We show diffusion phenomenon for linear abstract dissipative wave equations with time dependant coefficients of propagation speed and dissipation. Coefficients are decaying in time but not assumed to be monotone.
Let H be a separable Hilbert space with norm . Let A be a non-negative self-adjoint operator with domain and range . Then, for a non-negative number υ, the space becomes a Hilbert space with the graph-norm of denoted by
We write for every , if there exists a positive constant C such that
We write , if and .
For positive valued functions and on , we consider the difference between the solution of the initial value problem to the abstract dissipative wave equation
and the solution of the corresponding abstract parabolic equation (1.2):
where
If we take ,
where Ω is an exterior domain in with smooth boundary , or , and are bounded functions satisfying , , then (1.1) becomes the following initial boundary value problem:
If we replace by
where ν denotes the outer unit normal to and is a non-negative smooth function on , then Dirichlet conditions (1.6) becomes the Robin boundary value conditions. Here we note that if we take , then (1.7) stands for the Neumann boundary conditions.
Nishihara [17] () and Han and Milani [7] considered the Cauchy problem for quasilinear hyperbolic equations in , and showed that the norm of the difference between the solution of the quasilinear hyperbolic equation and that of the corresponding parabolic equation decays faster than each of the solutions does. This implies that the solution of the quasilinear hyperbolic equation behaves like the solution of the parabolic equation, which phenomenon is called diffusion phenomenon.
There are many results on diffusion phenomenon for (1.4)–(1.6) with : for example, see Karch [13], Ikehata [11], Nishihara [18], Narazaki [16], (see also Hayashi, Kaikina and Naumkin [8], Hosono and Ogawa [9], Ikeda, Inui and Wakasugi [10] for semilinear problem).
For the abstract dissipative wave equation (1.1) with , there are some results on diffusion phenomenon. Ikehata–Nishihara [12] first showed the diffusion phenomenon for the abstract dissipative wave equation (1.1). Chill–Haraux [3] improved their estimate to the following one:
and also showed that this estimate is optimal in the sense that the inequality
holds when 0 belongs to the essential spectrum of A. Radu, Todorova and Yordanov [19] (see also [20] for more general equations) improved the previous results by showing
for every , where denotes the semigroup generated by A.
Now we refer to the results for variable coefficients. Matsumura [14] considered the dissipative wave equation (1.4)–(1.6) with replaced by which satisfies
and some assumption on . Then he showed the energy decay of solutions:
where is a constant depending on . On the other hand, Mochizuki [15] proved that the energy does not decay in general and that every finite energy solution behaves like a solution of the free wave equation as , if the dissipative term is sufficiently weak in the sense that for some . Besides, Wirth [24] showed that if a function satisfies and , and either or , then the energy of tends to 0 as , where u is the solution of the abstract dissipative wave equation (1.1) with initial data in some classes. The above facts imply that is critical for the scattering.
When , the author [26] considered the abstract dissipative wave equation (1.1) with function on satisfying the following: there exists a positive constant θ and strictly monotone decreasing continuous function and monotone decreasing continuous function on for some satisfying such that the following inequalities for every .
Then we showed diffusion phenomenon by giving (1.8) type estimates of the difference between the solution u of abstract dissipative wave equation (1.1) and the solution v of corresponding parabolic equation (1.2):
for every () with a sketch of proof (see also [25] which gave the complete proof in the case ()).
In the case , and , Wirth [23] considered Cauchy problem of (1.4)–(1.5) with , in the case is a monotone positive valued function which satisfy the following assumption: and there exists positive valued monotone function satisfying
such that
Then, he showed that the norm of the low frequency part of of is dominated by , where u is the solution of the Cauchy problem of (1.4)–(1.5) in and v is the solution of the corresponding parabolic equation. Since the WKB representation of the solution is employed in the proof, the monotonicity of the functions b and γ is essentially. In [23], he gave an example (), which had the fastest decay on which the diffusion phenomenon occurs, as far as the author know. Wakasugi [22] showed the diffusion phenomenon for semilinear problem in the case .
When both and are time-dependent, the following results are known. The author [27] considered the singular limit problem for quasilinear abstract hyperbolic equation of Kirchhoff type. In order to solve the problem, the decay estimates like (1.8) was shown for (1.1) in the case () and . D’Abbicco–Ebert [4] gave sufficient condition on which the solution behaves like the solution of free wave equation. The estimate of the energy of the solutions of (1.4)–(1.5) in was given by Bui–Reissig [2] in the case is strictly monotone increasing, and by Ebert–Reissig [5] in the case is integrable, under some assumptions on and . The diffusion phenomenon was not shown in [2] and [5].
The purpose of this paper is to show the diffusion phenomenon for abstract dissipative wave equation (1.1) with time-dependent positive functions and in the following.
We give sufficient condition for diffusion phenomenon on and which may not be monotone and may converge to 0 as .
In the special case () in , we give estimates for by applying our theorems.
We note that in the case , our sufficient condition on for diffusion phenomenon is a generalization of conditions in [25,26] and [23] in the case decays.
We give some examples of and which satisfy the assumptions in Section 2.
Main results
Before stating our results, we list some notations.
For , we write .
For every , we abbreviate to , and -norm is denoted by .
Let denote the homogeneous Sobolev space equipped with the norm
Let and be positive valued functions on satisfying the following: There exist positive constants , , and strictly monotone increasing continuous function and monotone decreasing continuous function on satisfying the following.
for every .
for every .
.
.
Moreover, we assume “(A6)–(A7)”, or “(A6) and that is monotone decreasing”, or “(A8)–(A9)”.
for every .
.
.
.
Consider the case . Assume that there exists a positive constant such that the following (B1)–(B4) hold.
,
.
.
.
Then we see that assumptions (A1)–(A5), (A8) and (A9) hold by taking .
As we mention in Introduction, [23] considered diffusion phenomena in the case is monotone function. The monotone decreasing case is related to our results, since we consider decaying function . In this case, [23] showed diffusion phenomena when satisfies the assumptions (1.9), (1.10) and
If monotone decreasing function satisfies these condition above, then satisfies (B1)–(B4) with . Hence, condition (B1)–(B4) are more general than (1.9), (1.10) and (2.1), in the case is monotone decreasing.
In view of the correspondence in Theorem A in Section 3, there exists a σ-finite measure space , a unitary operator , and we may assume that and that for every , the operator is of the form
Putting
we have . Hence we may assume that without loss of generality.
We put where and f are as in (A1). We denote
Letandbefunctions onsatisfying (A1)–(A7). Then there exist a positive constant δ such that the following assertions hold. Letand. For every, letandbe the solutions of (
1.1
) and (
1.2
), respectively, and letwhich is the solution of
If, then we havefor every.
If, then we havefor every.
If A is strictly positive, that is, the spectrum of A is away from the origin, then the first term on the right-term on the right-hand side of (2.4) is not present and in this case the statement of the diffusion phenomenon is meaningless, since both the solution u of the abstract wave equation (1.1) and the solution v of the abstract parabolic equation (1.2) decay in the same order as the second term of the right-hand side of (2.4). Hence, the theorem means the diffusion phenomenon in the case that 0 is an eigenvalue of A or essential spectrum of A. Then by taking , (2.4) implies that the decay of the difference between the solutions u and v is faster than the decay of the solution of abstract parabolic equation (2.3). Also, (2.5) implies that the decay of the difference between the derivative of the solutions u and v is faster than the decay of the derivative of the solution of (2.3).
If is monotone decreasing, then the following holds without assuming (A7).
Letandbefunctions onsatisfying (A1)–(A6), and assume furthermore thatis monotone decreasing. Then there exists a positive constant δ such that the following assertions hold. Letand. For every, letandbe the solutions of (
1.1
) and (
1.2
), respectively. Letbe the function defined by (
2.2
).
If, we havefor every. If, we havefor every. For the derivative of solutions, if, we havefor every.
Next we assume (A8) and (A9), in place of (A6) and (A7).
Letandbefunctions onsatisfying (A1)–(A5), (A8) and (A9). Then there exist a positive constant δ such that the following assertions hold. Letand. For every, letandbe the solutions of (
1.1
) and (
1.2
), respectively. Let() be the function defined by (
2.2
). Let κ be an arbitrary nonnegative number.
If, we havefor every. If, we havefor every.
We give examples of Theorem 1–3. First we give two examples of Theorem 1.
Let , satisfy . Let and . Let
Then, we easily see that (A1)–(A7) with are satisfied, and thus we can apply Theorem 1. Since
inequalities (2.4) and (2.5) are
Let , . Let β and γ satisfy , , . Let satisfy (). Let
Then we see that the assumption (A1)–(A7) are satisfied with and , and thus, we can apply Theorem 1. Also, (2.14) and the following inequality hold.
Hence, inequalities (2.4) and (2.5) are
Let , , (). Let
Then assumptions (A1)–(A5), (A8) and (A9) are satisfied with , . In fact, we easily see that (A1)–(A3) are satisfied. Assumptions (A4), (A5), (A8) and (A9) are satisfied since
Hence, assumptions of Theorem 3 are satisfied. Furthermore, we have
Thus, inequalities (2.11) and (2.12) are
Assume moreover that and . Then is monotone decreasing and (A6) is also satisfied. Thus we can apply Theorem 2, and decay estimates of the second term of the right-hand sides of (2.15) and (2.16) get faster as
for every , where κ is an arbitrary positive constant.
Here we note that even if , in Examples 1, 2, 3 do not satisfy the assumption of [23] if , and in Example 3 does not satisfy the assumption of [25] and [26]. Furthermore, Example 3 does not satisfy the assumption of [23] when even if and .
In the case and () with , we give estimates of the difference of between the solutions.
Letandwith. Letandbefunctions onsatisfying (A1)–(A5), (A8) and (A9). Letandbe a number satisfying (
2.17
). Then there existssuch that the following holds. Let. For every, letandbe the solutions of (
1.1
) and (
1.2
), respectively.
In the case, the following hold: If, we havefor every. If, we havefor every.
In the case, the following hold for every: If, then we havefor every. If, then we havefor every.
Let and be functions in Example 3. Then the assumptions of Theorem 5 are satisfied, and (2.20) and (2.21) are
Reduction of the equations to ordinary differential equations
We first derive ordinary differential equations from (1.1) and (1.2) by employing Theorem A, the spectral theorem for self-adjoint operators, as in Chill and Haraux [3].
(Hall [6, Theorem 10.10, p. 207], Reed and Simon [21, Theorem VIII. 4, p. 260]).
Let A be a self-adjoint operator on a separable Hilbert space H with domain. Then there exists a σ-finite measure space, a unitary operatorand a measurable, real-valued functionon S which is finite and non-negative a.e. such that
if and only if.
If, then the equalityholds almost every.
For every , there exist a unique solution of the ordinary differential equation
and a unique solution of the equation
where ′ denotes the derivative with respect to t. Then and are solutions of (1.1) and (1.2) respectively.
From now on, and denote the solutions of (3.1) and (3.2), respectively. We also put
where is defined by (2.2).
The solution of the corresponding parabolic equation (3.2) has the following estimates.
Assume (A2) and (A3). Then for everyand, we have
We give some estimates in general form for later use: For every , and , we have
In fact, (A2) and (A3) yield
if t is sufficiently large, which implies (3.5).
Inequality (3.5) implies that if , and , then
for every , and thus,
for every . Therefore, we have
for every , if , and .
Definition (1.3) and inequality (3.7) with imply
Since
we have
that is, (3.3) is proved
Since is a solution of (3.2), we have
Hence, (3.4) follows from (3.3). □
Assume that 0 is an eigenvalue of A or essential spectrum of A. Then for every , we have
From (3.10) and (3.12), it follows that
Now we divide into three regions. Since f is strictly increasing, we can define the function T by
For each , we divide , where
Then is divided into , where
We give decay estimates of the difference between the solution of (1.1) and the solution of (1.2) on , and decay estimates of each solution of (1.1) and (1.2) on and . Solutions decay exponentially on . On , decay estimates of the solutions differ according to the assumptions of theorems.
Estimates on
Throughout this section, we assume (A1)–(A5) and that
We note that implies for every , since is monotone increasing.
Step 1. Transformation of (
3.1
). Let be the solution of the equation , that is,
Here we note that by the assumptions (A1) and (4.1), we have
and thus,
for every . We put
Then
We easily see that (3.1) is equivalent to
where
We put
Then the system (4.10)–(4.15) is equivalent to
It follows from the above equations
where
Step 2. Some basic estimates.
Definitions (4.14), (4.15) and (4.1) imply
for every such that .
By mean value theorem and the definition of (see (4.2)), we have
with some depending on t and ξ. Therefore, making use of (4.6), we obtain
In the same way, we can derive the following from (4.24) and (4.28):
We estimate . By definition (4.13), we have
From (4.31), (4.25)–(4.27), (4.5) and (A2), it follows that
Hence, employing (A2) and (4.4), we have
and
for every .
By definition, we have
and thus,
By (A5), (4.4) and (4.5), we have
Making use of this inequality and (4.5) in the right-hand side of (4.35), we obtain
The mean value theorem implies
with some depending on t and ξ, which together with (4.32) and (4.36) yields
for ().
In the same way, we can derive the following from (4.32) and (4.36)
and we can derive the following from (4.32) and (4.37)
for ().
Substituting (4.19) into (4.21), we have
By (4.28), (4.31) together with (4.5), we have
By (A2) and (A3), we can take such that for every . Hence, we obtain
for every , for a positive constant C. Substituting the inequality above, (4.33) and (4.34) into (4.41), we obtain
Since is monotone decreasing, we have
Substituting this inequality into (4.43), there is a positive constant C such that
In the same way, it follow from (4.20), (4.42) and (4.33) that
and thus,
We substitute (4.45) into (4.44). Then by Gronwall’s inequality, we have
which together with (A4) and (4.4) yields
for every . Inequalities (4.19), (4.45) and (4.47) yield
for every .
Step 3. By employing the estimates in Step 2, we estimate the difference between and the solution of (3.2).
The following estimates hold for everyand:
We estimate the left hand side of (4.49), divided into
where
First we estimate . By definitions (4.17) and (4.19), we have
Hence, we can write
Inequality (4.38) implies
From inequalities (4.25), (4.36), (4.50), (4.6) and (4.45), it follows that
By inequalities (4.5), (4.6), (4.36) and (4.46), we have
Inequalities (4.51) and (4.52) together with (4.22) and (4.23) yield
Next we estimate . Since
where
inequalities (4.6) and (4.29) imply
Inequality (3.7) with , implies . Hence integrating (1.3) by parts, we obtain
Operating U to the above equality and substituting the equality into (4.20), we have
By definition (4.14), we have
Inequalities (4.30), (4.36), (4.39), (4.40), (4.37), (4.33) and (4.32) together with (A2), (A4), (4.22) and the boundedness of yield
From (3.8) with , , and (3.7) with , , it follows that
Substituting the inequality above and (3.7) with , , into (4.57), we obtain
Assumption (A2) implies . Hence by (4.15) and (3.7) with , , we obtain
Inequality (3.6) with , implies
By this inequality and that , we have
Substituting (4.56), (4.58), (4.59), (4.60) and (4.22) into (4.55), we obtain
Substituting (4.61) and (3.9) into (4.54), we obtain
Inequalities (4.53) and (4.62) yield
Since f is monotone-increasing, (A1) implies . Hence, (4.63) yields (4.49). □
Step 4. We estimate itself.
Let() be functions ondefined byfor. Then the following estimate holds for everyand:
Let . Put
Integrating (4.18) on the interval , and multiplying to it, we obtain
Inequality (4.48) implies
for every , since . Making use of (4.28), (4.37), (4.36), (4.34) and (4.68) in (4.67), we obtain
where
We note that for . In fact,
and
Multiplying to (4.67), and making use of (4.6), (4.69), (4.70) and (4.71), we obtain
for . By inequalities (4.7) and (4.37), we have
Summing (4.72) and (4.73) up, and taking (4.22) and (4.23) into account, we obtain (4.66). □
Lemmas 2 and 3 yield the following corollary, which provides estimates of the difference between the solutions and its derivatives of (1.1) and (1.2) for .
Let. Letandare functions defined by (
4.64
) and (
4.65
), respectively. For every,and, the following holds.
First we prove (4.74). Since
Inequallity (4.49) with , (4.66) and (4.5) yield
for every and . Then in view of
for every , we obtain
for every and . By (A1) together with the assumption that f is monotone-increasing, we have
Substituting this inequality into (4.76), we obtain (4.74).
Next we prove (4.75). From (4.9) and the equality , it follows that
Hence, by (4.5), we have
Then making use of (4.49) with , (3.10), (3.9) and (4.66), we obtain
for every and , which yields (4.75) in the same way as in the proof of (4.74). □
Estimates on
Throughout this section, we assume that
We put
for solution u of (1.1).
Assume (A1)–(A3). Then there exist constantsandsuch that the following estimates hold. Let u be the solution of (
1.1
). Thenfor every. In particular,for every.
If we assume furthermore (A6), thenfor every.
First we note that
for every , that is, .
We define by
where is a positive number satisfying
By the Cauchy–Schwarz inequality together with (5.5) and (5.6), we have
for every , Thus we have
for every .
Since u is the solution of (3.1), we have
By (A2) and (A3), there exists such that
for every .
By the Cauchy–Schwarz inequality together with the assumptions (A1), (A2) and (5.5), we have
Since , we have
We make use of (5.7), (5.10) and (5.11) in (5.9) to obtain
Thus by (5.8) and the assumption that , we have
for every . Hence, by taking , we obtain
for every , which together with (5.8), we obtain
for every . That is, (5.2) holds in the case .
It remain the case . Since u is the solution of (3.1), we have
for every . Thus, we have
for every . Hence,
for every . This implies (5.2) in the case . In the case , substituting (5.15) with into (5.12) with , we obtain
and we complete the proof of (5.2).
Assume furthermore (A6). Then for every , we have
and thus
Making use of (5.16) in place of (5.5) in the estimate (5.7), we can prove (5.4) for every , by taking □
Assume (A1)–(A3), (A6) and (A7). Then there exists a positive constantindependent of ξ,andsuch that the following holds for every solution u of (
1.1
).for every, whereis defined by (
5.1
).
Assume (A1)–(A3), (A6) and thatis monotone decreasing on. Then the following holds for every solution u of (
1.1
),for every.
Assume (A1)–(A3), (A8) and (A9). Then the following holds for every solution u of (
1.1
):for every.
(i) By assumption (A7), there are positive numbers and such that
for every . By (5.13), (A2) and (A3), there exists such that
for every . Hence, we have
for every . Since we assume (A6), inequality (5.4) holds for . Substituting (5.21) into (5.4) with , and making use of (5.20), we obtain
for every . Substituting (5.14) with into (5.22), we see that (5.17) holds for every , by taking .
(ii) Inequality (5.13) together with the assumption that yields
Since we assume (A6), inequality (5.4) holds for . Substituting (5.23) into (5.4) with , we obtain (5.18).
(iii) Put
By assumptions (A2), (A3) and (A9), there exists such that
for every . Thus,
for every .
First assume that . Since , inequality (4.74) with and (A8) implies
By (A9),
Substituting (5.26) into (5.25), we have
This inequality and (3.3) yield
From (4.75) with and , it follows that
By (A9), we have
By (A8) and (A9), we have
Substituting (5.29) and (5.30) into (5.28), we obtain
By (3.4) and (A9), we have
Adding this inequality with to (5.31), we obtain
Inequalities (5.27) and (5.32) yield
Multiplying to (5.2) with , and substituting (5.33) to the inequality, we obtain
for every . If , we take in (5.24) to have
for every . Substituting this inequality into (5.34), we obtain
for every in the case .
If , substitute (5.14) into (5.2) with . Then we have
for every .
Inequality (5.19) follows from (5.35) and (5.36). □
Estimates on. For the estimates on , it suffices to assume (A1)–(A3). Hence, we can estimate solutions for in the same way under the assumptions of Theorem 1, 2 and 3.
Inequalities (5.3) and (3.5) with imply
and
if , and .
Since and are positive valued continuous functions, we have . Hence, (3.3) with and α replaced by and (5.5) yield
if and .
In the same way, making use of (3.4), (3.5) and (5.5), we obtain
if and .
Proof of (
2.4
). First we estimate difference between the solutions of (1.1) and (1.2) on . Inequality (5.20) yields
for every . From (6.5), (3.5) and (3.8), it follows that
for every . Since the left-hand side of the inequality above is bounded on compact set of , we have
for every . Hence (defined by (4.65)) satisfies
for every . Thus (4.74) with yields
if and .
Next we estimate each solutions of (1.1) and (1.2) on . Inequality (5.17) implies
for every and . Hence, we have
for every . In the case , it follows from (5.5) that
if . Hence, (6.9) yields
for every , if .
We make use of (6.9) in the case and (6.11) in the case . Then in view of (3.5) with , we obtain
if and .
Since , it follows from (3.3) with and α replaced by and (5.16) that
for every , since (A6) is assumed. Substituting (6.5) into (6.13), we obtain
for every . Inequality (3.3) with implies that (6.14) holds also for every . (6.14) holds if and .
For each , we take the -norm of (6.1) and (6.3), the -norm of (6.7) and the -norm of (6.12) and (6.14). Then we see that there exists a positive number satisfying (2.4).
Proof of (
2.5
) and (
2.6
). In the same way as in the proof of (6.6), we have
for every , which shows that
for every . Hence (4.75) with yields
if and .
Making use of (3.4) in place of (3.3), and proceeding in the same way as in the proof of (6.14), we obtain
if and . Thus by (3.5) with , we obtain
if and .
Take the -norm of (6.15) and (6.16), the -norm of (6.2) and (6.4), and the -norm of (6.8) and (6.17), for each . Then (2.5) and (2.6) hold for a positive constant δ. □
We make use of (4.74) and (4.75) with on , and (6.1)–(6.4) on . Hence we only have to give estimates on .
Proof of (
2.7
) and (
2.8
). Inequality (5.18) implies
for every and . Hence, we have
for every .
In the case , we take the -norm of (4.74) with , the -norm of (6.1) and (6.3), and the -norm of (6.19) and (6.13) for each . Then we see that there exists a positive constant δ satisfying (2.8).
In the case , inequalities (6.10) and (6.19) yield
if . For each , take the -norm of (4.74) with , the -norm of (6.1) and (6.3) and the -norm of (6.20) and (6.13). Then we see that (2.7) holds for a positive constant δ.
Proof of (
2.9
). In the same way as in the proof of (6.13) by making use of (3.4) in place of (3.3), we obtain
if and .
We take the -norm of (4.75), the -norm of (6.2) and (6.4), the -norm of (6.18) and (6.21), for each . Then we see that (2.9) holds for a positive constant δ.
Inequality (2.10) immediately follows from (2.9) and (3.4). □
By the same reason as in the proof of Theorem 2, we only have to give estimates on .
Proof of (
2.11
) Inequality (5.19) with α replaced by () implies
if and . Since by (5.5), the above inequality implies
which yields
if and , since . Dividing (6.22) by , we obtain
if and .
Assumptions (A2), (A3) and (A9) imply that
for every . In fact, we see that
if t is sufficiently large, which implies (6.24).
Inequality (6.23) together with (6.24) implies
if and .
By (3.3) with α replaced by , we have
Then by (5.5), we have
if and .
We take the -norm of (4.74) with , the -norm of (6.1) and (6.3), and the -norm of (6.25) and (6.26). Then in view of (A8) and (A9), there is a positive constant δ such that (2.11) holds.
Proof of (
2.12
) Making use of (3.4) in place of (3.3), and proceeding in the same way as in the proof of (6.26), we obtain
for every .
We take the -norm of (4.75) with , the -norm of (6.2) and (6.4), and the -norm of (6.22) and (6.27) for each , and take (A9) into account. Then we see that there is a positive constant δ satisfying (2.12).
Inequality (2.13) immediately follows from (2.12) and (3.4). □
First we show the following: there is a positive δ such that
for every .
If , then Sobolev’s inequality
implies
and thus, (7.2) follows from (2.4) of Theorem 1.
We show (7.2) for . By (7.1), we have
Thus, by (2.4), there is a positive δ such that
for every . Let be the second term of the right-hand side of the above. Then making use of (3.8) we have
Hence (7.2) holds also for .
We see that
holds for and (). In fact, this is trivial if . Thus, assume that . Let be the dual exponent of q, and let be the number defined by
Then, by Hölder’s inequality and Hausdorff–Young’ inequality, we have
By change of variables, we have
Substituting this inequality into (7.5), we obtain (7.4).
Substituting (7.4) with and into (7.3), we have
This inequality and (7.2) yield (2.18).
Making use of (2.6) in place of (2.4), we can prove (2.19) in the same way as above. □
In the case , we can prove (2.22) and (2.23) in the same way as in the proof of Theorem 4 by applying Theorem 3 in place of Theorem 1.
Now assume that . Let and . We employ Theorem 3 with replaced by for . Then there is a positive δ such that (2.11) with , and holds. Thus, in view of (3.5) and (3.8), we have
Hence, taking
in the above, and proceeding in the same way as in the proof of Theorem 4, we obtain
for every .
We take in the inequality above. Then in view of
we obtain (2.20) in the case .
Since , (2.13) with and implies
From this inequality, estimate (2.21) follows in the same way as in the proof of (2.20). □
Footnotes
Acknowledgement
The author expresses her sincere gratitude to referees for valuable advices. This work is partly supported by Grant-in-Aid for Scientific Research (C) 17K05338, Ministry of Education, Culture, Sports, Science and Technology, Japan.
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