Hyperbolic equations with non analytic coefficients well posed

in all Gevrey classes

Tamotu KINOSHITA and Sergio SPAGNOLO

In this talk, we consider weakly hyperbolic equations. For the second order equation ∂_{t}^{2}u = a_{2}(t)∂_{x}^{2}u + a_{1}(t)∂_{x}∂_{t}u with a_{2}(t) ∈ C^{k} (resp. C^{∞}, resp. A) and a_{1}(t) ≡ 0 under a hyperbolic condition a_{2}(t) ≥ 0, [CJS] showed the wellposedness in γ^{s} for 1 ≤ s < 1 + k/2 (resp. in γ^{∞}, resp. in C^{∞}). For the third order equation

(P)  \{ \begin{matrix}∂_{t}^{3}u = \{a_{3}(t)∂_{x}^{3} + a_{2}(t)∂_{x}^{2}∂_{t} + a_{1}(t)∂_{x}∂_{t}^{2}\}u, \\ ∂_{t}^{h}u(0, x) = u_{h}(x) (0 ≤ h ≤ 2), \end{matrix}

we need some additional conditions to get similar kind of results.

1. By [K] we get the wellposedness in γ^{∞} of (P) with a_{1}(t), a_{2}(t) ∈ C^{∞} and a_{3}(t) ≡ 0 under a hyperbolic condition a_{2}(t) ≥ 0.

2. By [S] we get the wellposedness in γ^{∞} of (P) with a_{2}(t), a_{3}(t) ∈ C^{∞} and a_{1}(t) ≡ 0 under a hyperbolic condition 4a_{2}(t)^{3} − 27a_{3}(t)^{2} ≥ ^{∃}ca_{3}(t)^{2}.

3. By [CO] we get the wellposedness in γ^{∞} of (P) with a_{1}(t), a_{3}(t) ∈ C^{∞} and a_{2}(t) ≡ 0 under a condition

λ_{i}^{2}(t) + λ_{j}^{2}(t) ≤ ^{∃}M (λ_{i}(t) − λ_{j}(t))^{2}, 1 ≤ i < j ≤ 3.

We denote that

\begin{matrix}Δ(t) = & ∏_{1≤i<j≤3}(λ_{j}(t) − λ_{i}(t))^{2} \\ = & 4a_{2}(t)^{3} − 27a_{3}(t)^{2} + a_{1}(t)^{2}a_{2}(t)^{2} − 4a_{1}(t)^{3}a_{3}(t) − 18a_{1}(t)a_{2}(t)a_{3}(t). \end{matrix}

When the roots are distict, there exists c > 0 such that

Δ(t) ≥ c.

Remark 1. If the condition in [CO] is satisfied, it follows that

Δ(t) ≥ \frac{(λ_{1}(t)^{2} + λ_{2}(t)^{2})(λ_{2}(t)^{2} + λ_{3}(t)^{2})(λ_{3}(t)^{2} + λ_{1}(t)^{2})}{M}.

Then we get the following: Theorem : If the coefficients are C^{k} (resp. C^{∞}, resp. analytic) and satisfy

(9a_{3}(t) + a_{1}(t)a_{2}(t))^{2} ≤ ^{∃}MΔ(t),

then the Cauchy problem (P) is well-posed in γ^{s} for

1 ≤ s < 1 + \frac{k}{4}

(resp. in γ^{∞}, resp. in C^{∞}).

Remark 2. For the simplicity we suppose that the spacial dimension n = 1. Remark 3. In case when the m-th order equation whose coefficients belong to C^{k} for k > 3(m + n + 2), [B] proved the wellposedness in γ^{s} of (P) for

1 ≤ s < 1 + \frac{1}{m − 1}.

For the proof, we shall improve the theory of quasi-symmetrizers (see [DS]).

References

[B] M.D. Bronštein, The Cauchy problem for hyperbolic operators with characteristics of variable multiplicity, Trudy Moskov. Mat. Obs̆c̆.41 (1980), 87-103 (Trans. Moscow Math. Soc., 1 (1982), 87-103).

[CJS] F. Colombini, E. Jannelli and S. Spagnolo, Wellposedness in the Gevrey classes of the Cauchy problem for a non strictly hyperbolic equation with coefficients depending on time, Ann. Scuola Norm Sup. Pisa, 10 (1983), 291-312.

[CO] F. Colombini and N. Orrú, Well posedness in C^{∞} for some weakly hyperbolic equations, J. Math. Kyoto. Univ., 39 (1999), 399-420.

[DS] P. D’Ancona and S. Spagnolo, Quasi-symmetrization of hyperbolic systems and propagation of the analytic regularity, Boll. Un. Mat. Ital., 1-B (1998), 169-185.

[K] T. Kinoshita, Gevrey wellposedness of the Cauchy problem for the hyperbolic equations of third order with coefficients depending only on time, Publ. RIMS Kyoto Univ., 34 (1998), 249-270.

[S]  S. Spagnolo, Hyperbolic systems well posed in all Gevrey classes, preprint.