Hyperbolic operators with non-Lipschitz ceofficients
In a joint paper of some years ago [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 6, 511-559 (1979; Zbl 0417.35049)], the first author, \textit{E. De Giorgi} and the reviewer investigated the solvability of a strictly hyperbolic Cauchy problem of the form \[ \partial^2_t u= \sum \partial_i(a_{ij}, \partial_j u),\quad u(0, x)= u_0(x),\quad \partial_t u(0, x)= u_1(x),\tag{\(*\)} \] where \(a_{ij}= a_{ji}\) and \(\sum a_{ij} \xi_i\xi_j\geq \lambda_0|\xi|^2\) \((\lambda_0> 0)\), in absence of the usual Lipschitz continuity in time assumption on the coefficients. In particular, it was proved that in the special case wherein the coefficients \(a_{ij}\equiv a_{ij}(t)\) are independent of \(x\), the problem \((*)\) is globally well posed in the Gevrey class \({\mathcal E}^s(\mathbb{R}^n)\) for \(1\leq s< (1- \alpha)^{- 1}\) whenever the \(a_{ij}\)'s are \(\alpha\)-Hölder continuous \((0< \alpha< 1)\), while \((*)\) is globally well posed in \({\mathcal C}^\infty(\mathbb{R}^n)\) as soon the \(a_{ij}\)'s satisfy, for small \(|\tau|\), the logarithmic Lipschitz condition \[ |a_{ij}(t+ \tau)- a_{ij}(t)|\leq C|\tau||\log|\tau||.\tag{LL} \] In the present paper, the general case wherein the coefficients \(a_{ij}\equiv a_{ij}(t, x)\) depend also on \(x\) is studied and the following results are proved: 1) If the \(a_{ij}(x, t)\) are \({\mathcal C}^\infty\) in \(x\) and satisfy (LL) with respect to time, then \((*)\) is globally well posed in \({\mathcal C}^\infty(\mathbb{R}^n)\). 2) If the \(a_{ij}(x, t)\) satisfy isotropically an (LL) condition with respect to both the variables \((x, t)\), then for all \((u_0, u_1)\in H^{1- \theta}\times H^\theta\) \((0< \theta< 1/4)\) there exists a unique local solution \(u(\cdot, t)\in H^{1- \theta- \beta t}\) with \(\partial_t u(\cdot, t)\in H^{- \theta- \beta t}\), for \(0\leq t< \beta\), where \(\beta\) is a positive constant depending only on the coefficients. The proof consists in a sort of microlocalization (via the Littlewood-Paley decomposition) of the approximate energy method introduced in the above-mentioned paper. Finally, the authors show, by suitable counterexamples, that the (LL) condition cannot be further weakened without loosing the existence of a distribution solution.
scientific article; zbMATH DE number 784718
scientific article