Article

# The Pointwise Estimates of Solutions for Semilinear Dissipative Wave Equation

04/2008;
Source: arXiv

ABSTRACT In this paper we focus on the global-in-time existence and the pointwise estimates of solutions to the initial value problem for the semilinear dissipative wave equation in multi-dimensions. By using the method of Green function combined with the energy estimates, we obtain the pointwise estimates of the solution. Comment: 20 pages

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ABSTRACT: We study the time-asymptotic behavior of solutions for the isentropic Euler equations with damping in multi-dimensions. The global existence and pointwise estimates of the solutions are obtained. Furthermore, we obtain the optimal Lp, 1<p⩽+∞, convergence rate of the solution when it is a perturbation of a constant state. Our approach is based on a detailed analysis of the Green function of the linearized system and some energy estimates.
Journal of Differential Equations. 01/2001;
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ABSTRACT: We consider the Cauchy problem for the damped wave equation with absorption The behavior of u as t→∞ is expected to be the same as that for the corresponding heat equation which has the similarity solution wa(t,x) with the form depending on a=lim|x|→∞|x|2/(p−1)f(x)⩾0 provided that p is less than the Fujita exponent pc(N):=1+2/N. In this paper, as a first step, if 1<p<pc(N) and the data (u0,u1)(x) decays exponentially as |x|→∞ without smallness condition, the solution is shown to decay with rates as t→∞,(∗) those of which seem to be reasonable, because the similarity solution wa(t,x) has the same decay rates as (∗). For the proof, the weighted L2-energy method will be employed with suitable weight, similar to that in Todorova and Yordanov [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464–489].
Journal of Mathematical Analysis and Applications. 01/2006;
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##### Article: Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption
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ABSTRACT: We consider the Cauchy problem for the damped wave equation with absorption(∗) The behavior of u as t→∞ is expected to be same as that for the corresponding heat equation ϕt−Δϕ+|ϕ|ρ−1ϕ=0, (t,x)∈R+×RN. In the subcritical case 1<ρ<ρc(N):=1+2/N there exists a similarity solution wb(t,x) with the form depending on b=lim|x|→∞|x|2/(ρ−1)f(|x|)⩾0. Our first aim is to show the decay rates(∗∗) provided 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 L2-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].
Journal of Differential Equations. 01/2006; 226(1):1-29.