Global Solvability and Lp Decay for the Semilinear Dissipative Wave Equations in Four and Five Dimensions

Ono, Kosuke

doi:10.1619/fesi.49.215

Cited by 11 publications

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References 21 publications

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“…(2.1), where we can see the same results with lower dimensions in [14,15] for N 3 and [17] for N 5 (also see Marcati and Nishihara [6], Nishihara [13] for N = 1, 3, cf. Milani and Han [8] for t 1).…”

Section: Proposition 21 Let M 0 Be Zero or An Integer Suppose Thatsupporting

confidence: 72%

L1 estimates for dissipative wave equations in exterior domains

Ono

2007

Journal of Mathematical Analysis and Applications

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We study the exterior initial-boundary value problem for the linear dissipative wave equation ( + ∂ t )u = 0 in Ω × (0, ∞) with (u, ∂ t u)| t=0 = (u 0 , u 1 ) and u| ∂Ω = 0, where Ω is an exterior domain in N -dimensional Euclidean space R N . We first show higher local energy decay estimates of the solution u(t), and then, using the cut-off technique together with those estimates, we can obtain the L 1 estimate of the solution u(t) when N 3, that is, u(t) L 1 (Ω) C( u 0 H n (Ω) + u 1 H n−1 (Ω) + u 0 W n,1 (Ω) + u 1 W n−1,1 (Ω) ) for t 0, where n = [N/2] is the integer part of N/2. Moreover, by induction argument, we derive the higher energy decay estimates of the solution u(t) for t 0.

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Section: Proposition 21 Let M 0 Be Zero or An Integer Suppose Thatsupporting

confidence: 72%

L1 estimates for dissipative wave equations in exterior domains

Ono

2007

Journal of Mathematical Analysis and Applications

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“…By starting with this less sharp rate, to obtain the asymptotic profile θ 0 G(t, x) in the supercritical case is the second aim. In fact, we will show There have been many studies of the non-absorbing semilinear term problem [7,8,[13][14][15][16][17]22,24,31,32,34], etc., about the small date global existence and blow-up in a finite time, etc. See also the survey papers by Levine [21], Deng and Levine [2], and the pioneering paper [5] by H. Fujita, concerning the semilinear heat equations.…”

Section: 2)mentioning

confidence: 88%

Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption

Ikehata

Nishihara

Zhao

2006

Journal of Differential Equations

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We consider the Cauchy problem for the damped wave equation with absorption u tt − u + u t + |u| ρ−1 u = 0, (t,x)∈ R + × R N .( * )The behavior of u as t → ∞ is expected to be same as that for the corresponding heat equation φ t − Δφ + |φ| ρ−1 φ = 0, (t, x) ∈ R + × R N . In the subcritical case 1 < ρ < ρ c (N )Our first aim is to show the decay ratesprovided that the initial data without initial data size restriction spatially decays with reasonable polynomial order. The decay rates ( * * ) are sharp in the sense that they are same as those of the similarity solution. The second aim is to show that the Gauss kernel is the asymptotic profile in the supercritical case, which has been shown in case of one-dimensional space by Hayashi, Kaikina and Naumkin [N. Hayashi, E.I. Kaikina, P.I. Naumkin, Asymptotics for nonlinear damped wave equations with large initial data, preprint, 2004]. We show this assertion in two-and three-dimensional space. To prove our results, both the weighted L 2 -energy method and the explicit formula of solutions will be employed. The weight is an improved one originally developed in [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].

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“…Later on, Todorova and Yordanov [65] and Zhang [68] determined the critical exponent as p = 1 + 2 n for all space dimensions. Moreover, Ono [55,56] derived L m -decay of solutions for 1 ≤ m ≤ 2n/(n−2) + . The results of [36] and [65] require that the initial data belongs to H 1 × L 2 and has the compact support.…”

Section: Introductionmentioning

confidence: 99%

The Cauchy problem for the nonlinear damped wave equation with slowly decaying data

Ikeda

Inui

Wakasugi

2017

Nonlinear Differ. Equ. Appl.

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We study the Cauchy problem for the nonlinear damped wave equation and establish the large data local well-posedness and small data global well-posedness with slowly decaying initial data. We also prove that the asymptotic profile of the global solution is given by a solution of the corresponding parabolic problem, which shows that the solution of the damped wave equation has the diffusion phenomena. Moreover, we show blow-up of solution and give the estimate of the lifespan for a subcritical nonlinearity. In particular, we determine the critical exponent for any space dimension.1 , min{2, n n−2 }] if 3 ≤ n ≤ 6. Finite time blow-up of local solutions was also obtained for any n ≥ 1 and 1 < p < 2r n . However, the above global well-posedness results are restricted to n ≤ 6 and there are no results for higher dimensional cases. Also, Narazaki [42] considered the slowly decaying data belonging to modulation spaces and proved the global existence when the nonlinearity has integer power.Concerning the asymptotic profile of global solutions, Gallay and Raugel [5] determined the asymptotic expansion up to the second order when n = 1 and the initial data belongs to the weighted Sobolev space H 1,1 × H 0,1 (see Section 1.2 for the definition). Using the expansion of solutions to the heat equation, Kawakami and Ueda [31] extended it to the case n ≤ 3. Hayashi, Kaikina and Naumkin [9] obtained the first order asymptotics for all n ≥ 1 and the initial data belonging to (H s,0 ∩ H 0,α ) × (H s−1,0 ∩ H 0,α ) with α > n 2 (particularly, belonging to L 1 ). Recently, Takeda [63, 64] determined the higher order asymptotic expansion of global solutions. Narazaki and Nishihara [43] studied the case of slowly decaying data and proved that if n ≤ 3 and the data behaves like (1 + |x|) −kn with 0 < k ≤ 1, then, the asymptotic profile of the global solution is given by G(t, x) * (1 + |x|) −kn , where G is the Gaussian and * denotes the convolution with respect to spatial variables.Related to the equation (1.1), systems of nonlinear damped wave equation were studied and the critical exponent and the asymptotic behavior of solutions were investigated (see [61,41,62,53,42,54,49,50,15,51]).In the present paper, we establish the large data local well-posedness and the small data global well-posedness for the nonlinear damped wave equation (1.1) with slowly decaying initial data. Our global well-posedness results extend those of [25,43] to all space dimensions, and generalize that of [9] to slowly decaying initial data. Moreover, we study the asymptotic profile of the global solution. This also extends those of [43] to all space dimensions. Considering the asymptotic behavior of solutions in weighted norms, we further extended the result of [9] to the asymptotics in L m -norm with m ≤ 2. Finally, we give an almost optimal lifespan estimate from both above and below. This is also an extension of [36,47,20], in which L 1 -data were treated.1.1. Main results. We say that u ∈ L ∞ (0, T ; L 2 (R n )) is a mild solution of (1.1) if u satisfies the ...

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Global Solvability and Lp Decay for the Semilinear Dissipative Wave Equations in Four and Five Dimensions

Cited by 11 publications

References 21 publications

L1 estimates for dissipative wave equations in exterior domains

L1 estimates for dissipative wave equations in exterior domains

Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption

The Cauchy problem for the nonlinear damped wave equation with slowly decaying data

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