Geodesic
Asymptotics for nonlinear damped wave equations with large initial data
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 4 (2007), pp. 249-277

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We study the one dimensional nonlinear damped wave equation \begin{equation} \begin{cases} u_{tt}+u_{t}-u_{xx}=\lambda|u|^{\sigma}u,\in\mathbf{R},\quad t>0,\\ u(0,x)=u_0(x), x\in\mathbf{R},\\ u_t(0,x)=u_1(x), x\in\mathbf{R}, \end{cases} \tag{0.1} \end{equation} where $\sigma>0$, $\lambda\in\mathbf R$. Our aim is to prove the large time asymptotic formulas for solutions of the Cauchy problem (0.1) without any restriction on the size of the initial data. 
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     author = {N. Hayashi and E. I. Kaikina and P. I. Naumkin},
     title = {Asymptotics for nonlinear damped wave equations with large initial data},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {249--277},
     publisher = {mathdoc},
     volume = {4},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2007_4_a14/}
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N. Hayashi; E. I. Kaikina; P. I. Naumkin. Asymptotics for nonlinear damped wave equations with large initial data. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 4 (2007), pp. 249-277. http://geodesic.mathdoc.fr/item/SEMR_2007_4_a14/