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Asymptotic profile of solution for the Cauchy problem of beam equation with variable coefficient
Accepted Manuscript Asymptotic profile of solution for the Cauchy problem of beam equation with variable coefficient Shuji Yoshikawa, Yuta Wakasugi
PII: DOI: Reference:
S0893-9659(17)30285-9 https://doi.org/10.1016/j.aml.2017.09.006 AML 5334
To appear in:
Applied Mathematics Letters
Received date : 23 July 2017 Revised date : 11 September 2017 Accepted date : 11 September 2017 Please cite this article as: S. Yoshikawa, Y. Wakasugi, Asymptotic profile of solution for the Cauchy problem of beam equation with variable coefficient, Appl. Math. Lett. (2017), https://doi.org/10.1016/j.aml.2017.09.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Asymptotic profile of solution for the Cauchy problem of beam equation with variable coefficient Shuji Yoshikawaa , Yuta Wakasugib a Division
of Mathematical Sciences, Faculty of Science and Technology, Oita University, 700 Dannoharu, Oita, 870-1192 JAPAN, Corresponding author: [email protected] b Department of Engineering for Production and Environment, Graduate School of Science and Engineering, Ehime University, Bunkyo-cho 3, Matsuyama, Ehime, 790-8577 JAPAN
Abstract We consider the Cauchy problem for the linear beam equation: utt + ut + uxxxx − a(t)uxx = 0,
(t, x) ∈ R+ × R,
where a(t) ∼ (1 + t)α . The purpose of this study is to clarify the behavior of solution depending on the rate α. Here we shall give the asymptotic behavior of the equation in the case α > −1/2, by using the method of scaling variables developed by Gallay and Raugel [3]. Keywords: asymptotic profile, beam, variable coefficient, scaling variables 2010 MSC: 35G10, 35B65, 35G25, 35E05
1. Introduction In this note we study the Cauchy problem for the following one-dimensional linear beam equation with time dependent coefficient: utt + ut + uxxxx − a(t)uxx = 0,
u(0, x) = u0 (x),
(t, x) ∈ R+ × R,
ut (0, x) = u1 (x),
x ∈ R.
(1) (2)
Suppose that the given function a(t) satisfies a(t) ∼ (t + 1)α ,
a(t) > 0,
|a0 (t)| ≤
Ca a(t)2 A(t) + 1
(3)
Rt for some constant Ca with A(t) := 0 a(τ )dτ . The last assumption in (3) looks intricate but not strange. Indeed, a rough calculation implies |a0 (t)| ≤ C(t + 1)α−1 , which is compatible with the first assumption. Our interest is directed toward the asymptotic behavior of solution u as t → ∞. In particular we aim at finding the thresholds of α. It is well-known that the solution of the Cauchy problem for damped wave equation: utt + ut − uxx = 0 behaves as the Preprint submitted to Applied Mathematics Letters
September 23, 2017
solution for the heat equation: ut − uxx = 0 asymptotically as t → ∞ (see e.g. [4, 5, 6, 8, 9, 11] etc.), because utt decays faster than ut as t → ∞. From the observation, the interesting problem is studied in e.g. [7, 10, 15, 16]. Namely, if we consider the equation: utt + b(t)ut − uxx = 0 with b(t) ∼ (1 + t)β , then the asymptotic profile of the solution for the equation changes depending on β. In the case β < −1, b(t)ut decays faster than utt , hence the solution behaves as the wave equation: utt − uxx = 0. On the other hand, in the case −1 < β < 1, utt decays faster than b(t)ut , hence the solution behaves as the heat equation: b(t)ut − uxx = 0. It is shown in [13] that the solution for the beam equation with constant coefficient: utt + ut + uxxxx − uxx = 0 behaves as the solution for a heat equation. However if we remove the second order term −uxx , the solution u decays like 4th order parabolic equation and the asymptotic profile is also the kernel of it. From the above observations, our purpose of this article is to compare the effect from a(t)uxx with uxxxx in (1). From the physical point of view, the equation (1) is regarded as a linearized model of the Woinovsky-Krieger equation with damping: utt + ut + uxxxx − a(ku(t)kL2 )uxx = 0, which represents the vibration of extensible beam proposed in [17]. For the mathematical results we refer the reader to e.g. [1, 12] and reference therein. Moreover, (1) is also regarded as the thermoelastic beam equation in the case of controllable temperature distributed uniformly with respect to the space variable (see [2, Chapter 5]). In this problem the non-uniformly distributed temperature case a = a(t, x) is more natural than the uniformly distributed case a = a(t). We shall give the result that the solution of (1)-(2) behaves as the solution of ut − a(t)uxx = 0 in the case α > −1/2. To show the result, we shall adopt the method proposed by Gallay-Raugel in [3]. This method allows us to show the asymptotic profile for the problem not only with time dependent coefficient a(t) but also with space dependent one a(t, x), because it is shown by using only the energy method, namely, integration by parts, not using the Fourier transform. In particular, if a(t, x) is decomposed to a0 (t) + a1 (t, x) and the second term a1 (t, x) is regarded as perturbation in some sense as in [14], then we can extend the result to the multi-dimensional case. Moreover, we can also apply the method to semilinear problems by adding several modifications. However, to avoid redundancy and to give a simplified proof, here we restrict ourselves to deal with only time dependent coefficient case. 2. Asymptotic Profile We first introduce the notation used throughout this paper. We denote by C a generic positive constant, which may change from line to line. Also, we frequently use the symbol a(t) ∼ b(t), which means that C −1 b(t) ≤ a(t) ≤ Cb(t) holds for some constant C ≥ 1. Let Lp and H m be the standard Lebesgue and Sobolev spaces on R, and H k,m be the weighted on R Pk Sobolev spaces m l equipped with the norm defined by kf kH k,m = k(1 + |x|) ∂ f (·)k L2 . x l=0 The existence of a unique global solution u satisfying u ∈ C([0, ∞), H 2,1 ) and ut ∈ C([0, ∞), H 0,1 ) for any (u0 , u1 ) ∈ H 2,1 × H 0,1 can be shown in a standard 2
way. Let us denote a heat kernel by K(t, x) := (4πt)−1/2 exp(−|x|2 /4t). Observe that K(A(t), x) satisfies the equation ut − a(t)uxx = 0. Our main theorem is the following. Theorem 2.1. Let α > −1/2. Then the solution u for (1)-(2) satisfies 1
λ
ku(t, ·) − m∗ K(A(t), ·)kL2 ≤ C(A(t) + 1)− 4 − 2 k(u0 , u1 )kH 2,1 ×H 0,1 2 for t ≥ 1 with some C > 0, m∗ ∈ R, λ ∈ (0, min{ 12 , α+1 , 4α+2 α+1 }).
Remark 2.2. It is easily seen that the solution u for (1)-(2) and G satisfy 1
1
ku(t, ·)kL2 ≤ CA(t)− 4 k(u0 , u1 )k(H 2 ∩L1 )×(L2 ∩L1 ) , kK(A(t), ·)kL2 = CA(t)− 4 . Although H 2,1 × H 0,1 ⊂ (H 2 ∩ L1 ) × (L2 ∩ L1 ) holds, it is known that the decay rate can not be accelerated anymore. Thus the theorem says that the solution u behaves as m∗ K(A(t), x) asymptotically as t → ∞. The main reason we use weighted Sobolev spaces for initial data is technical, and we will giveR the remark in the proof. As we shall see later, m∗ is given as m∗ := limt→∞ R u(t, x)dx. Moreover, in the same observation as many other results (see e.g. [8]), we shall also obtain more rapidly decaying estimates with higher order derivatives under corresponding adequate smooth initial data. However, for compactness we do not discuss it here. Proof of Theorem 2.1. Let us denote s := log(A(t)+1) and y := (A(t)+1)−1/2 x. Under the assumption α > −1/2, t → ∞ and s → ∞ are equivalent. For the new variables (s, y) we define the transformation into new unknowns (v, w) = (v(s, y), w(s, y)) from (u, ut )(t, x) by the relations: ! 1 x u(t, x) = p v log(A(t) + 1), p , A(t) + 1 A(t) + 1 ! a(t) x ut (t, x) = w log(A(t) + 1), p . (A(t) + 1)3/2 A(t) + 1 Then the equation (1) is rewritten as y v y 3 a0 e−s −s vs − vy − = w, ae ws − wy − w + w + w = vyy − vyyyy . 2 2 2 2 a a 1
From the definition of s, we see (t + 1) ∼ e α+1 s . Then it is easy to check that 2α+1 1 1 ae−s ∼ e− α+1 s , e−s /a ∼ e− α+1R s and |a0 |/a ≤ Ca ae−s ∼ e− α+1 s . For the functions: m(s) := R v(s, y)dy, φ(y) := K(1, y) = (4π)−1/2 exp(−y 2 /4) and ψ(y) := φyy (y), we decompose solutions as follows: v(s, y) = m(s)φ(y) + f (s, y),
w(s, y) = ms (s)φ(y) + m(s)ψ(y) + g(s, y).
3
We expect that the functions m(s)φ(y) and ms (s)φ(y)+m(s)ψ(y) are the asymptotic profiles of v and w, respectively. Substituting these into the above equation, we deduce the equations for the residue terms f and g: y f y 3 a0 e−s −s fs − fy − = g, ae gs − gy − g + g + g = fyy − fyyyy + h, (4) 2 2 2 2 a a y 3 e−s a0 h := −2ae−s ms ψ + ae−s m ψy + ψ − mψyy − mψ, 2 2 a a where we have used yφy /2 + φ/2 = −ψ and a0 −s −s (5) ae mss = ae − − 1 ms . a R R R It is easy to check that R f (s, y)dy = R g(s, y)dy = R h(s, y)dy = 0. By using the classical energy method, we shall derive decay estimates for f and g. Let us define energies Eij = Eij (s) (i = 0, 1 and j = 1, 2) by Z Z 2 1 e−s 2 F 2 −s 2 −s E01 := Fy + F + ae G dy, E02 := + ae F G dy, a yy 2 R 2 R Z Z f2 1 e−s 2 fyy + ae−s g 2 dy, E12 := + ae−s f g dy, E11 := fy2 + a 2 R R 2 Ry Ry where F (s, y) := −∞ f (s, z)dz and G(s, y) := −∞ g(s, z)dz. Here, we note that F and G make sense as L2 -functions by the following well-known Hardy type inequality. To define F and G, we choose the weighted Sobolev spaces for the initial data. For the proof we refer to e.g. [14, Lemma 3.9]. R Lemma 2.3. Suppose that f = f (y) ∈ H 0,1 satisfies R f (y)dy = 0, and let Ry R R F (y) = −∞ f (z)dz. Then, we have R F (y)2 dy ≤ 4 R y 2 f (y)2 dy. From the equations (4), F and G satisfy
y Fs − Fy = G, 2 where H(s, y) :=
y a0 e−s ae−s Gs − Gy − G + G + G = Fyy − Fyyyy + H, 2 a a
Ry
−∞
ae−s
s
h(s, z)dz. It holds that =
a0 − ae−s , a
e−s a
s
=−
a0 e−s . − 3 a a
Then, by a straightforward calculation, we have Z Z Z Z d 1 a0 a0 2 E01 + G2 dy = E01 − 3 Fyy dy − G2 dy + GHdy, ds 2 2a R 2a R R R Z Z 1 d E02 + E02 + 2E01 = 2ae−s G2 dy + F Hdy. ds 2 R R 4
(6)
It follows from the assumptions in (3) that Z Z Z Z −s a0 Ca −s a0 Ca e 2 Fyy dy ≤ − G2 dy ≤ ae G2 dy, − 3 F 2 dy ≤ Ca E01 . 2a R 2 2a R 2 R a yy R Setting E0 := E01 + Ca2+1 E02 , from the Young inequality we have for any ε > 0 Z d 1 E0 (s) + E0 (s) ≤ ε F 2 (s, y)dy + C(ε)kH(s, ·)k2L2 , s ≥ s0 , (7) ds 2 R R where we have used the fact that the term Cae−s R G2 dy can be absorbed into R G2 dy in the left hand side for s ≥ s0 with large s0 , since ae−s is decreasing R function. Analogously, for E11 and E12 we have Z Z Z Z 3 a0 a0 d 2 E11 + g 2 dy = E11 − 3 fyy dy − g 2 dy + ghdy, ds 2 2a R 2a R R R Z Z d 1 E12 + 2E11 + E12 = E12 + 2ae−s g 2 dy + f hdy. ds 2 R R R R Observe that E12 ≤ C R f 2 dy+Cae−s R g 2 dy from the Young inequality. Then in a similar manner to above, by setting E1 := E11 + Ca2+2 E12 , we obtain Z d 1 E1 (s) + E1 (s) ≤ C f 2 (s, y)dy + Ckh(s, ·)k2L2 , s ≥ s0 . (8) ds 2 R Therefore, calculating (8) + C0 × (7) with a large constantR C0 and a small R constant ε, we can absorb the terms ε R F 2 (s, y)dy and C R f 2 (s, y)dy into E01 and E02 in the left-hand side, and hence we obtain for some λ ∈ (0, 1/2): d (C0 E0 + E1 )(s) + λ(C0 E0 + E1 )(s) ≤ C(kh(s, ·)k2L2 + kH(s, ·)k2L2 ), ds
s ≥ s0 .
d Next, putting Em0 := 21 ae−s m2s , we obtain ds Em0 + 12 Em0 + m2s = ae−s m2s . 1 Then Em0 (s) ≤ e− 2 s Em0 (s0 ) for s ≥ s0 with large s0 . Set Em1 := 21 m2 + d ae−s mms . It follows from (5) and (6) that ds Em1 = 2Em0 (s). We thus obtain Em1 (s) ≤ Em1 (s0 ) + CEm0 (s0 ), which implies the boundedness of |m|(s). With the help of Lemma 2.3, we can easily check that −s 2 ! e 2 2 −s 2 2 −s 2 (Em0 (s0 ) + Em1 (s0 )). kHkL2 + khkL2 ≤ C|ae | |ms | + C |ae | + a
e Let us define E(s) and E(s) by E(s) := C0 E0 (s) + E1 (s) + Em0 (s) and 2α+1 1 e E(s) := E(s) + Em1 (s). Thus, applying ae−s ∼ e− α+1 s , e−s /a ∼ e− α+1 s , we have 2 d 1 e 0 ) + Ce− 4α+2 e 0 ), α+1 s E(s E(s) + λE(s) + m2s ≤ Ce− α+1 s E(s ds 2
5
whence 1 2 d λs e 0 ) + Ce−( 4α+2 e 0 ). α+1 −λ)s E(s e E(s) + eλs m2s ≤ Ce−( α+1 −λ)s E(s ds 2
2 − λ > 0 and Thanks to α > −1/2, there exists some λ ∈ (0, 1/2) satisfying α+1 4α+2 −λs e − λ > 0. Therefore, we obtain E(s) ≤ Ce E(s ) for s ≥ s 0 0 . From the α+1 R s λσ 2 e estimate we also have s0 e ms (σ) dσ ≤ C E(s0 ). Then one has
|m(s) − m(s0 )|2 ≤
Z
s
s0
e−λσ dσ
Z
s
s0
0 e 0 ), eλσ ms (σ)2 dσ ≤ Ce−λs E(s
s0 ≤ s0 ≤ s,
which assures that the limit m∗ := lims→+∞ m(s) exists and |m(s) − m∗ |2 ≤ e 0 ). Finally, we have Ce−λs E(s e 0 ). kv(s, ·) − m∗ φk2L2 ≤ 2kf (s, ·)k2L2 + 2|m(s) − m∗ |2 kφk2L2 ≤ Ce−λs E(s
e 0 ), we easily see that E(s e 0 ) ≤ Ck(u0 , u1 )k2 2,1 0,1 . From the definition of E(s H ×H Recalling the relation K(A(t) + 1, x) = (A(t) + 1)−1/2 φ((A(t) + 1)−1/2 x), we complete the proof of Theorem 2.1. Acknowledgments. This work was partially supported by JSPS KAKENHI Grant Numbers JP16K05234, JP16K17625. The authors express their sincere gratitude for referees’ kind and careful comments. References [1] J. M. Ball, Stability theory for an extensible beam, J. Differential Equations, 14 (1973), 399–418. [2] M. Brokate, J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996. [3] Th. Gallay, G. Raugel, Scaling variables and asymptotic expansions in damped wave equations, J. Differential Equations, 150 (1998), 42–97. [4] P. Marcati, M. Mei, Convergence to nonlinear diffusion waves for solutions of the initial boundary problem to the hyperbolic conservation laws with damping, Quart. Appl. Math., 58 (2000), 763–784. [5] P. Marcati, M. Mei, B. Rubino, Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping, J. Math. Fluid Mech., 7 (2005), suppl. 2, S224–S240. [6] P. Marcati, K. Nishihara, The Lp -Lq estimates of solutions to onedimensional damped wave equations and their application to the compressible flow through porous media, J. Differential Equations, 191 (2003), 445– 469. 6
[7] K. Mochizuki, Scattering theory for wave equations with dissipative terms, Publ. Res. Inst. Math. Sci., 12 (1976), 383–390. [8] T. Narazaki, Lp -Lq estimates for damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan, 56 (2004), 586–626. [9] K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differential Equations, 131 (1996), 171–188. [10] K. Nishihara, Asymptotic profile of solutions for 1-D wave equation with time-dependent damping and absorbing semilinear term, Asymptot. Anal., 71 (2011), 185–205. [11] T. Ogawa, H. Takeda, Large time behavior of solutions for a system of nonlinear damped wave equations, J. Differential Equations, 251 (2011), 3090–3113. [12] R. Racke, S. Yoshikawa, Singular limits in the Cauchy problem for the damped extensible beam equation, J. Differential Equations, 259 (2015), 1297–1322. [13] H. Takeda, S. Yoshikawa, On the initial value problem of the semilinear beam equation with weak damping II: Asymptotic profiles, J. Differential Equations, 253 (2012), 3061–3080. [14] Y. Wakasugi, Scaling variables and asymptotic profiles for the semilinear damped wave equation with variable coefficients, J. Math. Anal. Appl., 447 (2017), 452–487. [15] J. Wirth, Wave equations with time-dependent dissipation I. Non-effective dissipation, J. Differential Equations, 222 (2006), 487–514. [16] J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation, J. Differential Equations, 232 (2007), 74–103. [17] S. Woinovsky-Krieger, The effect of axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35–36.
7
PII: DOI: Reference:
S0893-9659(17)30285-9 https://doi.org/10.1016/j.aml.2017.09.006 AML 5334
To appear in:
Applied Mathematics Letters
Received date : 23 July 2017 Revised date : 11 September 2017 Accepted date : 11 September 2017 Please cite this article as: S. Yoshikawa, Y. Wakasugi, Asymptotic profile of solution for the Cauchy problem of beam equation with variable coefficient, Appl. Math. Lett. (2017), https://doi.org/10.1016/j.aml.2017.09.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Asymptotic profile of solution for the Cauchy problem of beam equation with variable coefficient Shuji Yoshikawaa , Yuta Wakasugib a Division
of Mathematical Sciences, Faculty of Science and Technology, Oita University, 700 Dannoharu, Oita, 870-1192 JAPAN, Corresponding author: [email protected] b Department of Engineering for Production and Environment, Graduate School of Science and Engineering, Ehime University, Bunkyo-cho 3, Matsuyama, Ehime, 790-8577 JAPAN
Abstract We consider the Cauchy problem for the linear beam equation: utt + ut + uxxxx − a(t)uxx = 0,
(t, x) ∈ R+ × R,
where a(t) ∼ (1 + t)α . The purpose of this study is to clarify the behavior of solution depending on the rate α. Here we shall give the asymptotic behavior of the equation in the case α > −1/2, by using the method of scaling variables developed by Gallay and Raugel [3]. Keywords: asymptotic profile, beam, variable coefficient, scaling variables 2010 MSC: 35G10, 35B65, 35G25, 35E05
1. Introduction In this note we study the Cauchy problem for the following one-dimensional linear beam equation with time dependent coefficient: utt + ut + uxxxx − a(t)uxx = 0,
u(0, x) = u0 (x),
(t, x) ∈ R+ × R,
ut (0, x) = u1 (x),
x ∈ R.
(1) (2)
Suppose that the given function a(t) satisfies a(t) ∼ (t + 1)α ,
a(t) > 0,
|a0 (t)| ≤
Ca a(t)2 A(t) + 1
(3)
Rt for some constant Ca with A(t) := 0 a(τ )dτ . The last assumption in (3) looks intricate but not strange. Indeed, a rough calculation implies |a0 (t)| ≤ C(t + 1)α−1 , which is compatible with the first assumption. Our interest is directed toward the asymptotic behavior of solution u as t → ∞. In particular we aim at finding the thresholds of α. It is well-known that the solution of the Cauchy problem for damped wave equation: utt + ut − uxx = 0 behaves as the Preprint submitted to Applied Mathematics Letters
September 23, 2017
solution for the heat equation: ut − uxx = 0 asymptotically as t → ∞ (see e.g. [4, 5, 6, 8, 9, 11] etc.), because utt decays faster than ut as t → ∞. From the observation, the interesting problem is studied in e.g. [7, 10, 15, 16]. Namely, if we consider the equation: utt + b(t)ut − uxx = 0 with b(t) ∼ (1 + t)β , then the asymptotic profile of the solution for the equation changes depending on β. In the case β < −1, b(t)ut decays faster than utt , hence the solution behaves as the wave equation: utt − uxx = 0. On the other hand, in the case −1 < β < 1, utt decays faster than b(t)ut , hence the solution behaves as the heat equation: b(t)ut − uxx = 0. It is shown in [13] that the solution for the beam equation with constant coefficient: utt + ut + uxxxx − uxx = 0 behaves as the solution for a heat equation. However if we remove the second order term −uxx , the solution u decays like 4th order parabolic equation and the asymptotic profile is also the kernel of it. From the above observations, our purpose of this article is to compare the effect from a(t)uxx with uxxxx in (1). From the physical point of view, the equation (1) is regarded as a linearized model of the Woinovsky-Krieger equation with damping: utt + ut + uxxxx − a(ku(t)kL2 )uxx = 0, which represents the vibration of extensible beam proposed in [17]. For the mathematical results we refer the reader to e.g. [1, 12] and reference therein. Moreover, (1) is also regarded as the thermoelastic beam equation in the case of controllable temperature distributed uniformly with respect to the space variable (see [2, Chapter 5]). In this problem the non-uniformly distributed temperature case a = a(t, x) is more natural than the uniformly distributed case a = a(t). We shall give the result that the solution of (1)-(2) behaves as the solution of ut − a(t)uxx = 0 in the case α > −1/2. To show the result, we shall adopt the method proposed by Gallay-Raugel in [3]. This method allows us to show the asymptotic profile for the problem not only with time dependent coefficient a(t) but also with space dependent one a(t, x), because it is shown by using only the energy method, namely, integration by parts, not using the Fourier transform. In particular, if a(t, x) is decomposed to a0 (t) + a1 (t, x) and the second term a1 (t, x) is regarded as perturbation in some sense as in [14], then we can extend the result to the multi-dimensional case. Moreover, we can also apply the method to semilinear problems by adding several modifications. However, to avoid redundancy and to give a simplified proof, here we restrict ourselves to deal with only time dependent coefficient case. 2. Asymptotic Profile We first introduce the notation used throughout this paper. We denote by C a generic positive constant, which may change from line to line. Also, we frequently use the symbol a(t) ∼ b(t), which means that C −1 b(t) ≤ a(t) ≤ Cb(t) holds for some constant C ≥ 1. Let Lp and H m be the standard Lebesgue and Sobolev spaces on R, and H k,m be the weighted on R Pk Sobolev spaces m l equipped with the norm defined by kf kH k,m = k(1 + |x|) ∂ f (·)k L2 . x l=0 The existence of a unique global solution u satisfying u ∈ C([0, ∞), H 2,1 ) and ut ∈ C([0, ∞), H 0,1 ) for any (u0 , u1 ) ∈ H 2,1 × H 0,1 can be shown in a standard 2
way. Let us denote a heat kernel by K(t, x) := (4πt)−1/2 exp(−|x|2 /4t). Observe that K(A(t), x) satisfies the equation ut − a(t)uxx = 0. Our main theorem is the following. Theorem 2.1. Let α > −1/2. Then the solution u for (1)-(2) satisfies 1
λ
ku(t, ·) − m∗ K(A(t), ·)kL2 ≤ C(A(t) + 1)− 4 − 2 k(u0 , u1 )kH 2,1 ×H 0,1 2 for t ≥ 1 with some C > 0, m∗ ∈ R, λ ∈ (0, min{ 12 , α+1 , 4α+2 α+1 }).
Remark 2.2. It is easily seen that the solution u for (1)-(2) and G satisfy 1
1
ku(t, ·)kL2 ≤ CA(t)− 4 k(u0 , u1 )k(H 2 ∩L1 )×(L2 ∩L1 ) , kK(A(t), ·)kL2 = CA(t)− 4 . Although H 2,1 × H 0,1 ⊂ (H 2 ∩ L1 ) × (L2 ∩ L1 ) holds, it is known that the decay rate can not be accelerated anymore. Thus the theorem says that the solution u behaves as m∗ K(A(t), x) asymptotically as t → ∞. The main reason we use weighted Sobolev spaces for initial data is technical, and we will giveR the remark in the proof. As we shall see later, m∗ is given as m∗ := limt→∞ R u(t, x)dx. Moreover, in the same observation as many other results (see e.g. [8]), we shall also obtain more rapidly decaying estimates with higher order derivatives under corresponding adequate smooth initial data. However, for compactness we do not discuss it here. Proof of Theorem 2.1. Let us denote s := log(A(t)+1) and y := (A(t)+1)−1/2 x. Under the assumption α > −1/2, t → ∞ and s → ∞ are equivalent. For the new variables (s, y) we define the transformation into new unknowns (v, w) = (v(s, y), w(s, y)) from (u, ut )(t, x) by the relations: ! 1 x u(t, x) = p v log(A(t) + 1), p , A(t) + 1 A(t) + 1 ! a(t) x ut (t, x) = w log(A(t) + 1), p . (A(t) + 1)3/2 A(t) + 1 Then the equation (1) is rewritten as y v y 3 a0 e−s −s vs − vy − = w, ae ws − wy − w + w + w = vyy − vyyyy . 2 2 2 2 a a 1
From the definition of s, we see (t + 1) ∼ e α+1 s . Then it is easy to check that 2α+1 1 1 ae−s ∼ e− α+1 s , e−s /a ∼ e− α+1R s and |a0 |/a ≤ Ca ae−s ∼ e− α+1 s . For the functions: m(s) := R v(s, y)dy, φ(y) := K(1, y) = (4π)−1/2 exp(−y 2 /4) and ψ(y) := φyy (y), we decompose solutions as follows: v(s, y) = m(s)φ(y) + f (s, y),
w(s, y) = ms (s)φ(y) + m(s)ψ(y) + g(s, y).
3
We expect that the functions m(s)φ(y) and ms (s)φ(y)+m(s)ψ(y) are the asymptotic profiles of v and w, respectively. Substituting these into the above equation, we deduce the equations for the residue terms f and g: y f y 3 a0 e−s −s fs − fy − = g, ae gs − gy − g + g + g = fyy − fyyyy + h, (4) 2 2 2 2 a a y 3 e−s a0 h := −2ae−s ms ψ + ae−s m ψy + ψ − mψyy − mψ, 2 2 a a where we have used yφy /2 + φ/2 = −ψ and a0 −s −s (5) ae mss = ae − − 1 ms . a R R R It is easy to check that R f (s, y)dy = R g(s, y)dy = R h(s, y)dy = 0. By using the classical energy method, we shall derive decay estimates for f and g. Let us define energies Eij = Eij (s) (i = 0, 1 and j = 1, 2) by Z Z 2 1 e−s 2 F 2 −s 2 −s E01 := Fy + F + ae G dy, E02 := + ae F G dy, a yy 2 R 2 R Z Z f2 1 e−s 2 fyy + ae−s g 2 dy, E12 := + ae−s f g dy, E11 := fy2 + a 2 R R 2 Ry Ry where F (s, y) := −∞ f (s, z)dz and G(s, y) := −∞ g(s, z)dz. Here, we note that F and G make sense as L2 -functions by the following well-known Hardy type inequality. To define F and G, we choose the weighted Sobolev spaces for the initial data. For the proof we refer to e.g. [14, Lemma 3.9]. R Lemma 2.3. Suppose that f = f (y) ∈ H 0,1 satisfies R f (y)dy = 0, and let Ry R R F (y) = −∞ f (z)dz. Then, we have R F (y)2 dy ≤ 4 R y 2 f (y)2 dy. From the equations (4), F and G satisfy
y Fs − Fy = G, 2 where H(s, y) :=
y a0 e−s ae−s Gs − Gy − G + G + G = Fyy − Fyyyy + H, 2 a a
Ry
−∞
ae−s
s
h(s, z)dz. It holds that =
a0 − ae−s , a
e−s a
s
=−
a0 e−s . − 3 a a
Then, by a straightforward calculation, we have Z Z Z Z d 1 a0 a0 2 E01 + G2 dy = E01 − 3 Fyy dy − G2 dy + GHdy, ds 2 2a R 2a R R R Z Z 1 d E02 + E02 + 2E01 = 2ae−s G2 dy + F Hdy. ds 2 R R 4
(6)
It follows from the assumptions in (3) that Z Z Z Z −s a0 Ca −s a0 Ca e 2 Fyy dy ≤ − G2 dy ≤ ae G2 dy, − 3 F 2 dy ≤ Ca E01 . 2a R 2 2a R 2 R a yy R Setting E0 := E01 + Ca2+1 E02 , from the Young inequality we have for any ε > 0 Z d 1 E0 (s) + E0 (s) ≤ ε F 2 (s, y)dy + C(ε)kH(s, ·)k2L2 , s ≥ s0 , (7) ds 2 R R where we have used the fact that the term Cae−s R G2 dy can be absorbed into R G2 dy in the left hand side for s ≥ s0 with large s0 , since ae−s is decreasing R function. Analogously, for E11 and E12 we have Z Z Z Z 3 a0 a0 d 2 E11 + g 2 dy = E11 − 3 fyy dy − g 2 dy + ghdy, ds 2 2a R 2a R R R Z Z d 1 E12 + 2E11 + E12 = E12 + 2ae−s g 2 dy + f hdy. ds 2 R R R R Observe that E12 ≤ C R f 2 dy+Cae−s R g 2 dy from the Young inequality. Then in a similar manner to above, by setting E1 := E11 + Ca2+2 E12 , we obtain Z d 1 E1 (s) + E1 (s) ≤ C f 2 (s, y)dy + Ckh(s, ·)k2L2 , s ≥ s0 . (8) ds 2 R Therefore, calculating (8) + C0 × (7) with a large constantR C0 and a small R constant ε, we can absorb the terms ε R F 2 (s, y)dy and C R f 2 (s, y)dy into E01 and E02 in the left-hand side, and hence we obtain for some λ ∈ (0, 1/2): d (C0 E0 + E1 )(s) + λ(C0 E0 + E1 )(s) ≤ C(kh(s, ·)k2L2 + kH(s, ·)k2L2 ), ds
s ≥ s0 .
d Next, putting Em0 := 21 ae−s m2s , we obtain ds Em0 + 12 Em0 + m2s = ae−s m2s . 1 Then Em0 (s) ≤ e− 2 s Em0 (s0 ) for s ≥ s0 with large s0 . Set Em1 := 21 m2 + d ae−s mms . It follows from (5) and (6) that ds Em1 = 2Em0 (s). We thus obtain Em1 (s) ≤ Em1 (s0 ) + CEm0 (s0 ), which implies the boundedness of |m|(s). With the help of Lemma 2.3, we can easily check that −s 2 ! e 2 2 −s 2 2 −s 2 (Em0 (s0 ) + Em1 (s0 )). kHkL2 + khkL2 ≤ C|ae | |ms | + C |ae | + a
e Let us define E(s) and E(s) by E(s) := C0 E0 (s) + E1 (s) + Em0 (s) and 2α+1 1 e E(s) := E(s) + Em1 (s). Thus, applying ae−s ∼ e− α+1 s , e−s /a ∼ e− α+1 s , we have 2 d 1 e 0 ) + Ce− 4α+2 e 0 ), α+1 s E(s E(s) + λE(s) + m2s ≤ Ce− α+1 s E(s ds 2
5
whence 1 2 d λs e 0 ) + Ce−( 4α+2 e 0 ). α+1 −λ)s E(s e E(s) + eλs m2s ≤ Ce−( α+1 −λ)s E(s ds 2
2 − λ > 0 and Thanks to α > −1/2, there exists some λ ∈ (0, 1/2) satisfying α+1 4α+2 −λs e − λ > 0. Therefore, we obtain E(s) ≤ Ce E(s ) for s ≥ s 0 0 . From the α+1 R s λσ 2 e estimate we also have s0 e ms (σ) dσ ≤ C E(s0 ). Then one has
|m(s) − m(s0 )|2 ≤
Z
s
s0
e−λσ dσ
Z
s
s0
0 e 0 ), eλσ ms (σ)2 dσ ≤ Ce−λs E(s
s0 ≤ s0 ≤ s,
which assures that the limit m∗ := lims→+∞ m(s) exists and |m(s) − m∗ |2 ≤ e 0 ). Finally, we have Ce−λs E(s e 0 ). kv(s, ·) − m∗ φk2L2 ≤ 2kf (s, ·)k2L2 + 2|m(s) − m∗ |2 kφk2L2 ≤ Ce−λs E(s
e 0 ), we easily see that E(s e 0 ) ≤ Ck(u0 , u1 )k2 2,1 0,1 . From the definition of E(s H ×H Recalling the relation K(A(t) + 1, x) = (A(t) + 1)−1/2 φ((A(t) + 1)−1/2 x), we complete the proof of Theorem 2.1. Acknowledgments. This work was partially supported by JSPS KAKENHI Grant Numbers JP16K05234, JP16K17625. The authors express their sincere gratitude for referees’ kind and careful comments. References [1] J. M. Ball, Stability theory for an extensible beam, J. Differential Equations, 14 (1973), 399–418. [2] M. Brokate, J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996. [3] Th. Gallay, G. Raugel, Scaling variables and asymptotic expansions in damped wave equations, J. Differential Equations, 150 (1998), 42–97. [4] P. Marcati, M. Mei, Convergence to nonlinear diffusion waves for solutions of the initial boundary problem to the hyperbolic conservation laws with damping, Quart. Appl. Math., 58 (2000), 763–784. [5] P. Marcati, M. Mei, B. Rubino, Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping, J. Math. Fluid Mech., 7 (2005), suppl. 2, S224–S240. [6] P. Marcati, K. Nishihara, The Lp -Lq estimates of solutions to onedimensional damped wave equations and their application to the compressible flow through porous media, J. Differential Equations, 191 (2003), 445– 469. 6
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