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Small analytic solutions to nonlinear weakly hyperbolic systems. (English) Zbl 0858.35079

It is considered the first-order system \(\partial_tu= f(u,\partial_1u,\dots, \partial_nu)\), \(u(0,x)= \varepsilon\phi(x)\), when it is weakly hyperbolic at \(u=0\), i.e., when \(\sum^n_{h=1} \zeta_h{{\partial f}\over{\partial z_h}}(0)\) has real eigenvalues for all \(\zeta=(\zeta_1,\dots, \zeta_n)\in\mathbb{R}^n\). Let \(f(0)=0\) and \(|\partial_x^\alpha \phi(x)|\leq C\rho_0^{-|\alpha|}\alpha\) \(\forall \alpha\in\mathbb{N}^n\), \(x\in\mathbb{R}^n\) for some \(C\), \(\rho_0>0\). Then it is proved:
(i) For \(\varepsilon\to 0\), the lifespan \(T_\varepsilon\) of the solution \(u_\varepsilon\) tends to infinity, and for all \(T>0\) the sequence \(u_\varepsilon\) tends to 0 in the class of analytic functions on \([0,T]\times \mathbb{R}^n\).
(ii) The lifespan \(T_\varepsilon\) admits the asymptotic estimate (for \(\varepsilon\to0\)) \(T_\varepsilon\geq \mu(\log{1\over\varepsilon})^{1/N}\). In the special case when \(f(0)= \partial_y f(0)=0\), in particular for the system \(\partial_tu= f(\partial_1u,\dots, \partial_nu)\), we have the stronger estimate \(T_\varepsilon\geq \mu({1\over\varepsilon})^{1/N}\). It is proved that these estimates are optimal.
Reviewer: L.G.Vulkov (Russe)

MSC:

35L60 First-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs

Keywords:

lifespan
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References:

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