On the Gevrey wellposedness of the Cauchy problem

for weakly hyperbolic equations of higher order

Ferruccio COLOMBINI and Tamotu KINOSHITA

We shall consider the Cauchy problem in [0, T] × R_{x}^{1}

(1)  \{ \begin{matrix}∂_{t}^{m}u = ∑_{j+k=m, j<m} a_{j,k}(t)∂_{t}^{j}∂_{x}^{k}u + ∑_{j+k≤l} a_{j,k}(t)∂_{t}^{j}∂_{x}^{k}u \\ ∂_{t}^{j}u(0, x) = u_{j}(x) (j = 0, 1, ⋯, m − 1), \end{matrix}

where 0 ≤ l ≤ m − 1 and a_{j,k}(t) ∈ C^{0}[0, T] for j, k satisfying j + k ≤ l.

Now we assume that the characteristic roots τ_{1}(t)ξ, τ_{2}(t)ξ, ⋯, τ_{m}(t)ξ of the characteristic equation τ^{m} = ∑_{j+k=m, j<m} a_{j,k}(t)τ^{j}ξ^{k} are real valued and satisfy the following conditions

(2) τ_{1}(t), ⋯, τ_{r}(t) coincide at the same points such that for any 1 ≤ i, p, q ≤ r(i ≠ p, q) |τ_{i}(t) − τ_{p}(t)| ≤ ^{∃}C|τ_{i}(t) − τ_{q}(t)| for t ∈ [0, T] and belong to C^{α}[0, T] (0 ≤ α ≤ 1),

(3) τ_{r+1}(t), ⋯, τ_{m}(t) are di stinct and belong to C^{β}[0, T] (0 ≤ β ≤ 1),

where 1 ≤ r ≤ m. In particular when r = 1, τ_{1}(t) also becomes a distinct root. Therefore (2) and (3) imply that

(4) τ_{1}(t), τ_{2}(t), ⋯, τ_{m}(t) are distinct and belong to C^{β}[0, T] (0 ≤ β(= α) ≤ 1).

When m = 2, l = 1 and r = 2 (resp. r = 1), F. Colombini, E. Jannelli, S. Spagnolo and E. De Giorgi assumed a_{1,1}(t) ≡ 0 and a_{0,2}(t) ∈ C^{γ}[0, T] (γ ≥ 0) which means that the characteristic roots satisfy (2) with α = \frac{γ}{2} (resp. (4) with β = γ), and showed that the Cauchy problem (1) is wellposed in G^{s}, provided 1 ≤ s < 1 + α (resp. 1 ≤ s < 1 + \frac{β}{1−β} )(see [CDS] and [CJS]).

When m ≥ 2, l = m − 1 and r ≥ 2, Y. Ohya and S. Tarama assumed a_{j,k}(t) ∈ C^{γ}[0, T] (0 ≤ gamma ≤ 2) for j, k satisfying j + k = m. In their case we remark that the characteristic roots don’t always multiply at the same points. Rougly speaking their assumption means that the characteristic roots satisfy (2) with α = \frac{γ}{r} and (3) with β = γ(= rα) from the properties of hyperbolic polynomials(see [B]). Then they showed that the Cauchy problem (1) is wellposed in G^{s}, provi ded 1 ≤ s < 1 + min\{\frac{γ}{r} , \frac{1}{r−1}\} = 1 + min\{α, \frac{1}{r−1}\}(see [OT]).

Theorem. Let T 0, 0 ≤ l ≤ m − 1 and 2 ≤ r ≤ m(res p. r = 1). Assume that

(2) and (3)(resp. (4)). Then for any u_{j}(x) ∈ G^{s}(R)(j = 0, 1, ⋯, m − 1), the Cauchy problem (1) has a unique solution u ∈ C^{m}([0, T]; G^{s}(R_{x}^{1})), provided

(5)  1 ≤ s < 1 + min\{α, \frac{β}{r − β} , \frac{m − l}{r − 1} \} ( resp. 1 ≤ s < 1 + \frac{β}{1 − β} ).

We shall give the typical example to apply our theorem. We consider the Cauchy problem for the weakly hyperbolic equation of 4th order

(6)  \{ \begin{matrix}∂_{t}^{4}u = \{a(t) + b(t)\}∂_{t}^{2}∂_{x}^{2}u − a(t)b(t)∂_{x}^{4}u \\ ∂_{t}^{j}u(0, x) = u_{j}(x) (j = 0, 1, 2, 3), \end{matrix}

where a(t) and b(t) belong to C^{2α}[0, T] and C^{β}[0, T] respectively and satisfy a(t) ≥ 0 and b(t) − a(t) ≥ ^{∃}δ0 wh ich imply that the multiplicity r = 2. Since the coefficients belong to C^{γ}[0, T] where γ = min{2α, β}, according to [OT] the Cauchy problem (6) is wellposed in G^{s}, provided

1 ≤ s < 1 + \frac{γ}{2} = 1 + min\{α, \frac{β}{2} \}.

Noting that b(t) is strictly positive, we see that τ_{1}(t) ≡ −\sqrt{a(t)} ∈ C^{α}[0, T], τ_{2}(t) ≡ \sqrt{a(t)} ∈ C^{α}[0, T], τ_{3}(t) ≡ −\sqrt{b(t)} ∈ C^{β}[0, T] and τ_{4}(t) ≡ \sqrt{b(t)} ∈ C^{β}[0, T]. Applying our theorem, we find that the Cauchy problem (7) is wellposed in G^{s}, provided

1 ≤ s < 1 + min\{α, \frac{β}{2 − β} \}.

NOTATIONS G^{s}(R)(s ≥ 1) is the space of Gevrey functions f(x) satisfying for any

compact set K ⊂ R, sup_{x∈K} |D^{α}f(x)| ≤ C_{K}ρ_{K}^{α}α!^{s} for ^{∀}α ∈ N.

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).

[CDS] F. Colombini, E. De Giorgi and S. Spagnolo, Sur les é quations hyperboliques avec des coefficients qui ne dépendent que du temps, Ann. Scuola Norm Sup. Pisa, 6 (1979), 511-559.

[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.

[KWY] K. Kajitani, S. Wakabayashi and K. Yagdjian, The C^{∞}-wellposed Cauchy problem for hyperbolic operators with multiple characteristics vanishing with the different speeds, preprint.

[OT] Y. Ohya and S. Tarama, Le problème de Cauchy à caractéristiques multiples -coefficients hölderiens en t- , (Proc. Taniguchi Intern. Sympos. on Hyperbolic Equations and Related Topics 1984), Kinokuniya, 1986, 273-306.