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

Abstract

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 ...

Large time behavior of solutions to a nonlinear hyperbolic relaxation system with slowly decaying data

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