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):
(2.1)
with the initial data in (1.2), where . We apply the Fourier transform to (2.1). This yields
(2.2)
The corresponding initial values are given as
(2.3)
The characteristic equation of (2.2) is
(2.4)
Let be the corresponding eigenvalues of (2.4), we obtain
(2.5)
The solution to the problem (2.2), (2.3) in the Fourier space is then given explicitly in the form
(2.6)
where
(2.7)
and
(2.8)
We define and by
(2.9)
and
(2.10)
respectively, where denotes the inverse Fourier transform. Then, applying to (2.6), we obtain
(2.11)
By the Duhamel principle, we obtain the solution formula to (1.1), (1.2),
(2.12)
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 2.1 The solution of the problem (2.2), (2.3) satisfies
(2.13)
for and , where .
Proof Multiplying (2.2) by and taking the real part yields
(2.14)
Multiplying (2.2) by and taking the real part, we obtain
(2.15)
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 2.2 Let and be 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
(2.19)
and
(2.20)
for and , where .
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 2.3 Let and be 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 number such that if and , we have the following estimates:
(2.21)
(2.22)
and
(2.23)
Proof For sufficiently small ξ, using the Taylor formula, we get
(2.24)
We rewrite in (2.6) as
(2.25)
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 2.4 Let and be the fundamental solutions of (2.1), which are given in (2.9) and (2.10), respectively. Let , and let k, j, and l be nonnegative integers. Then we have
(2.26)
(2.27)
for , where in (2.26). Similarly, we have
(2.28)
(2.29)
for in (2.28) and in (2.29).
Proof We only prove (2.26). By the Plancherel theorem and (2.19), the Hausdorff-Young inequality, we obtain
For the term , letting , we have
where we used the Hölder inequality with and the Hausdorff-Young inequality for . On the other hand, we can estimate the term simply as
where .
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 2.1 Let and be the fundamental solution of (2.1), which are given in (2.8) and (2.9), respectively. Let , and let k, j, and l be nonnegative integers. Then we have
(2.30)
for and . Also we have
(2.31)
for .
Lemma 2.5 Let be the fundamental solution of (2.1), given in (2.8) and let be the fundamental solution of (2.1), given in (2.8). Let , and let k, j, and l be nonnegative integers. Then we have
(2.32)
for . Similarly,
(2.33)
for .
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. □