an:01270287
Zbl 0933.35119
Kinoshita, Tamotu
On the wellposedness in the Gevrey classes of the Cauchy problem for weakly hyperbolic systems with H??lder continuous coefficients in time
EN
Osaka J. Math. 35, No. 4, 735-750 (1998).
00055166
1998
j
35L40 35A05
energy estimates
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 H??lder 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 \textit{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 \break linear operator in \(L^2_{p_1,k, \nu}(\mathbb{R}^n)\), which ensures the existence and unicity of \(u_l\). Moreover \break \(\{e^{\rho_1 \langle\xi_l (D)\rangle^k_\nu}u_l\}\) is bounded on \(L^2\), and its weak limit is a solution of (*), with the required regularity.
G.Gussi (Bucure??ti)
Zbl 0583.35074