Damped wave equation with a critical nonlinearity

Abstract

Abstract. We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearitywhere ε > 0, and space dimensions n = 1, 2, 3. Assume that the initial data, weighted Sobolev spaces areThen we prove that there exists a positive ε 0 such that the Cauchy problem above has a unique global solution u ∈ C [0, ∞) ; H δ,0 satisfying the time decay property

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

“…Since he has used the Fourier transform method in the proof, his results do not have any information on the L p decay estimates for 1 a p < 1 þ a of the solutions. Also, Hayashi, Kaikina, and Naumkin [5] have obtained the global solutions in any dimensions for initial data on suitable weighted Soblev spaces.…”

Asymptotics of solutions to the periodic problem for the nonlinear damped wave equation

“…without smallness assumption on the initial data) have the same large time asymptotics as that for the linear heat equation. The critical case σ = 2 was considered in paper [12], where it was proved that the large time decay estimate of solutions of the Cauchy problem has an additional logarithmic correction. The large time asymptotics of solutions to the Cauchy problem with small initial data in the subcritical case σ ∈ (0, 2) , when σ is close to 2, was obtained in paper [14].…”

Global existence of solutions for 2‐D semilinear wave equations with dissipation localized near infinity in an exterior domain

“…We are concerned with the following mixed problem for the damped wave equation in the 2-dimensional exterior domain ⊂ R 2 with a smooth boundary @ u tt (t; x) − u(t; x) + a(x)u t (t; x) = |u(t; x)| p ; (t; x) ∈ (0; ∞) × (1) u(0; x) = u 0 (x); u t (0; x) = u 1 (x); x∈ (2) Without loss of generality, one may assume 0 ∈ and @ ⊂ B 0 = {x ∈ R N : |x|¡ 0 } for some 0 ¿0.…”

“…There are many previous results concerning global in time existence results for the Cauchy problem in R N of Equation (1) with a constant coe cient a(x) ≡ 1, and for these results we refer the reader to Hayashi et al [1] (N ¿1), Hosono and Ogawa [2] (N = 2), Ikehata et al [3] (N = 1; 2), Ikehata and Tanizawa [4] (N ¿1), Narazaki [5] (N = 4; 5), Nishihara [6] (N = 3) and Todorova and Yordanov [7] (N ¿1). All these results deal with the case when the power p of the non-linear term for (1) is larger than the so-called Fujita exponent p * (N ) = 1 + 2=N .…”

“…Note that only the (pioneering) result due to Reference [7] is developed in the framework of compactly supported solutions. The basic precise decay estimates to the linear equation (1) (without non-linear term) played essential roles in these results, and that was developed by Matsumura [8]. For blowup results in the region 1¡p6p * (N ) we can cite excellent works due to Li and Zhou [9], Todorova and Yordanov [7] and Zhang [10].…”