Abstract
This paper is concerned with the Cauchy problem for a class of fourthorder wave equations in an ndimensional space. Based on the decay estimate of solutions to the corresponding linear equation, a solution space is defined. We prove the global existence and optimal decay estimate of the solution in the corresponding Sobolev spaces by the contraction mapping principle provided that the initial value is suitably small.
MSC: 35L30, 35L75.
Keywords:
fourthorder wave equation; global existence; decay estimate1 Introduction
We investigate the Cauchy problem for a class of fourthorder wave equations
with the initial value
Here is the unknown function of and , and is a constant. The nonlinear term is a smooth function with for .
Equation (1.1) is reduced to the classical CahnHilliard equation if (see [1]), which has been widely studied by many authors. Galenko et al.[25] proposed to add inertial term to the classical CahnHilliard equation in order to model nonequilibrium decompositions caused by deep supercooling in certain glasses. For more background, we refer to [46] and references therein. It is obvious that (1.1) is a fourthorder wave equation. For global existence and asymptotic behavior of solutions to more higher order wave equations, we refer to [714] and references therein.
Very recently, global existence and asymptotic behavior of solutions to the problem (1.1), (1.2) were established by Wang and Wei [7] under smallness condition on the initial data. When , , they obtained the following decay estimate:
for and . The main purpose of this paper is to refine the result in [7] and prove the following decay estimate for the solution to the problem (1.1), (1.2) for with data,
for and . Here is assumed to be small. We also establish the decay estimate for the solution to the problem (1.1), (1.2) for with data,
for and . Here is assumed to be small. Compared to the result in [7], we obtain a better decay estimate of solutions for small initial data.
The paper is organized as follows. In Section 2, we study the decay property of the solution operators appearing in the solution formula. We prove global existence and asymptotic behavior of solutions for the Cauchy problem (1.1), (1.2) in Sections 3 and 4, respectively.
Notations We introduce some notations which are used in this paper. Let denote the Fourier transform of u defined by
We denote its inverse transform by . For , denotes the usual Lebesgue space with the norm . The usual Sobolev space of order s is defined by with the norm . The corresponding homogeneous Sobolev space of order s is defined by with the norm ; when , we write and . We note that for .
For a nonnegative integer k, denotes the totality of all the kth order derivatives with respect to . Also, for an interval I and a Banach space X, denotes the space of ktimes continuously differential functions on I with values in X.
Throughout the paper, we denote every positive constant by the same symbol C or c without confusion. is the Gauss symbol.
2 Decay property
The aim of this section is to derive the solution formula to the Cauchy problem (1.1), (1.2). Without loss of generality, we take . We first study the linearized equation of (1.1),
with the initial data in (1.2). Taking the Fourier transform, we have
The corresponding initial value are given as
The characteristic equation of (2.2) is
Let be the corresponding eigenvalues, we obtain
The solution to the problem (2.2), (2.3) is given in the form
where
and
We define and by and , respectively, where denotes the inverse Fourier transform. Then, applying to (2.5), we obtain
By the Duhamel principle, we obtain the solution formula to (1.1), (1.2)
The aim of this section is to establish decay estimates of the solution operators and appearing in the solution formula (2.8).
Lemma 2.1The solution of the problem (2.2), (2.3) verifies the estimate
Proof We apply the energy method in the Fourier space to prove (2.10). Such an energy method was first developed in [15]. We multiply (2.2) by and take the real part. This yields
Multiplying (2.2) by and taking the real part, we obtain
Combining (2.11) and (2.12) yields
where
and
A simple computation implies that
where
Note that
It follows from (2.14) that
where
Using (2.13) and (2.15), we get
Thus
which together with (2.14) proves the desired estimates (2.10). Then we have completed the proof of the lemma. □
Lemma 2.2Letandbe the fundamental solutions to (2.1) in the Fourier space, which are given in (2.6) and (2.7), respectively. Then we have the estimates
and
Proof If , from (2.5), we obtain
Substituting the equalities into (2.10) with , we get (2.16).
In what follows, we consider , it follows from (2.5) that
Substituting the equalities into (2.10) with , we get (2.17). The lemma is proved. □
Lemma 2.3Letandbe the fundamental solutions to (2.1) in the Fourier space, which are given in (2.6) and (2.7), respectively. Then there exists a small positive numbersuch that ifand, we have the following estimate:
and
Proof For sufficiently small ξ, using the Taylor formula, we get
and
It follows from (2.6) and (2.7) that
Equations (2.18) and (2.19) follow from (2.20)(2.22). The proof of Lemma 2.3 is completed. □
and
Proof By the property of the Fourier transform and (2.16), we obtain
where is a positive constant in Lemma 2.3, and and .
By a straight computation, we get
It follows from the HausdorffYoung inequality that
Combining (2.29)(2.31) yields (2.23). Similarly, we can prove (2.24)(2.28). Thus we have completed the proof of the lemma. □
3 Global existence and decay estimate (I)
The purpose of this section is to prove global existence and asymptotic behavior of solutions to the Cauchy problem (1.1), (1.2) with data. We need the following lemma, which comes from [16] (see also [17]).
Lemma 3.1Assume thatis smooth, whereis a vector function. Suppose that (is an integer) when. Then, for the integer, ifand, , then. Furthermore, the following inequalities hold:
and
Based on the decay estimates of solutions to the linear problem (2.1), (1.2), we define the following solution space:
where
The GagliardoNirenberg inequality gives
Theorem 3.1Assume that, (). Put
Ifis suitably small, the Cauchy problem (1.1), (1.2) has a unique global solutionsatisfying
Moreover, the solution satisfies the decay estimate
and
Proof Let us define the mapping
Using (2.23), (2.24), (2.27), (3.1) and (3.3), for , we obtain
Thus
It follows from (3.6) that
By exploiting (3.8), (2.25), (2.26), (2.28), (3.1) and (3.3), for , we have
The above inequality implies
Combining (3.7) and (3.9) and taking and R suitably small, we get
By (2.27), (3.2) and (3.3), for , we infer that
which implies
Similarly, for , from (3.11), (2.28) and (3.2), (3.3), we deduce that
which gives
Combining (3.12) and (3.13) and taking R suitably small yields
From (3.10) and (3.14), we know that Φ is a strictly contracting mapping. Consequently, we conclude that there exists a fixed point of the mapping Φ, which is a solution to (1.1), (1.2). We have completed the proof of the theorem. □
4 Global existence and decay estimate (II)
In the previous section, we have proved global existence and asymptotic behavior of solutions to the Cauchy problem (1.1), (1.2) with data. The purpose of this section is to establish global existence and asymptotic behavior of solutions to the Cauchy problem (1.1), (1.2) with data. Based on the decay estimates of solutions to the linear problem (2.1), (1.2), we define the following solution space:
where
Thanks to the GagliardoNirenberg inequality, we get
Theorem 4.1Suppose that, (). Put
Ifis suitably small, the Cauchy problem (1.1), (1.2) has a unique global solutionsatisfying
Moreover, the solution satisfies the decay estimate
and
Proof Let the mapping Φ be defined in (3.6).
For , (2.23), (2.24), (2.27), (3.1) and (4.1) give
Thus we get
By using (2.25), (2.26), (2.28), (3.1), (4.1), for , we have
This yields
Combining (4.4) and (4.6) and taking and R suitably small, we obtain
It follows from (2.27), (3.2) and (4.1) for that
which implies
Similarly, for , from (4.5), (2.28), (3.2) and (4.1), we infer that
which implies
Using (4.9) and (4.10) and taking R suitably small yields
It follows from (4.7) and (4.11) that Φ is a strictly contracting mapping. Consequently, we infer that there exists a fixed point of the mapping Φ, which is a solution to (1.1), (1.2). We have completed the proof of the theorem. □
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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