Strictly hyperbolic Cauchy problems with coefficients low-regular in time and space

Abstract

We consider the strictly hyperbolic Cauchy problem \begin{align*} &D_t^m u - \sum\limits_{j = 0}^{m-1} \sum\limits_{|γ|+j = m} a_{m-j,\,γ}(t,\,x) D_x^γD_t^j u = 0, \newline &D_t^{k-1}u(0,\,x) = g_k(x),\,k = 1,\,\ldots,\,m, \end{align*} for $(t,x)\in [0,T]\times {\mathbb{R}}^n$ with coefficients belonging to the Zygmund class $C_{\ast }^s$ in $x$ and having a modulus of continuity below Lipschitz in $t$. Imposing additional conditions to control oscillations, we obtain a global (on $[0,T]$) $L^2$ energy estimate without loss of derivatives for $s\ge \{1+\epsilon ,\frac{2m_0}{2-m_0}\}$, where $m_0$ is linked to the modulus of continuity of the coefficients in time.