On the Cauchy problem for hyperbolic operators with non-regular coefficients.

Ferruccio Colombini

We present some new results on the well–posedness of the Cauchy problem for a class of hyperbolic operators. Let T > 0. We are concerned with the equation

u_{tt} − ∑_{i,j=1}^{n} a_{ij}(t)u_{x_{i}x_{j}} + ∑_{i=1}^{n} b_{i}(t)u_{x_{i}} + c(t)u = 0 in [0, T] × R^{n},  (1)

with initial data

u(0, x) = u_{0}(x), u_{t}(0, x) = u_{1}(x) in R^{n},  (2)

where the coefficients b_{i} and c are measurable and bounded and (a_{ij}) is a real symmetric matrix such that

a(t, ξ) = ∑_{i,j=1}^{n} a_{ij}(t)ξ_{i}ξ_{j}/|ξ|^{2} ≥ λ_{0} > 0, (strictly hyperbolic case)

or

a(t, ξ) = ∑_{i,j=1}^{n} a_{ij}(t)ξ_{i}ξ_{j}/|ξ|^{2} ≥ 0, (weakly hyperbolic case)

for all t and for all ξ ≠ 0.

Our main hypothesis is the following: we suppose that there exist t̄ ∈ [0, T] and there exist q, C > 0 such that a_{ij} ∈ C^{1}([0, T] \ {t̄}) for all i, j = 1, …, n, and

|a′(t, ξ)| ≤ C|t − t̄|^{−q},  (3)

for all (t, ξ) ∈ ([0, T]\{t̄})×(R^{n} \{0}) (here ′ denotes the derivative with respect to the variable t).

Theorem 1 Consider the equation (1) and suppose that the condition (3) holds with q = 1.

Then the Cauchy problem (1), (2) is C^{∞}–well–posed in the strictly hyperbolic case and it is γ^{(s)}–well–posed for all s < 3/2 in the weakly hyperbolic case.

Our second result is the following.

Theorem 2 Let q > 1 and 0 ≤ p < 1, with p ≤ q − 1. Consider the equation (1) and suppose that the condition (3) holds. Suppose moreover that there exists C′ > 0 such that

|a(t, ξ)| ≤ C′|t − t̄|^{−k},  (4)

for all (t, ξ) ∈ ([0, T] \ {t̄}) × (R^{n} \ {0}).

Then (1), (2) is γ^{(s)}–well–posed for all s < (q − k)/(q − 1) in the strictly hyperbolic case and it is γ^{(s)}–well–posed for all s < (q − 3k + 2)/(q − 2k + 1) in the weakly hyperbolic case.

Our third theorem gives a link between the results for operators having Hölder–continuous coefficients in the principal part

Theorem 3 Let q > 1 and 0 < α < 1. Consider the equation (1) and suppose that a_{ij} ∈ C^{0,α}([0, T]) for all i, j = 1, …, n. Suppose moreover that the condition (3) holds.

Then (1), (2) is γ^{(s)}–well–posed for all s < (q/(q − 1))(1/(1 − α))in the strictly hyperbolic case and it is γ^{(s)}–well–posed for all s < (q + 2)(α + 2)/(2(q + α + 2)) in the weakly hyperbolic case.

The results of the Theorems 1, 2 and 3 in the case of the strictly hyperbolicity are in some sense optimal. In fact we can construct some sharp counter examples.

References

[1] F. Colombini, D. Del Santo and T. Kinoshita, On the Cauchy problem for hyperbolic operators with non-regular coefficients, to appear.

[2] F. Colombini, D. Del Santo and T. Kinoshita, Well–posedness of the Cauchy problem for a hyperbolic equation with non–Lipschitz coefficients, to appear.

[3] F. Colombini, D. Del Santo and T. Kinoshita, Gevrey-well-posedness for weakly hyperbolic operators with non-regular coefficients, to appear.