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

Abstract

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].

Asymptotic behavior of solutions for the damped wave equation with slowly decaying data

“…The formula of S N (t)g is referred in Courant and Hilbert [2]. The decompositions are proposed in Marcati and Nishihara [16] (N = 1), Ikehata, Nishihara and Zhao [10] (N = 2) and Nishihara [20] (N = 3). In the case N = 2 see Hosono and Ogawa [9].…”

On the critical nonlinear damped wave equation with large initial data

“…In [17,27] the following weighted energy type estimates were proved. (See also papers [15,16,28,31].)…”

“…Recently in paper [28] it was obtained an optimal time decay estimate of the L p -norm of solutions to the Cauchy problem (1.1) in the subcritical case σ ∈ (0, 2 n ) under the condition that the initial data decay exponentially at infinity without any restriction on the size. In the sequential paper [17] the condition of the exponential decay on the initial data was relaxed to the algebraic decay. Here we use a priori weighted energy type estimates proved in papers [17,27] via modification of the method of papers [15,28].…”

“…In the sequential paper [17] the condition of the exponential decay on the initial data was relaxed to the algebraic decay. Here we use a priori weighted energy type estimates proved in papers [17,27] via modification of the method of papers [15,28]. Then we follow the approach of papers [12] and [14] to find the large time asymptotics of solutions.…”

Global well-posedness for semilinear hyperbolic equations with dissipative term

“…we can find a host of literature [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. Recently in [9], problem (1) was studied.…”