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On the wellposedness in the Gevrey classes of the Cauchy problem for weakly hyperbolic systems with Hölder continuous coefficients in time. (English) Zbl 0933.35119
The author considers systems of the form \[ \partial_tu=\sum A_h(t)\partial_h u+B(t)u\tag{*} \] on \([0,T]\times\mathbb{R}^n_\eta\) with the initial condition \(u(0,x)=u_0(x)\), where \(A_h(t)\), \(B(t)\) are \(N\times N\) matrices, \(A_h\) is Hölder continuous, \(B\) is continuous, and the system (*) is weakly hyperbolic. Two cases are considered: First, when no condition is imposed (besides the weak hyperbolicity) and second, when there exists a non-singular matrix \(P(t,\xi)\) such that \[ P(t,\xi) A(t,\xi)P(t,\xi)^{-1}=\text{diag} \{D_1,D_2, \dots,D_k\} \quad\text{for some }1\leq k\leq N \] and the \(D_j\) are triangular matrices of size \(m_j\times m_j\), whose diagonal elements are real, and moreover \[ \bigl |P(t,\xi)\bigr |+ \bigl|P(t,\xi) \bigr|^{-1} \leq C\text{ for any }t\in [0,T],\quad|\xi|=1. \] The result proved by the author, which generalizes previous results of E. Jannelli [Ann. Mat. Pura Appl., IV. Ser. 140, 133-145 (1985; Zbl 0583.35074)], is the following: Let \(0<p_0< \infty\) and \(\nu_0>0\). Then there exists \(\nu>0\) such that for any \(u_0\in L^2_{\rho,k, \nu_0}(\mathbb{R}^n)\) the Cauchy problem has an unique solution \(u\in C^1([0,T]\), \(L^2_{\rho_1,k,\nu}(\mathbb{R}^n))\) provided \(0<p_1<\rho_0\), \(1<s< {\mu(1 +\sigma^{-1}) \over\mu (1+\sigma^{-1}) -1}\) where \(\mu=N\) (case 1) or \(\mu= \max_{1\leq i\leq k}m_i\) (case 2) and \(s={1\over k}\). Here \(L^2_{p,k, \nu} (\mathbb{R}^n) =\{u\in L^2 (\mathbb{R}^n)\), \(e^{\rho\langle \xi\rangle^k_\nu} \widehat \mu \in L^2(\mathbb{R}^n)\}\), where \(\langle\xi \rangle_\nu= (|\xi |^2+ \nu^2)^{1 /2}\). The proof uses energy estimates and some nontrivial algebraic lemmas. For the existence part the author considers an associated system of the form \[ \partial_t u_l=\sum A_h(t)il\sin(Dh/l)u_l+B(t)u_l\tag{**} \] and since \(il\sin(Dh/l)\) belong to \(OPS^\circ\) for any fixed \(l\) the right hand side of (**) is a bounded
linear operator in \(L^2_{p_1,k, \nu}(\mathbb{R}^n)\), which ensures the existence and unicity of \(u_l\). Moreover
\(\{e^{\rho_1 \langle\xi_l (D)\rangle^k_\nu}u_l\}\) is bounded on \(L^2\), and its weak limit is a solution of (*), with the required regularity.

35L40 First-order hyperbolic systems
35A05 General existence and uniqueness theorems (PDE) (MSC2000)