Strictly Hyperbolic Cauchy Problems with Coefficients Low-Regular in Time and Space

Abstract

We consider the strictly hyperbolic Cauchy problem

$$\displaystyle \begin {cases} D_t^m u - \sum \limits _{j = 0}^{m-1} \sum \limits _{|\gamma |+j = m} a_{m-j,\,\gamma }(t,\,x) D_x^\gamma D_t^j u = 0,\\ D_t^{k-1}u(0,\,x) = g_k(x),\,k = 1,\,\ldots ,\,m, \end {cases} $$

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 $s \geq \max \{1+\varepsilon ,\,\frac {2m_0}{2-m_0}\}$, where m ₀ is linked to the modulus of continuity of the coefficients in time.