable set under the flow of the problem (1)–(5).
Lemma 17. Let , , , and be the solution of problem (1)–(5) with the initial data and . Assume that the nonlinear viscoelastic terms and satisfyand the initial data satisfy (121) andwhere is the coefficient of Poincaré inequality and is the coefficient of Poincaré inequality . Then all solutions of problem (1)–(5) with belong to , provided .
Proof. We prove . If it is false, let be the first time such that ; that is, , for and . Now let be defined by (123) above. Hence from Lemma 16, we get that and are strictly increasing on the interval . And then by (131), we have Moreover, from the continuity of in , we obtainOn the other hand, by (37) and (39) we can obtainNote that ; hence we haveUsing the Poincaré inequality we haveBy (138), we deduce (136) towhich meansIt is obvious that (140) contradicts (131).
Theorem 18 (finite time blow-up under the case of and ). Let , , and (A1)–(A4) hold. Assume that the nonlinear viscoelastic terms and satisfy (105), (106), and (130) and the initial data satisfy (121), (131), and . Then the solution of problem (1)–(5) with and blows up in finite time.
Proof. Recalling the auxiliary function defined as (81) and the proof of Theorem 14, we haveSince (141) holds for any , we can choose ; then (141) becomesThen from Lemma 17 and Poincaré inequality, we conclude thatwhich means . Similar to the proof of Theorem 14, by the concavity argument, we conclude the result.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (11071033 and 11101102), the PhD Start-Up Fund of Liaoning Province of China (nos. 20141137 and 20141139), the Natural Science Foundation of Liaoning Province of China (no. 20170540004), and the Liaoning BaiQianWan Talents Program (no. 2013921055).