## 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

lifespan
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### References:

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