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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)
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References:
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