Abstract
In this paper, we investigate the initial value problem for the semilinear beam equation. Under a small condition on the initial value, we prove the global existence and optimal decay estimate of solutions. Moreover, we show that as time tends to infinity, the solution is asymptotic to a diffusion wave, which is given explicitly in terms of the solution of parabolic equation.
MSC: 35L30, 35L75.
Keywords:
beam equation; decay estimate; asymptotic profile; diffusion wave1 Introduction
We investigate the initial value problem for the following semilinear beam equation:
with the initial value
Here is the unknown function of and , , and are constants. The nonlinear term is a given smooth function of . More precisely,
This initial value problem was studied by [1,2] when f satisfies
with , so that . Here C is independent of v, , and . [1] proved that there exists a global solution to the problem (1.1), (1.2) under smallness condition on the initial data and . In particular, they showed the decay estimates:
and
In addition to the above assumptions, suppose that the initial data , Takeda and Yoshikawa [2] established the following asymptotic profile of global solution:
where
and
The main purpose of our present paper is twofold: first, we try to recover all the results about global existence and decay estimate of solution in Takeda and Yoshikawa [1] under some assumptions on the initial data and the nonlinear function f, which is much weaker than those needed in Takeda and Yoshikawa’s arguments. More precisely, the condition , and (1.3) have been relaxed to , () and the nonlinear function f satisfies in this paper. Second, we show that the solution is asymptotic to a diffusion wave, given explicitly in terms of the solution of parabolic equation that is different from the one in [2]. For the details, we refer to Theorem 4.1. Moreover, under some additional assumptions on the initial data, we also prove that the convergence rates of our new asymptotic profile are better than that obtained by [2]. For details, we refer to Theorem 4.2.
The study of the global existence and asymptotic behavior of solutions to hyperbolictype equations has a long history. We refer to [3,4] for hyperbolic equations, [57] for the damped wave equation and [816] for various aspects of dissipation of the plate equation.
The paper is organized as follows. In Section 2, we study the decay property of the solution to the linear problem. Then, in Section 3, we prove the global existence and decay estimate of the solutions. Finally, we prove that the solution is asymptotic to a diffusion wave, which is given explicitly in terms of the solution of the parabolic equation in Section 4.
Notations We give some notations which are used in this paper. Let denote the Fourier transform of u defined by
and we denote its inverse transform by .
For , denotes the usual Lebesgue space with the norm . The usual Sobolev space of s is defined by with the norm .
Finally, in this paper, we denote every positive constant by the same symbol C or c which will not lead to any confusion. is the Gauss symbol.
2 Linear problem
2.1 Solution formula
The aim of this subsection is to derive the solution formula for the problem (1.1), (1.2). We first investigate the linearized equation of (1.1):
with the initial data in (1.2), where . We apply the Fourier transform to (2.1). This yields
The corresponding initial values are given as
The characteristic equation of (2.2) is
Let be the corresponding eigenvalues of (2.4), we obtain
The solution to the problem (2.2), (2.3) in the Fourier space is then given explicitly in the form
where
and
and
respectively, where denotes the inverse Fourier transform. Then, applying to (2.6), we obtain
By the Duhamel principle, we obtain the solution formula to (1.1), (1.2),
where is a smooth function; it satisfies .
2.2 Decay property
The aim of this subsection is to establish decay estimates of the solution operators and appearing in (2.9) and (2.10), respectively.
Lemma 2.1The solution of the problem (2.2), (2.3) satisfies
Proof Multiplying (2.2) by and taking the real part yields
Multiplying (2.2) by and taking the real part, we obtain
Multiplying both sides of (2.14) by 2 and summing up the resulting equation and (2.15) yields
where
and
A simple computation implies that
where
Note that
It follows from (2.17) that
Using (2.16) and (2.18), we get
Thus
which together with (2.17) proves the desired estimates (2.13). Then we have completed the proof of the lemma. □
Lemma 2.2Letandbe the fundamental solution of (2.1) in the Fourier space, which are given in (2.7) and (2.8), respectively. Then we have the estimates
and
Proof Firstly, we investigate the problem (2.1), (1.2) with , from (2.6), we obtain
Substituting the equalities into (3.1) with , we get (2.19).
In what follows, we consider the problem (2.1), (1.2) with ; it follows from (2.6) that
Substituting the equalities into (2.13) with , we get the desired estimate (2.20). The lemma is proved. □
Let
be the fundamental solution to .
Lemma 2.3Letandbe the fundamental solution of (2.1) in the Fourier space, which are given in (2.6) and (2.7), respectively. Then there is a small positive numbersuch that ifand, we have the following estimates:
and
Proof For sufficiently small ξ, using the Taylor formula, we get
For sufficiently small ξ, from (2.24) and (2.25), we immediately obtain
For sufficiently small ξ, from (2.7) and (2.24), we immediately get
Thus we get (2.21). The other estimates are proved similarly and we omit the details. The proof of Lemma 2.3 is completed. □
Lemma 2.4Letandbe the fundamental solutions of (2.1), which are given in (2.9) and (2.10), respectively. Let, and letk, j, andlbe nonnegative integers. Then we have
for, wherein (2.26). Similarly, we have
Proof We only prove (2.26). By the Plancherel theorem and (2.19), the HausdorffYoung inequality, we obtain
For the term , letting , we have
where we used the Hölder inequality with and the HausdorffYoung inequality for . On the other hand, we can estimate the term simply as
Combining the above three inequalities yields (2.26). This completes the proof of Lemma 2.4. □
From Lemma 2.4, we immediately have the following corollary.
Corollary 2.1Letandbe the fundamental solution of (2.1), which are given in (2.8) and (2.9), respectively. Let, and letk, j, andlbe nonnegative integers. Then we have
Lemma 2.5Letbe the fundamental solution of (2.1), given in (2.8) and letbe the fundamental solution of (2.1), given in (2.8). Let, and letk, j, andlbe nonnegative integers. Then we have
Proof The proof of Lemma 2.5 is similar to the proof of Lemma 2.4. By employing (2.21) and (2.23), we can prove Lemma 2.5. We omit the details. □
3 Global existence and asymptotic behavior of solutions to (1.1), (1.2)
The purpose of this section is to prove global existence and optimal decay estimate of solutions to the initial value problem (1.1), (1.2). We need the following lemma, which comes from [17] (see also [18]).
Lemma 3.1Assume thatis a smooth function. Suppose that (is an integer) when. Then for integer, ifand, the following inequalities hold:
Theorem 3.1Let. Suppose thatis a smooth function and. Assume that, . Put
Then there exists a positive constantsuch that if, and the initial value problem (1.1), (1.2) has a unique global solutionsatisfying
Moreover, the solution satisfies the decay estimate
and
Proof The existence and uniqueness of small solutions can be proved by the contraction mapping principle. Here we only show the decay estimates (3.2) and (3.3) for the solution u of (2.12) satisfying with some . To this end, we introduce the quantity
Here we note that
provided that . This follows from the GagliardoNirenberg inequality, and the definition of in (3.4).
Applying to (2.12) and taking the norm, we obtain
Firstly, we estimate . We apply (2.26) with , and ( for , for ). This yields
where . Similarly, applying (2.27) with , , and to the term , we have
We estimate the nonlinear term J. We divide J into two parts and write , where and are corresponding to the time intervals and , respectively. For the term , we apply (2.30) with , , and . This yields
Here we see that by Lemma 3.1. Thus we have . Therefore we can estimate the term as
On the other hand, we have by Lemma 3.1. Therefore, using (3.5), we find that . Consequently, we can estimate the term as
Finally, we estimate the term on the time interval . Applying (2.30) with , , and , and using , we obtain
Thus we have shown that
Substituting all these estimates into (3.6), we obtain
for . Consequently, we have , from which we can deduce , provided that is suitably small. This proves the decay estimate (3.2).
In what follows, we prove the decay estimate (3.3) for the time derivative . For this purpose we differentiate (2.12) with respect to t to obtain
Applying to (3.11) and taking the norm, we have
where . For the term , we apply (2.28) with , , and to get
Also, for the term , applying (2.29) with , and , we have
To estimate the nonlinear term , we rewrite , where and correspond to the time intervals and , respectively. For the term , we apply (2.31) with , , and . This yields
Since as before, we can estimate the term as
Also, the term is estimated similarly as before and we have . Finally, we estimate the term by applying (2.31) with , , and and obtain
where we used the estimate . Consequently we have shown that
Substituting all these estimates together with the previous estimate into (3.12), we arrive at the desired estimate (3.3) for . This completes the proof of Theorem 3.1. □
The above proof of Theorem 3.1 shows that the solution u to the integral equation (2.12) is asymptotic to the linear solution given by the formula in (2.11) as . This result is stated as follows.
Corollary 3.1Assume the same conditions of Theorem 3.1. Then the solutionuof the problem (1.1), (1.2), which is constructed in Theorem 3.1, can be approximated by the solutionto the linearized problem (2.1), (1.2) as. More precisely, we have the following asymptotic relations:
4 Asymptotic profile
In this section, our aim is to establish an asymptotic profile to our global solution that is constructed in Theorem 4.1. In the previous section, we have shown that the solution u to the problem (1.1), (1.2) can be approximated by the linear solution . In what follows, we shall derive a simpler asymptotic profile of the linear solution .
Let v be the solution to the initial data problem
Then
gives a asymptotic profile of the linear solution . In fact we have the following.
Lemma 4.1Let. Assume thatand put. Letbe the linear solution and letvbe defined by (4.2). Then we have
Proof Note that , so for the proof of (4.3), it suffices to show the following estimates:
where . These estimates can be obtained by (2.27) and (2.32). Here we omit the details. □
the diffusion wave with the amount . Obviously, satisfies the following problem:
It is not difficult to prove the following lemma.
Combining Corollary 3.1, Lemma 4.1, and Lemma 4.2, we immediately have the following.
Theorem 4.1Under the same assumption as Theorem 3.1, and also assuming thatand, we letube the global solution to the problem (1.1), (1.2), which is constructed in Theorem 3.1 and we letbe the diffusion wave defined by (4.4). Then we have
We have and , where . We consider the initial value problem
and
When , by applying Lemma 4.2, we have
We call
the diffusion wave with the amount . Obviously, satisfies
From (4.11) and (4.13), we immediately obtain the following lemma.
Corollary 3.1, Lemma 4.1, and Lemma 4.3 immediately give the following result.
Theorem 4.2Under the same assumption as Theorem 3.1, also assuming that, , and, where, we letube the global solution to the problem (1.1), (1.2), which is constructed in Theorem 3.1 and we letbe the diffusion wave defined by (4.12). Then we have
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed to each part of this work equally and read and approved the final manuscript.
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