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Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients. (English) Zbl 1098.35094
The Cauchy problem for
$u_{tt} - \sum_1^n\sum_1^n a_{ij}(t)u_{x_ix_j} + \sum_1^n b_i(t)u_{x_i} + c(t)u=0 \text{ in } [0,T]\times \mathbb R^n$ with initial data
$u(0,x) = u_0(x)\;,\;u_t(0,x)=u_1(x) \text{ in }\mathbb R^n$ is considered under the basic assumption of strict hyperbolicity. The first purpose of the paper is to weaken the Lipschitz continuity assumption on the coefficients of the principal part of the operator - it is assumed that $$a_{ij}$$ are $$C^1$$ functions on $$[0,T]\setminus\bar{t}$$ and $|a^{\prime}_{ij}(t)|\leq C|t-\bar{t}|^{-q}\;,\;\forall t\in [0,T]\setminus\{\bar{t}\}$ where $$\bar{t}$$ may be 0.
The following results are proved: If the above condition holds with $$q=1$$, then the problem is $$C^\infty$$ well posed; if $$q>1$$ and $$a_{ij}$$ are bounded, then the problem is $$\gamma^{(s)}$$ well posed, where $$\gamma^{(s)}$$ is the Gevrey space of order $$s$$, $$s<q/(q-1)$$; if $$q>1$$ and $$a_{ij}$$ are Hölder continuous of exponent $$\alpha$$ then $$s<q((q-1)(1-\alpha))^{-1}$$. By means of counter-examples it is proved that these results cannot be improved. For $$n=1$$ well posedness can be proved also for the case $$c(t,x)$$ instead of $$c(t)$$.

##### MSC:
 35L15 Initial value problems for second-order hyperbolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
##### Keywords:
strict hyperbolicity
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##### References:
 [1] F. Colombini - E. De Giorgi - S. Spagnolo, Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temp, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 6 (1979), 511-559. Zbl0417.35049 MR553796 · Zbl 0417.35049 · numdam:ASNSP_1979_4_6_3_511_0 · eudml:83819 [2] F. Colombini - D. Del Santo - T. Kinoshita, On the Cauchy problem for hyperbolic operators with non-regular coefficients, to appear in Proceedings of the Conference “À la mémoire de Jean Leray” Karlskrona 2000, M. de Gosson - J. Vaillant (eds.), Kluwer, New York. Zbl1036.35122 MR2051477 · Zbl 1036.35122 [3] F. Colombini - N. Lerner, Hyperbolic operators with non-Lipschitz coefficients, Duke Math. J. 77 (1995), 657-698. Zbl0840.35067 MR1324638 · Zbl 0840.35067 · doi:10.1215/S0012-7094-95-07721-7 [4] F. Colombini - S. Spagnolo, Some examples of hyperbolic equations without local solvability, Ann. Sci. École Norm. Sup. (4) 22 (1989), 109-125. Zbl0702.35146 MR985857 · Zbl 0702.35146 · numdam:ASENS_1989_4_22_1_109_0 · eudml:82242 [5] L. Hörmander, “Linear Partial Differential Operators”, Springer-Verlag, Berlin, 1963. Zbl0108.09301 · Zbl 0108.09301 [6] E. Jannelli, Regularly hyperbolic systems and Gevrey classes, Ann. Mat. Pura Appl. 140 (1985), 133-145. Zbl0583.35074 MR807634 · Zbl 0583.35074 · doi:10.1007/BF01776846 [7] T. Nishitani, Sur les équations hyperboliques à coefficients höldériens en $$t$$ et de classe de Gevrey en $$x$$, Bull. Sci. Math. 107 (1983), 113-138. Zbl0536.35042 MR704720 · Zbl 0536.35042
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