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On wave equations with supercritical nonlinearities
被引：0
作者：
P. Brenner
P. Kumlin
机构：
[1] School of Mathematical and Computing Sciences,
[2] Chalmers University of,undefined
[3] Technology and Göteborg University,undefined
[4] S-41296 Göteborg,undefined
[5] Sweden,undefined
来源：
Archiv der Mathematik
|
2000年
/
74卷
关键词：
Wave Equation;
Nonlinear Wave;
Lipschitz Mapping;
Nonlinear Wave Equation;
Solution Operator;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
We prove that the solution operators \documentclass[12pt]{minimal}
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${\cal e}_t (\phi , \psi )$\end{document} for the nonlinear wave equations with supercritical nonlinearities are not Lipschitz mappings from a subset of the finite-energy space \documentclass[12pt]{minimal}
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$(\dot {H}^1 \cap L_{\rho +1}) \times L_2$\end{document} to \documentclass[12pt]{minimal}
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$\dot {H}^s_{q'}$\end{document} for \documentclass[12pt]{minimal}
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$t\neq 0$\end{document}, and \documentclass[12pt]{minimal}
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$0\leq s\leq 1,$\end{document}\documentclass[12pt]{minimal}
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$(n+1)/(1/2-1/q')= 1$\end{document}. This is in contrast to the subcritical case, where the corresponding operators are Lipschitz mappings ([3], [6]). Here \documentclass[12pt]{minimal}
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${\cal e}_t(\phi , \psi )=u(\cdot , t)$\end{document}, where u is a solution of \documentclass[12pt]{minimal}
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$$\left\{\matrix {\partial ^2_tu-\Delta _xu+ m^2u+|u|^{\rho -1}u=0, \, t>0, \, x \in {\Bbb R}^n,\cr u\vert _{t=0}(x)=\phi (x),\hfill\cr \partial _tu\vert _{t=0}(x)=\psi (x). \hfill}\right.$$\end{document} where \documentclass[12pt]{minimal}
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$n \geq 4, m\geq 0$\end{document} and \documentclass[12pt]{minimal}
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$\rho >\rho ^\ast =(n+2)/(n-2)$\end{document} in the supercritical case.
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页码：129 / 147
页数：18
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