Asymptotic profile of solutions to the semilinear beam equation
© Zhang and Li; licensee Springer. 2014
Received: 26 January 2014
Accepted: 1 April 2014
Published: 17 April 2014
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.
Keywordsbeam equation decay estimate asymptotic profile diffusion wave
where for .
The main purpose of our present paper is two-fold: first, we try to recover all the results about global existence and decay estimate of solution in Takeda and Yoshikawa  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 . 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 . For details, we refer to Theorem 4.2.
The study of the global existence and asymptotic behavior of solutions to hyperbolic-type equations has a long history. We refer to [3, 4] for hyperbolic equations, [5–7] for the damped wave equation and [8–16] 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.
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
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.
for and , where .
which together with (2.17) proves the desired estimates (2.13). Then we have completed the proof of the lemma. □
for and , where .
Substituting the equalities into (3.1) with , we get (2.19).
Substituting the equalities into (2.13) with , we get the desired estimate (2.20). The lemma is proved. □
be the fundamental solution to .
Thus we get (2.21). The other estimates are proved similarly and we omit the details. The proof of Lemma 2.3 is completed. □
for in (2.28) and in (2.29).
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.
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  (see also ).
where , .
where in (3.2) and in (3.3).
provided that . This follows from the Gagliardo-Nirenberg inequality, and the definition of in (3.4).
for . Consequently, we have , from which we can deduce , provided that is suitably small. This proves the decay estimate (3.2).
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.
for and , respectively, where is the linear solution.
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 .
gives a asymptotic profile of the linear solution . In fact we have the following.
where . These estimates can be obtained by (2.27) and (2.32). Here we omit the details. □
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.
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.
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