Scattering and asymptotic order for the wave equations with the scale-invariant damping and mass

Inui, Takahisa; Mizutani, Haruya

doi:10.1007/s00030-020-00671-7

Cited by 4 publications

(8 citation statements)

References 29 publications

(32 reference statements)

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“…We consider (DW) and give an extension of their result by the density and the better estimate for L r -data. This extension also gives an improvement of the asymptotic order which was obtained by our previous paper [4] for H 1 ×L 2 -data. 1.2.…”

supporting

confidence: 63%

“…Recently, do Nascimento, Palmieri, and Reissig [5] gave a better estimate with respect to time t of the L 2 -norm of the solution than that in [1] when the initial data belongs to L r (R d ) for 1 ≤ r < 2. Thus, if the initial data belongs to L r (R d ), then we expect to improve the asymptotic order combining our previous argument in [4] and the estimate in [5]. We will also get the scattering result in the case which was not treated in the previous paper [4].…”

mentioning

confidence: 62%

“…Thus, if the initial data belongs to L r (R d ), then we expect to improve the asymptotic order combining our previous argument in [4] and the estimate in [5]. We will also get the scattering result in the case which was not treated in the previous paper [4]. For example, we can get the scattering in the case of µ 1 > 2 and µ 2 = 0, which implies µ < 0.…”

mentioning

confidence: 67%

“…In the previous paper [4], we considered the linear wave equation with the scaleinvariant damping and mass (DW), showed the scattering result (see also [8,10,6]), and obtained the asymptotic order. Namely, there exists…”

mentioning

confidence: 99%

“…Main results. As in the previous paper [4], by the Liouville transfrom v := (1 + t) µ 1 2 u, we transform (DW) into the following Klein-Gordon equation with the scale-invariant mass:…”

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confidence: 99%

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Remarks on asymptotic order for the linear wave equation with the scale-invariant damping and mass with $L^r$-data

Inui¹,

Mizutani²

2020

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In the present paper, we consider the linear wave equation with the scale-invariant damping and mass. It is known that the global behavior of the solution depends on the size of the coefficients in front of the damping and mass at initial time t = 0. Indeed, the solution satisfies the similar decay estimate to that of the corresponding heat equation if it is large and to that of the modified wave equation if it is small. In our previous paper, we obtain the scattering result and its asymptotic order for the data in the energy space H 1 × L 2 when the coefficients are in the wave regime. In fact, the threshold of the coefficients relies on the spatial decay of the initial data. Namely, it varies depending on r when the initial data is in L r (1 ≤ r < 2). In the present paper, we will show the scattering result and the asymptotic order in the wave regime for L r -data, which is wider than the wave regime for the data in the energy space. Moreover, we give an improvement of the asymptotic order obtained in our previous paper for the data in the energy space.

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supporting

confidence: 63%

mentioning

confidence: 62%

mentioning

confidence: 67%

mentioning

confidence: 99%

“…Main results. As in the previous paper [4], by the Liouville transfrom v := (1 + t) µ 1 2 u, we transform (DW) into the following Klein-Gordon equation with the scale-invariant mass:…”

mentioning

confidence: 99%

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