Cauchy problem for hyperbolic operators with triple effective characteristics on the initial plane

TL;DR

This paper proves well-posedness of the Cauchy problem for effectively hyperbolic operators with triple characteristics on the initial plane, removing previous restrictive conditions and extending prior results.

Contribution

It establishes well-posedness without assuming the previously required condition (E), advancing the understanding of hyperbolic operators with triple characteristics.

Findings

Well-posedness of the Cauchy problem for operators with triple characteristics.

Removal of the condition (E) from previous assumptions.

Extension of results to more general hyperbolic operators.

Abstract

We study effectively hyperbolic operators $P$ with triple characteristics points lying on $t= 0$. Under some conditions on the principal symbol of $P$ one proves that the Cauchy problem for $P$ in $[0, T] \times U$ is well posed for every choice of lower order terms. Our results improves those in [11] since we don't assume the condition (E) of [11] satisfied.

Equations

$$P(t,x,D_t,D_x)=\sum _{k+|\alpha |\le 3}{c}_{k,\alpha }(t,x)D_t^k{D}_x^{\alpha },D_t=-i{\partial }_t,D_{x_j}=-i{\partial }_{x_j}$$

$$p(t,x,\tau ,\xi )=\sum _{k+|\alpha |=3}{c}_{k,\alpha }(t,x){\tau }^k{\xi }^{\alpha }={\tau }^3+q_1(t,x,\xi ){\tau }^2+q_2(t,x,\xi )\tau +q_3(t,x,\xi )$$

$$\begin{array}{r}P=(D_t-\mathrm{Op}(\phi )⟨D⟩{)}^3+\mathrm{Op}(a)⟨D⟩(D_t-\mathrm{Op}(\phi )⟨D⟩{)}^2-\mathrm{Op}(b)⟨D{⟩}^2(D_t-\mathrm{Op}(\phi )⟨D⟩)\\ +\mathrm{Op}(c)⟨D{⟩}^3-\sum _{j=0}^2\mathrm{Op}({\tilde{b}}_{1j})⟨D{⟩}^j(D_t-\mathrm{Op}(\phi )⟨D⟩{)}^{2-j}\end{array}$$

$$q_{1}\langle{\xi}\rangle^{-1}+3\varphi,\quad-\big{(}q_{2}\langle{\xi}\rangle^{-2}+2\varphi q_{1}\langle{\xi}\rangle^{-1}+3\varphi^{2}\big{)},\quad q_{3}\langle{\xi}\rangle^{-3}+\varphi q_{2}\langle{\xi}\rangle^{-2}+\varphi^{2}\langle{\xi}\rangle^{-1}+\varphi^{3},$$

$$s(t,x,D)u=({\rm Op}^{w}(s)u)(x)=(2\pi)^{-n}\int\int e^{i(x-y)\xi}s\Bigl{(}t,\frac{x+y}{2},\xi\Bigr{)}u(y)dyd\xi.$$

$$p(t,x,\tau ,\xi )=(\tau -\phi ⟨\xi ⟩{)}^3+a⟨\xi ⟩(\tau -\phi ⟨\xi ⟩{)}^2-b⟨\xi {⟩}^2(\tau -\phi ⟨\xi ⟩)+c⟨\xi {⟩}^3$$

$$\frac{{\partial }^{2}p}{\partial t\partial \tau }(\rho )<0.$$

$$\frac{\Delta}{\langle\xi\rangle^{6}}\geq\delta t\Bigl{(}\frac{\Delta_{0}}{\langle\xi\rangle^{2}}\Bigr{)}^{2}.$$

$$\frac{\Delta }{⟨\xi {⟩}^6}\ge \delta t^2\frac{{\Delta }_0}{⟨\xi {⟩}^2}.$$

$$\begin{array}{rl}& b(t,x,\xi )\ge {\delta }_{1}t,\\ & c=\mathcal{O}(b^2),⟨\xi {⟩}^{\alpha }{\partial }_{\xi }^{\alpha }{\partial }_x^{\beta }c=\mathcal{O}(b),|\alpha +\beta |=1,⟨\xi {⟩}^{\alpha }{\partial }_{\xi }^{\alpha }{\partial }_x^{\beta }c=\mathcal{O}(\sqrt{b}),|\alpha +\beta |=2,\\ & {\partial }_{t}c=\mathcal{O}(b),⟨\xi {⟩}^{\alpha }{\partial }_{\xi }^{\alpha }{\partial }_x^{\beta }(ac)=\mathcal{O}(\sqrt{b}),|\alpha +\beta |=3.\end{array}$$

$$\begin{split}b(t,x,\xi)\geq\delta_{1}t,\;\;\langle\xi\rangle^{\alpha}\partial_{t}^{\gamma}\partial_{\xi}^{\alpha}\partial_{x}^{\beta}c={\mathcal{O}}\big{(}b^{2-|\alpha+\beta|/2-|\gamma|}\big{)}\;\;\text{for}\;\;|\alpha+\beta+\gamma|\leq 3,\;\;\gamma=0,1.\end{split}$$

$$b^2(a^2+4b)\ge \delta t(a^2+3b{)}^2.$$

$$b^2(a^2+4b)\ge \delta t^2(a^2+3b)$$

$$p=({\tau }^2+2d(t,x,\xi )\tau +f(t,x,\xi ))(\tau -\lambda (t,x,\xi ))$$

$$p=\big{(}(\tau-\lambda)^{2}+2(\lambda+d)(\tau-\lambda)+\lambda^{2}+2d\lambda+f\big{)}(\tau-\lambda)$$

$$p={\tau }^3-(t+\alpha (x,\xi ))⟨\xi {⟩}^2\tau -(t^2{b}_2+t{b}_1+b_0)⟨\xi {⟩}^3,$$

$$p=(\tau+b_{1}\langle\xi\rangle)^{3}-3b_{1}\langle\xi\rangle(\tau+b_{1}\langle\xi\rangle)^{2}-\big{(}t+\alpha-3b_{1}^{2}\big{)}\langle\xi\rangle^{2}(\tau+b_{1}\langle\xi\rangle)$$

$$-\big{[}t^{2}b_{2}+b_{0}-b_{1}\alpha+b_{1}^{3}\big{]}\langle\xi\rangle^{3}.$$

$$h_{p,q}(\zeta ,\bar{\zeta })=\frac{p(\zeta )q(\bar{\zeta })-p(\bar{\zeta })q(\zeta )}{\zeta -\bar{\zeta }}=\sum _{i,j=1}^m{h}_{ij}{\zeta }^{i-1}{\bar{\zeta }}^{j-1}$$

$$A_{p}=\left(\begin{array}[]{ccccc}0&1&\cdots&0\\ \vdots&\vdots&\ddots\\ 0&0&\cdots&1\\ -a_{m}&-a_{m-1}&\cdots&-a_{1}\end{array}\right).$$

$${\sigma }_{\ell ,k}=\sum _{1\le j_1<\cdots <j_{\ell }\le m,j_p\ne k}{\lambda }_{j_1}\cdots {\lambda }_{j_{\ell }}.$$

$$h_{ij}=\sum _{k=1}^m(-1{)}^{i+j}{\sigma }_{m-i,k}{\sigma }_{j-1,k}.$$

$$R=\left(\begin{array}[]{ccccc}1&1&\cdots&1\\ \lambda_{1}&\lambda_{2}&\cdots&\lambda_{m}\\ \vdots&\vdots&\ddots&\vdots\\ \lambda_{1}^{m-1}&\lambda_{2}^{m-1}&\cdots&\lambda_{m}^{m-1}\end{array}\right).$$

$$A_{p}R=R\left(\begin{array}[]{ccc}\lambda_{1}&\\ &\ddots&\\ &&\lambda_{m}\end{array}\right).$$

$$r_{ij}=c_{ij}({\lambda }_1,\dots ,{\lambda }_{i-1},{\lambda }_{i+1},\dots ,{\lambda }_m){\Delta }_i.$$

$$m-j=m(m-1)/2-j+1-(m-1)(m-2)/2.$$

$$c_{ij}=(-1{)}^{i+j}{\sigma }_{m-j,i}.$$

$$C{A}_p{C}^{-1}=D^{-1}(R^{-1}{A}_{p}R)D.$$

$$C={\rm diag}\,\Big{(}\pm\prod_{k\neq 1}(\lambda_{i}-\lambda_{k}),\pm\prod_{k\neq 2}(\lambda_{i}-\lambda_{k}),\ldots,\pm\prod_{k\neq m}(\lambda_{i}-\lambda_{k})\Big{)}R^{-1}$$

$$p_{\varepsilon}(\zeta)=\Bigl{(}1+\varepsilon\frac{\partial}{\partial\zeta}\Bigr{)}^{m-1}p(\zeta),\>\varepsilon\neq 0.$$

Full text

1. Introduction

In this paper we study the Cauchy problem for a differential operator

[TABLE]

of order $3$ with smooth coefficients $c_{k,\alpha}(t,x),\>t\in{\mathbb{R}},\>x\in\Omega,\>c_{3,0}\equiv 1.$ Denote by

[TABLE]

the principal symbol of $P$. With a real symbol $\varphi\in S^{0}_{1,0}$ one can write

[TABLE]

which is a differential operator in $t$. Here the symbols $a,\,b,\,c\in S^{0}_{1,0}$ coincide with

[TABLE]

respectively, ${\tilde{b}}_{1j}\in S^{0}_{0,1},\>j=0,1,2$ (see [3]), and $\langle{D}\rangle$ has symbol $\langle\xi\rangle=(1+|\xi|^{2})^{1/2}.$ Throughout the paper we work with symbols $s(t,x,\xi)$ which depend smoothly on $t\in[0,T]$ and we use the Weyl quantization

[TABLE]

We will use the notation $S^{m}_{0,1}$ for the class of symbols (see [3]) and we abbreviate $S^{m}_{1,0}$ to $S^{m}$ and ${\rm Op}^{w}(s)$ to ${\rm Op}(s)$. First we assume that the principal symbol

[TABLE]

is hyperbolic, that is the roots of equation $p=0$ with respect to $\tau$ are real for $(t,x,\xi)\in[0,T]\times\Omega\times{\mathbb{R}}^{n},$ where $\Omega\subset{\mathbb{R}}^{n}$ is an open set. Second we assume also that $p$ has triple characteristic points only if $t=0$ and $P$ is effectively hyperbolic (see [4], [11]) at every triple characteristic points $\rho=(0,x,\tau,\xi)$ which is equivalent (see [4]) to the condition

[TABLE]

Recall that an operator is effectively hyperbolic if the fundamental matrix $F_{p}(z)$ of the principal symbol $p$ has two non-vanishing eigenvalues $\pm\mu(z)$ at every point $z$ where $dp(z)=0$. An effectively hyperbolic operator may have triple characteristics only for $t=0$ or $t=T$ (see [4]). Consequently, at a triple characteristic point $\rho_{0}=(0,x_{0},0,\xi_{0})$, assuming $\varphi(0,x_{0},\xi_{0})=0$, we have $b_{t}(0,x_{0},\xi_{0})>0.$ Moreover, at $\rho_{0}$ we have $a(0,x_{0},\xi_{0})=b(0,x_{0},\xi_{0})=c(0,x_{0},\xi_{0})=0$.

Our purpose is to study the Cauchy problem for $P$ and to prove that under some conditions on $p$ this problem is well posed for every choice of lower order terms. This property is called strong hyperbolicity and the effective hyperbolicity of $P$ is a necessary condition for it ([4]). For operators having only double characteristics every effectively hyperbolic operator is strongly hyperbolic and we refer to [9] for the related works. The conjecture is that effectively hyperbolic operators with triple characteristic points on $t=0$ are strongly hyperbolic (see [4],[1], [11]). On the other hand, for some class of hyperbolic operators with triple characteristics the above conjecture has been proved in [6], [1], [11], but the general case is still an open problem.

In [11] the strong hyperbolicity was established under the condition (E) saying that for some $\delta>0$ we have the lower bound

[TABLE]

Here $\Delta\in S^{6}$ is the discriminant of the equation $p=0$ with respect to $\tau$, while $\Delta_{0}\in S^{2}$ is the discriminant of the equation $\frac{\partial p}{\partial\tau}=0$ with respect to $\tau.$ In [11] it was introduced also a weaker condition (H) saying that with some constant $\delta>0$ we have

[TABLE]

We can consider a microlocal version of the conditions (E) and (H) assuming the above inequalities hold for $(t,x,\xi)$ in a small conic neighborhood $W_{0}$ of every triple characteristic point $(0,x_{0},\xi_{0}).$ The main goal of this paper will be stated in Theorem 4.1 and Corollary 4.5 which improve the results in [11] and show that we have a strong hyperbolicity for some operators for which (E) is not satisfied, but (H) holds. In particular, we cover the case of operators whose principal symbol $p$ admits a microlocal factorization with one smooth root under the condition that there are no double characteristic points of $p$ converging to a triple characteristic point $(0,x,0,\xi)$ (see Example 1.1).

Concerning the symbols $a(t,x,\xi),\>b(t,x,\xi),\>c(t,x,\xi),$ we assume the existence of $\delta_{1}>0$ such that

[TABLE]

It is clear that the condition (1.3) are satisfied if

[TABLE]

In fact, we assume a slightly weaker microlocal conditions formulated in (3.11) and Theorem 4.1.

Below we present two examples of operators with triple characteristics on $t=0$ satisfying the above assumptions.

Example 1.1**.**

Assume $c\equiv 0.$ Then the symbol $p$ becomes $p=((\tau-\varphi\langle{\xi}\rangle)^{2}+a\langle\xi\rangle(\tau-\varphi\langle{\xi}\rangle)-b\langle\xi\rangle^{2})(\tau-\varphi\langle{\xi}\rangle)$. Let $\rho=(0,x_{0},\varphi(0,x_{0},\xi_{0})\langle\xi_{0}\rangle,\xi_{0}),$ be a triple characteristic point. For small $t>0$ we have $b(t,x_{0},\xi_{0})>0.$ If for some $(y,\eta)$ sufficiently close to $(x_{0},\xi_{0})$ we have $b(0,y,\eta)<0$, then there exists $z=(t^{*},x^{*},\xi^{*})$ with $t^{*}>0$ such that $b(z)=0$ and the equation $(\tau-\varphi\langle{\xi}\rangle)^{2}+a\langle{\xi}\rangle(\tau-\varphi\langle{\xi}\rangle)-b\langle{\xi}\rangle^{2}=0$ has a root $\varphi(z)\langle{\xi^{*}}\rangle$ for $z$. This implies the existence of a double characteristic point $(t^{*},x^{*},\varphi(z)\langle\xi^{*}\rangle,\xi^{*})$ of $p$. We exclude this possibility, assuming $b(0,x,\xi)\geq 0$ for $(x,\xi)$ close to $(x_{0},\xi_{0}).$

Remark 1.1**.**

For the operator in Example $1.1$, the discriminant of the equation $p=0$ has the form $\Delta=b^{2}(a^{2}+4b)\langle\xi\rangle^{6},$ while $\Delta_{0}=4(a^{2}+3b)\langle\xi\rangle^{2}.$ Therefore the condition $(E)$ is reduced to

[TABLE]

If $b={\mathcal{O}}(t),$ this inequality yields $b^{2}a^{2}+4b^{3}\geq\delta ta^{4}$ and hence $a^{2}\leq{\mathcal{O}}(t^{2})/\delta t={\mathcal{O}}(t)$ which is not satisfied in any small neighborhood of a triple characteristic point $(0,x_{0},\varphi(0,x_{0},\xi_{0})\langle\xi_{0}\rangle,\xi_{0})$, unless $a(0,x,\xi)=0$ for all $(0,x,\xi)$ close to the point $(0,x_{0},\xi_{0})$. On the other hand, the inequality

[TABLE]

obviously holds ($b\geq\delta_{1}t$ is assumed), hence $(H)$ is satisfied.

The Example 1.1 covers the case when the principal symbol $p$ admits a factorization

[TABLE]

with $C^{\infty}$ smooth real root $\lambda(t,x,\xi)$ and $p$ has not double characteristic points in a neighborhood of $(0,x_{0},\xi_{0})$. In fact, we may write

[TABLE]

and taking $\varphi=\lambda\langle{\xi}\rangle^{-1}$ we reduce the symbol to Example 1.1. Notice that effectively hyperbolic operators with principal symbols admitting above factorization have been studied by V. Ivrii in [6] who proved the strong hyperbolicity constructing parametrix. Here we present another proof based on energy estimates with weight $t^{-N},$ assuming $P$ strictly hyperbolic for small $t>0$.

Example 1.2**.**

Consider the operator with principal symbol

[TABLE]

where $\alpha,b_{0},b_{1},b_{2}$ are zero order pseudo-differential operators and $\alpha\geq 0.$ This class of operators has been studied in [11] under the condition $(E)$. We write $p$ as follows

[TABLE]

[TABLE]

Choosing $\varphi=-b_{1}(t,x,\xi)$ one reduces the symbol $p$ to the form $(\ref{eq:1.2})$ with $a=-3b_{1},\>b=t+\alpha-3b_{1}^{2},\>c=-(t^{2}b_{2}+b_{0}-b_{1}\alpha+b_{1}^{3}).$ If $\alpha\geq 3b_{1}^{2},\>b_{0}=b_{1}\alpha-b_{1}^{3},$ the condition (1.4) is satisfied, while for $\alpha=3b_{1}^{2},\>b_{0}=b_{1}\alpha-b_{1}^{3}$ the condition $(E)$ is not satisfied for $b_{1}$, unless $b_{1}(0,x,\xi)\equiv 0$. It is easy to see that with the above choice of $b_{0}$ and $b_{1}$, the condition $(H)$ holds.

Notice that if $\rho=(t,x,\tau,\xi)$ with $t>0$ is a double characteristic point for $p$, one has $\Delta(\rho)=0$ and $\Delta_{0}(\rho)>0$. Therefore the condition (H) is not satisfied and the analysis of this case is a difficult open problem. The proofs in this work are based on energy estimates with weight $t^{-N}$ with $N\gg 1$ leading to estimates with big loss of regularity. This phenomenon is typical for effectively hyperbolic operators with multiple characteristics (see [4], [1], [11]). We follow the approach in [11] reducing the problem to the one for first order pseudo-differential system. In Section 2 we construct a symmetrizer $S$ for the principal symbol of the system following a general result (see Lemma 2.1) which has independent interest. Moreover, $\det S=\frac{1}{27}\Delta$ and under our assumptions one shows that $\det S\geq\delta b^{2}(a^{2}+4b),\>\delta>0.$ Therefore $\Delta\geq\varepsilon t^{2}(a^{2}+4b),\>\varepsilon>0,$ and in general the condition (E) is not satisfied. This leads to difficulties in Section 3, where a more fine analysis of the matrix pseudo-differential operators is needed. In Section 4 we show that the microlocal conditions (1.3) are sufficient for the energy estimates in Theorems 4.1 and 4.2.

2. Symmetrizer

First we recall a general result concerning the existence of a symmetrizer. Let $p(\zeta)=\zeta^{m}+a_{1}\zeta^{m-1}+\cdots+a_{m}$ be a monic hyperbolic polynomial of degree $m$ and let $q(\zeta)=p^{\prime}(\zeta)$. Here $a_{j}(t,x,\xi)$ depend on $(t,x,\xi)$ but we omit this in the notations below. Let

[TABLE]

be the Bézout form of $p$ and $q$. It is well known that the matrix $H=(h_{ij})$ is nonnegative definite (see for example [5]).

Consider the Sylvester matrix $A_{p}$ corresponding to $p(\zeta)$ which has the form

[TABLE]

One has the following result [10] and for the sake of completeness we present the proof.

Lemma 2.1**.**

([10, Lemma 2.3.1])* $H$ symmetrizes $A_{p}$ and ${\rm det}\,H=\Delta^{2}$ where $\Delta$ is the difference-product of the roots of $p(\tau)=0$.*

Proof.

We first treat the case when $p(\zeta)$ is a strictly hyperbolic polynomial. Let $\lambda_{j}$, $j=1,\ldots,m$ be the different roots of the equation $p(\zeta)=0$. Write $p(\zeta)=\prod_{j=1}^{m}(\zeta-\lambda_{j})$ and set

[TABLE]

Since $p^{\prime}(\zeta)=\sum_{k=1}^{m}\prod_{j=1,j\neq k}^{m}(\zeta-\lambda_{j})=\sum_{i=1}^{m}(-1)^{m-i}\sigma_{m-i,k}\zeta^{i-1}$ it is easy to see

[TABLE]

Denote by $R$ the Vandermonde’s matrix having the form

[TABLE]

Since $\lambda_{i}\neq\lambda_{j}$, $i\neq j$, the matrix $R$ is invertible and $|{\rm det}\,R|=|\Delta|$. It is clear that

[TABLE]

Denote by ${}^{co}\!R=(r_{ij})$ the cofactor matrix of $R$ and by $\Delta(\lambda_{1},\ldots,\lambda_{k})$ the difference-product of $\lambda_{1},\ldots,\lambda_{k}$. It is easily seen that $r_{ij}$ is divisible by $\Delta_{i}=\Delta(\lambda_{1},\ldots,\lambda_{i-1},\lambda_{i+1},\ldots,\lambda_{m})$, hence

[TABLE]

Since $r_{ij}$ and $\Delta_{i}$ are alternating polynomials in $(\lambda_{1},\ldots,\lambda_{i-1},\lambda_{i+1},\ldots,\lambda_{m})$ of degree $m(m-1)/2-j+1$ and $(m-1)(m-2)/2$ respectively, then $c_{ij}$ is a symmetric polynomial of degree

[TABLE]

Therefore $c_{ij}$ is a polynomial in fundamental symmetric polynomials of $(\lambda_{1},\ldots,\lambda_{i-1},\lambda_{i+1},\ldots,\lambda_{m})$. Noting that $\Delta_{i}$ is of degree $m-2$ and $r_{ij}$ ($j\neq m$) is of degree $m-1$ respectively with respect to $\lambda_{\ell}$ ($\ell\neq i$), one concludes that $c_{ij}$ is of degree $1$ with respect to $\lambda_{\ell}$ ($\ell\neq i$) which proves that

[TABLE]

Thus denoting $C=(c_{ij})$ we have ${}^{t}CC=(h_{ij})=H$. In particular, this shows that the symmetric matrix $H$ is nonnegative definite as it was mentioned above.

Set $D={\rm diag}\,(\Delta_{1},\ldots,\Delta_{m})$ and note that $D$ is invertible. Moreover it follows from (2.1) that $C=D^{-1}(^{co}\!R)=({\rm det}R)D^{-1}R^{-1}$ and hence

[TABLE]

It is clear that $CA_{p}C^{-1}$ is a diagonal matrix because both $R^{-1}A_{p}R$ and $D$ are diagonal matrices. Then $CA_{p}C^{-1}={{}^{t}}C^{-1}\,{{}^{t}}\!A_{p}{{}^{t}}C$ yields ${{}^{t}}CCA_{p}={{}^{t}}\!A_{p}{{}^{t}}CC$ which proves that $HA_{p}$ is symmetric. From $C=({\rm det}R)D^{-1}R^{-1}$ it follows that

[TABLE]

and hence $|{\rm det}\,C|=|\prod_{j=1}^{m}\prod_{k\neq j}^{m}(\lambda_{k}-\lambda_{j})|/|\Delta|=|\Delta|$. Consequently, ${\rm det}\,H=\Delta^{2}$ and this completes the proof for strictly hyperbolic polynomial $p(\zeta).$

Passing to the general case, introduce the polynomial

[TABLE]

According to [12], $p_{\varepsilon}(\zeta)$ is strictly hyperbolic and let $H_{\varepsilon}=$ ${}^{t}C_{\varepsilon}C_{\varepsilon}$ be the symmetrizer for $A_{p_{\varepsilon}}$ constructed above. Obviously, as $\varepsilon\to 0$, we have $A_{p_{\varepsilon}}\rightarrow A_{p}$ since the coefficients of $p_{\varepsilon}(\zeta)$ go to the ones of $p(\zeta)$. The roots of $p(\zeta)$ depend continuously on the coefficients and this yields $\lambda_{j,\varepsilon}\rightarrow\lambda_{j}$, $\lambda_{j,\varepsilon}$ being the roots of $p_{\varepsilon}(\zeta)=0.$ The equalities (2.2) imply $C_{\varepsilon}\rightarrow C$ and passing to the limit $\varepsilon\to 0$, we obtain the result. ∎

Note that $H$ is different from the Leray’s symmetrizer ([7]) since if $B$ is the Leray’s symmetrizer, then ${\rm det}\,B=\Delta^{2(m-1)}$. Now consider

[TABLE]

Corollary 2.1**.**

Let $J=(\delta_{i,m+1-j}),$ where $\delta_{ij}$ is the Kronecker’s delta. Then ${\tilde{H}}=JH\,{{}^{t}}\!J$ symmetrizes ${\tilde{A}}_{p}$ and ${\rm det}\,{\tilde{H}}=\Delta^{2}$.

Proof.

Since ${\tilde{A}}_{p}=JA_{p}\,{{}^{t}}\!J$ and ${{}^{t}}\!JJ=I$ the proof is immediate. ∎

With U={{}^{t}}\big{(}(D_{t}-{\rm Op}(\varphi)\langle{D}\rangle)^{2}u,\langle{D}\rangle(D_{t}-{\rm Op}(\varphi)\langle{D}\rangle)u,\langle{D}\rangle^{2}u\big{)} the equation $Pu=f$ is reduced

[TABLE]

where $F={{}^{t}}(f,0,0)$ and

[TABLE]

where $b_{ij}\in S^{0}_{1,0}$.

Introduce

[TABLE]

which is a representation matrix (conjugated by $J$ in Corollary 2.1) of the Bézout form of $p(\tau)=\tau^{3}+a\tau^{2}-b\tau+c$ and $p^{\prime}(\tau)$ (see for example [5], [8]). Therefore $S$ symmetrizes $A$ so that

[TABLE]

Note that when $c=0$ one has

[TABLE]

and hence

[TABLE]

.

Lemma 2.2**.**

There exist $\bar{\varepsilon}>0$ and $\delta>0$ such that

[TABLE]

if $|ac|\leq\bar{\varepsilon}\,b^{2}$ and $|c|\leq\bar{\varepsilon}\,b^{3/2}$.

Proof.

Note that

[TABLE]

Since

[TABLE]

choosing $\bar{\varepsilon}=1/50$ for instance, the assertion is clear. ∎

Lemma 2.3**.**

There exist $\bar{\varepsilon}>0$ and $\varepsilon_{1}>0$ such that

[TABLE]

provided $|ac|\leq\bar{\varepsilon}\,b^{2}$ and $|c|\leq\bar{\varepsilon}\,b^{3/2}$.

Proof.

Since

[TABLE]

one obtains

[TABLE]

Indeed

[TABLE]

Noting $b\geq\delta_{1}t$, one gets the above representation and we deduce $\det(3S-\varepsilon_{1}tJ)\geq 0$ for small $\varepsilon_{1}.$ In the same way one treats the principal minors of order 2. For example

[TABLE]

[TABLE]

[TABLE]

since all terms involving $\varepsilon_{1}t$ can be compensated by $a^{2}b^{2}+2b^{3}.$ ∎

Lemma 2.4**.**

Assume $\langle\xi\rangle^{\alpha}c_{(\beta)}^{(\alpha)}={\mathcal{O}}(\sqrt{b})$ for $|\alpha+\beta|=2$ and $\langle\xi\rangle^{\alpha}(ac)_{(\beta)}^{(\alpha)}={\mathcal{O}}(\sqrt{b})$ for $|\alpha+\beta|=3.$ There exists $C>0$ such that for $U\in C^{\infty}({\mathbb{R}}_{t}:C_{0}^{\infty}({\mathbb{R}}^{n}))$ we have

[TABLE]

Proof.

We will follow the argument of [11, Section 3] and we use the notation $\partial_{\xi}^{\alpha}D_{x}^{\beta}Q=Q^{(\alpha)}_{(\beta)}.$ Recall that we have the representation

[TABLE]

with $R\in S^{-2}_{1/2,0}$ and real symbols $\psi_{\alpha,\beta}\in S^{(|\alpha|-|\beta|)/2},$ where $Q_{F}$ is the Freidrichs part of $Q$ (see [11, Appendix], [2]) and hence $(Q_{F}U,U)\geq 0$.

Notice that $b$ is real, hence $({\rm Op}(b)U_{3},U_{3})={\mathsf{Re}}\,({\rm Op}(b)U_{3},U_{3}).$ Setting $Q=S-2\varepsilon_{1}tJ,$ we have

[TABLE]

and it is enough to prove

[TABLE]

Indeed if this is true, then we have

[TABLE]

Thus we conclude the assertion.

To prove (2.6), consider ${\mathsf{Re}}({\rm Op}(\psi_{\alpha\beta}Q_{(\beta)}^{(\alpha)})U,U)$ with $|\alpha+\beta|=2$. Setting $g=b^{2}-\varepsilon tb-2ac,$ one has

[TABLE]

Here and below $S^{m}$ denotes some symbol in the class $S^{m}$. This yields

[TABLE]

and hence

[TABLE]

Let $T=\psi_{\alpha\beta}g^{(\alpha)}_{(\beta)}\langle{\xi}\rangle.$ Then $\psi_{\alpha\beta}g^{(\alpha)}_{(\beta)}={\mathsf{Re}}\,(T\#\langle{\xi}\rangle^{-1})+S^{-2}$ and

[TABLE]

Note that $\|{\rm Op}(T)U_{3}\|^{2}=({\rm Op}(T\#T)U_{3},U_{3})$ and $T\#T=T^{2}+S^{-2}$. Therefore there exists $C>0$ such that

[TABLE]

because $\langle\xi\rangle^{\alpha}c^{(\alpha)}_{(\beta)}={\mathcal{O}}(\sqrt{b})$ and \langle\xi\rangle^{\alpha}\Bigl{(}b(b-\varepsilon_{1}t)\Bigr{)}^{(\alpha)}_{(\beta)}={\mathcal{O}}(\sqrt{b}) and $b\geq\delta t$. Applying the Fefferman-Phong inequality for the operator with symbol $Cb-T^{2},$ one proves the assertion.

For the case $|\alpha+\beta|=3$ with $T_{1}=\psi_{\alpha\beta}g^{(\alpha)}_{(\beta)}\langle{\xi}\rangle^{3/2}$ we have the inequality

[TABLE]

with some $C>0$. Indeed, $\langle\xi\rangle^{\alpha}(ac)^{(\alpha)}_{(\beta)}={\mathcal{O}}(\sqrt{b})$ and \langle\xi\rangle^{\alpha}\Bigl{(}b(b-\varepsilon_{1}t)\Bigr{)}^{(\alpha)}_{(\beta)}={\mathcal{O}}(\sqrt{b}). Repeating the above argument, we complete the proof. ∎

Corollary 2.2**.**

Let ${\tilde{S}}=S+\lambda\,t^{-1}\langle{\xi}\rangle^{-2}I$. Then there exists $\lambda_{0}>0$ such that for $\lambda\geq\lambda_{0}$ we have

[TABLE]

Corollary 2.3**.**

There exist $\delta_{2}>0$ and $\lambda_{0}>0$ such that

[TABLE]

Proof.

Since there exists $\delta_{1}>0$ such that $b\geq\delta_{1}t$ from the Fefferman-Phong inequality for the scalar symbol $b-\delta_{1}t$ one deduces

[TABLE]

which proves the assertion thanks to Corollary 2.2. ∎

3. Energy estimates

Consider the energy $(t^{-N}e^{-\gamma t}{\rm Op}({\tilde{S}})U,U)$, where $(\cdot,\cdot)$ is the $L^{2}({\mathbb{R}}^{n})$ inner product and $N>0$, $\gamma>0$ are positive parameters. Then one has

[TABLE]

Consider $S\#A\#\langle{\xi}\rangle-\langle{\xi}\rangle\#A^{*}\#S$. Note that

[TABLE]

Writing $S=(s_{ij})$ one has

[TABLE]

because $c^{(\beta)}_{(\alpha)}={\mathcal{O}}(\sqrt{b})$ for $|\alpha+\beta|=2$. Then

[TABLE]

Denoting the third term on the right-hand side by $K_{2}$, repeating the same arguments as before, it is easy to see

[TABLE]

Now we turn to the term with $|\alpha+\beta|=1$. Note

[TABLE]

since $c^{(\beta)}_{(\alpha)}={\mathcal{O}}(\sqrt{b})$ and $b^{(\alpha)}_{(\beta)}={\mathcal{O}}(\sqrt{b})$ for $|\alpha+\beta|=1$ and hence

[TABLE]

The same arguments proves

[TABLE]

Consider $A^{*}\#S$. We have the representation

[TABLE]

Repeating similar arguments, one gets

[TABLE]

Since $A^{*}S=SA$, taking (2.4) into account, we see

[TABLE]

Summarizing the above estimates, we obtain the following

Lemma 3.5**.**

Assume $\langle\xi\rangle^{\alpha}c_{(\beta)}^{(\alpha)}={\mathcal{O}}(\sqrt{b})$ for $|\alpha+\beta|\leq 2$. There is $C>0$ such that

[TABLE]

Consider $S\#\varphi\#\langle{\xi}\rangle-\langle{\xi}\rangle\#\varphi\#S$, where $\varphi\in S^{0}$ is scalar. Recall

[TABLE]

For $|\alpha+\beta|=2$ one has

[TABLE]

and hence

[TABLE]

Denoting the third term on the right-hand side by $K_{2},$ we have the same estimate as (3.2). Similarly one has

[TABLE]

Consider the term with $|\alpha+\beta|=1$ and observe that

[TABLE]

with $g=b^{2}-2ac$. Therefore

[TABLE]

because $c^{(\alpha)}_{(\beta)}={\mathcal{O}}(b)$ for $|\alpha+\beta|=1$ and then

[TABLE]

Similar arguments are applied to $|({\rm Op}(\varphi^{(\alpha)}_{(\beta)}S^{(\beta)}_{(\alpha)})U,U)|$. Finally, since

[TABLE]

we obtain

Lemma 3.6**.**

Assume $\langle\xi\rangle^{\alpha}c_{(\beta)}^{(\alpha)}={\mathcal{O}}(b)$ for $|\alpha+\beta|=1$ and $\langle\xi\rangle^{\alpha}c_{(\beta)}^{(\alpha)}={\mathcal{O}}(\sqrt{b})$ for $|\alpha+\beta|=2.$ Then there exists $C>0$ such that

[TABLE]

Combining Lemmas 3.5, 3.6 and Corollary 2.2, one concludes that for sufficiently large $N_{1}>0$ we have

[TABLE]

Now we pass to the analysis of the term involving $\partial_{t}S$.

Lemma 3.7**.**

Assume $\partial_{t}c={\mathcal{O}}(b)$. For $\varepsilon>0$ sufficiently small we have

[TABLE]

Proof.

Since $\partial_{t}c={\mathcal{O}}(b)$, one has

[TABLE]

It is not difficult to see that

[TABLE]

because $t={\mathcal{O}}(b)$. ∎

Lemma 3.8**.**

Assume $\partial_{t}c={\mathcal{O}}(b)$, $\langle\xi\rangle^{\alpha}c_{(\beta)}^{(\alpha)}={\mathcal{O}}(\sqrt{b})$ for $|\alpha+\beta|=2$ and $\langle\xi\rangle^{\alpha}(ac)_{(\beta)}^{(\alpha)}={\mathcal{O}}(\sqrt{b})$ for $|\alpha+\beta|=3.$ There exist $\varepsilon>0$ and $C>0$ such that for $U\in C^{\infty}({\mathbb{R}}_{t}:C_{0}^{\infty}({\mathbb{R}}^{n}))$ we have

[TABLE]

Proof.

Denoting $Q=S-2\varepsilon t\,\partial_{t}S$, it suffices to prove

[TABLE]

Consider {\mathsf{Re}}\big{(}{\rm Op}\big{(}\psi_{\alpha\beta}Q_{(\beta)}^{(\alpha)}\big{)}U,U\big{)} with $|\alpha+\beta|=2$. Note that

[TABLE]

where $g=b^{2}-2ac$. Consequently, one deduce

[TABLE]

Setting

[TABLE]

we obtain {\mathsf{Re}}\,\big{(}\psi_{\alpha\beta}\big{(}g^{(\alpha)}_{(\beta)}-\varepsilon t(\partial_{t}g)^{(\alpha)}_{(\beta)}\big{)}\big{)}=T\#\langle{\xi}\rangle^{-1}+S^{-2}. Therefore

[TABLE]

Note that $\|{\rm Op}(T)U_{3}\|^{2}=({\rm Op}(T\#T)U_{3},U_{3})$ and $T\#T=T^{2}+S^{-2}$. There is $C>0$ such that

[TABLE]

because $t={\mathcal{O}}(b)$ and $\langle\xi\rangle^{\alpha}c^{(\alpha)}_{(\beta)}={\mathcal{O}}(\sqrt{b})$ so that $Cb-T^{2}\geq 0$. Then applying the Fefferman-Phong inequality, we prove the assertion. Let $|\alpha+\beta|=3$ then with T_{1}=\big{(}\psi_{\alpha\beta}\big{(}g^{(\alpha)}_{(\beta)}-\varepsilon t(\partial_{t}g)^{(\alpha)}_{(\beta)}\big{)}\big{)}\#\langle{\xi}\rangle^{3/2}

[TABLE]

with some $C>0$ since $t={\mathcal{O}}(b)$ and $\langle\xi\rangle^{\alpha}(ac)^{(\alpha)}_{(\beta)}={\mathcal{O}}(\sqrt{b})$ and the proof is similar. ∎

From (3.5) setting $N_{2}=\varepsilon^{-1}$ and dividing by $\varepsilon$, one deduces

[TABLE]

and applying Corollary 2.2, this implies

[TABLE]

Fixing $\varepsilon$ and $N_{2}$, we choose $N_{3}$ sufficiently large and we arrange the right hand side of the above inequality to be negative.

Next we turn to the analysis of $2{\mathsf{Im}}({\rm Op}({\tilde{S}}){\rm Op}(B)U,U).$ Recall that $({\rm Op}({\tilde{S}})U,U)\gg 0$ by Corollary 2.3. Consequently,

[TABLE]

Lemma 3.9**.**

There exists $N_{4}>0$ depending on $T$ and $B$ such that for $0\leq t\leq T$ and any $\varepsilon>0$ there exists $D_{\varepsilon}>0$ such that

[TABLE]

Proof.

Recall

[TABLE]

which proves $3S-\varepsilon t^{2}B^{*}SB\gg 0$ with some $\varepsilon=\varepsilon(T)>0$. To justify this, notice that the terms $\varepsilon{\mathcal{O}}(t^{2}b),\>\varepsilon{\mathcal{O}}(t^{2}c),\>\varepsilon{\mathcal{O}}(t^{2}a^{2}),\>\varepsilon{\mathcal{O}}(t^{4}a)$ can be absorbed by $\det S$ because $b\geq\delta_{1}t$. For example,

[TABLE]

Choosing $\varepsilon(T)$ small enough, we obtain the result. Then the rest of the proof is just a repetition of the proof of Lemma 3.8. ∎

According to Lemma 3.9 and (3.8), one has

[TABLE]

Combining the estimates (3.4), (3.7), (3.9), it follows that

[TABLE]

Note that

[TABLE]

Denote $N^{*}=N_{1}+N_{2}+N_{3}+2N_{2}^{1/2}+2$ and we choose $0<\varepsilon\leq\varepsilon_{1}$. We fix $\varepsilon$ and $\lambda>2C_{\varepsilon}$. Next we fix $N_{4}$ so that

[TABLE]

Then the term with $t^{-N-2}e^{-\gamma t}\|\langle{D}\rangle^{-1}U\|^{2}$ is absorbed. Finally we choose $N>N^{*}$ and $\gamma$ such that $\gamma-D_{\varepsilon}-C_{1}\lambda-C\lambda N_{4}^{-1/2}T\geq 0$. Then we have

[TABLE]

Integrating (3.10) in $\tau$ from $\varepsilon>0$ to $t$ and taking Corollary 2.3 into account, one obtains

Proposition 3.1**.**

Assume that

[TABLE]

hold globally where $\bar{\varepsilon}$ is given in Lemmas 2.2 and $\ref{lem:G2}.$ Then there exist $\delta_{2}>0,\gamma_{0}>0,N\in{\mathbb{N}}$ and $C>0$ such that for $\gamma\geq\gamma_{0}$ and $0<\varepsilon\leq t\leq T$ we have for any $U\in C^{\infty}({\mathbb{R}}_{t}:C_{0}^{\infty}({\mathbb{R}}^{n}))$

[TABLE]

4. Microlocal energy estimates

First we prove the following

Lemma 4.10**.**

Assume that (1.3) is satisfied in $[0,T]\times{\tilde{W}}$ where ${\tilde{W}}$ is a conic neighborhood of $(x_{0},\xi_{0})$. Then there exist extensions ${\tilde{a}}(t,x,\xi)\in S^{0}$, ${\tilde{b}}(t,x,\xi)\in S^{0}$ and ${\tilde{c}}(t,x,\xi)\in S^{0}$ of $a$, $b$ and $c$ such that (3.11) holds globally.

Proof.

Assume that (1.3) is satisfied in $[0,T]\times{\tilde{W}}$. Choose conic neighborhoods $U$, $V$, $W$ of $(x_{0},\xi_{0})$ such that $U\Subset V\Subset W\Subset{\tilde{W}}$. Take $0\leq\chi(x,\xi)\in S^{0}$, $0\leq{\tilde{\chi}}(x,\xi)\in S^{0}$ such that $\chi=1$ on $V$ and $\chi=0$ outside $W$ and ${\tilde{\chi}}=0$ on $U$ and ${\tilde{\chi}}=1$ outside $V$. Choosing $W$ and $T$ small one can assume that $\chi b$ is small as we please in $[0,T]\times{\mathbb{R}}^{2n}$ because $b(0,x_{0},\xi_{0})=0$. We define the extensions of $a$, $b$, $c$ by

[TABLE]

where $M>0$ is a positive constant which we will choose below. Note that

[TABLE]

taking $a(0,x_{0},\xi_{0})=0$, $b(0,x_{0},\xi_{0})=0$ into account and choosing $W$ small.

If $(x,\xi)\in V$ then ${\tilde{b}}(t,x,\xi)=b+M{\tilde{\chi}}\geq\delta_{1}t$ and if $(x,\xi)$ is outside $V$ then ${\tilde{b}}(t,x,\xi)=\chi^{2}b+M\geq\delta_{1}t$ for $[0,T]\times{\mathbb{R}}^{2n}$ choosing $M$ so that $M\geq\delta_{1}T$. Thus we have

[TABLE]

We turn to estimate derivatives of ${\tilde{c}}$ and ${\tilde{a}}{\tilde{c}}$. For $|\alpha+\beta|=1$ it is clear that

[TABLE]

Similarly for $|\alpha+\beta|=2$ one sees

[TABLE]

For $|\alpha+\beta|=3$, taking $\langle\xi\rangle^{\alpha}(ac)^{(\alpha)}_{(\beta)}={\mathcal{O}}(\sqrt{b})$ into account, one has

[TABLE]

Since $|\partial_{t}{\tilde{c}}|=|\chi^{3}\partial_{t}c|\leq C\chi^{3}b\leq C{\tilde{b}}$ is obvious the proof is complete. ∎

Remark 4.2**.**

In the proof of Lemma 4.10 replacing ${\tilde{b}}$ by $\chi^{2}b+M{\tilde{\chi}}+M^{\prime}\chi_{0}(\xi)$ where $\chi_{0}(\xi)\in C_{0}^{\infty}({\mathbb{R}}^{n})$ which is $1$ near $\xi=0$ and $M^{\prime}>0$ is a suitable positive constant it suffices to assume that (1.3) is satisfied in $[0,T]\times{\tilde{W}}$ for $|\xi|\geq 1$.

Let $V\Subset V_{1}\Subset\Omega$ and $u\in C^{\infty}({\mathbb{R}}_{t}:C_{0}^{\infty}(V))$ Let $\{\chi_{\alpha}\}$ be a finite partition of unity with $\chi_{\alpha}(x,\xi)\in S^{0}$ so that

[TABLE]

where $\chi(x)=1$ on $\overline{V}$ and ${\rm supp}\,\chi\subset V_{1}$. We can suppose that ${\rm supp}\,\chi_{\alpha}\subset V_{1}$. We repeat the argument in [11, Section 4], studying a system

[TABLE]

with $U_{\alpha}=$ {}^{t}\big{(}(D_{t}-{\rm Op}(\varphi)\langle{D}\rangle)^{2}\chi_{\alpha}u,\langle{D}\rangle(D_{t}-{\rm Op}(\varphi)\langle{D}\rangle)\chi_{\alpha}u,\langle{D}\rangle^{2}\chi_{\alpha}u). One extends the coefficients $a$, $b$, $c$ and $\varphi$ outside the support of $\chi_{\alpha}$ and one can assume that (3.11) are satisfied globally. Thus we obtain the following

Theorem 4.1**.**

Let $Y\Subset\Omega$. Assume that for every point $(x_{0},\xi_{0})\in T^{*}\Omega\setminus\{0\}$ there exist a conic neighborhood $W\subset T^{*}\Omega\setminus\{0\}$ and $T(x_{0},\xi_{0})>0$ such that the estimates (3.11) are satisfied for $0\leq t\leq T(x_{0},\xi_{0})$ and $(x,\xi)\in W$. Then there exist $c>0,\>T_{0}>0,\>\gamma_{0}>0,C>0$ and $N\in{\mathbb{N}}$ such that for $\gamma\geq\gamma_{0},\>0<\varepsilon<t\leq T_{0}$ we have for any $U\in C^{\infty}({\mathbb{R}}_{t}:C_{0}^{\infty}(Y))$

[TABLE]

Corollary 4.4**.**

Let $Y\Subset\Omega$. Assume that for every point $(x_{0},\xi_{0})\in T^{*}\Omega\setminus\{0\}$ there exist a conic neighborhood $W\subset T^{*}\Omega\setminus\{0\}$ and $T(x_{0},\xi_{0})>0$ such that the estimates (1.3) are satisfied for $0\leq t\leq T(x_{0},\xi_{0})$ and $(x,\xi)\in W$. Then the same assertion as in Theorem $4.1$ holds.

The same argument can be applied for the adjoint operator $P^{*}$. With

[TABLE]

the equation $P^{*}v=g$ is reduced to

[TABLE]

with $G={{}^{t}}(g,0,0).$ Here the principal symbol is the same, while the lower order terms change. To study the Cauchy problem for $P^{*}$ in $0<t<T$ with initial data on $t=T$ one considers

[TABLE]

Repeating the argument of Section 3, one obtains the following

Theorem 4.2**.**

Let $Y\Subset\Omega$. Assume that for every point $(x_{0},\xi_{0})\in T^{*}\Omega\setminus\{0\}$ there exist a conic neighborhood $W\subset T^{*}\Omega\setminus\{0\}$ and $T(x_{0},\xi_{0})>0$ such that the estimates (3.11) are satisfied for $0\leq t\leq T(x_{0},\xi_{0})$ and $(x,\xi)\in W$. Then there exist $c>0,\>T_{0}>0,\>\gamma_{0}>0,C>0$ and $N\in{\mathbb{N}}$ such that for $\gamma\geq\gamma_{0},\>0<\varepsilon<t\leq T_{0}$ we have for any $V\in C^{\infty}({\mathbb{R}}_{t}:C_{0}^{\infty}(Y))$

[TABLE]

Following the argument in [11], we may absorb the weight $\tau^{-N}$ and obtain energy estimates with a loss of derivatives. For the sake of completeness we recall this argument. Consider $Pu=f$ for $u\in C^{\infty}({\mathbb{R}}_{t}:C_{0}^{\infty}({\mathbb{R}}^{n}))$. Assume $u(\varepsilon,x)=u_{t}(\varepsilon,x)=u_{tt}(\varepsilon,x)=0$. Differentiating $Pu=f$ with respect to $t$, we determine the functions $D_{t}^{j}u(\varepsilon,x)=u_{j}(x)\in C_{0}^{\infty}({\mathbb{R}}^{n})$ and set

[TABLE]

Therefore $w=u-u_{M}\in C^{\infty}({\mathbb{R}}_{t}:C_{0}^{\infty}({\mathbb{R}}^{n}))$ satisfies $Pw=f_{M}$ with

[TABLE]

Consequently, from Theorem 4.1 one deduce the existence of $N\in{\mathbb{N}}$ and $C>0$ such that for $\varepsilon>0$, $Z\Subset\Omega$ and a solution of $Pu=f\in C_{0}^{\infty}([0,T_{0}]\times Z)$ for $0<\varepsilon\leq t\leq T_{0}$, $x\in Z$ with

[TABLE]

we have

[TABLE]

where $C$ is independent on $\varepsilon$. We can obtain a similar estimates for higher order derivatives.

By applying the estimate (4.5) and the fact that under the assumptions of Theorem 4.1 the symbol $p$ is strictly hyperbolic for $0<t\leq T_{0}$ one can obtain the existence of a solution of the Cauchy problem in $[0,T_{0}]\times Z$ repeating the argument in [3, Theorem 25.4.5]. The fact that $p$ is strictly hyperbolic for $0<t\leq T_{0}$, is equivalent to $\Delta>0$ for $0<t\leq T_{0},$ $\Delta$ being the discriminant of the equation $p=0$ with respect to $\tau$. On the other hand, $\Delta=27\det S$ (see also Corollary 2.1) and $\det S>0$ for $t>0$ by Lemma 2.2. The local uniqueness of the solution of the Cauchy problem for $P$ can be obtained taking into account Theorem 4.2 for the adjoint operator $P^{*}$ and using the argument of [3, Theorem 25.4.5]. We leave the details to the reader.

Finally, we deduce

Corollary 4.5**.**

Under the assumptions of Theorem 4.1 the Cauchy problem for $P$ is $C^{\infty}$ well posed in $[0,T_{0}]\times Z$ for all lower order terms.