Estimates for Besov and Lizorkin–Triebel norms of solutions of the second-order linear hyperbolic equations

L. V. Kapitanskii

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 20, Tome 171 (1989), pp. 106-162

Résumé

We study the nonhomogeneous hyperbolic equations $$ \partial^2_tu+iB(t)\partial_tu+A(t)u=h\qquad{(1)} $$ on $[0,T]\times\mathfrak{M}$, where $\mathfrak{M}=\mathbb{R}^n$ or $\mathfrak{M}$ is a smooth closed manifold, $A(t)$ and $B(t)$ are the time-dependent pseudodifferential operators on $\mathfrak{M}$ of orders 2 and 1, resp. For the solutions of (1) we obtain the estimates of the form \begin{multline*} ||\partial_t^lu(t,\cdot)||_{G_{p,q_2}^{r-l}}\leqslant c\left\{\sigma_{\nu,p,n}(t)(||u(0,\cdot)||_{E_{p',q_1}^{r+\nu}}+\right.\\ +\left.||\partial_t u(0,\cdot)||_{E_{p',q_1}^{r+\nu-1}})+\int_0^t\sigma_{\nu,p,n}(t-\tau)||h(\tau,\cdot)||_{E_{p',q_1}^{r+\nu-1}}d\tau\right\} \end{multline*} with arbitrary real $r$ and integer $l\geqslant0$, where $G.^\cdot,.$ and $E.^\cdot,.$ are the corresponding Besov spaces $B.^\cdot,.(\mathfrak{M})$ or Lizorkin–Triebel spaces $F.^\cdot,.(\mathfrak{M})$. The admissible choice of these spaces as well as the choice of the scalar function $\sigma_{\nu,p,n}(t)$ depends on the values of $n$, $\nu$, $p$, $q_1$, $q_2$ and “the Brenner's number” $m$, defined by the principal symbols of operators $A(0)$ and $B(0)$. Another class of estimates obtained in this paper, the estimates of the form $$ \left(\int_0^T ||\partial_t^lu(t,\cdot)||_{G_{p,q_1}^{r-l}}^{q_2}dt\right)^{1/q_2}\leqslant c\left\{||u(0,\cdot)||_{H^s}+||\partial_tu(0,\cdot)||_{H^{s-1}}+\int_0^T||h(t,\cdot)||_{H^{s-1}}dt\right\}, $$ characterize the space-time integrability properties and the “smoothing” (for $t>0$) of the solutions of (1).