Abstract
The theory of nonlinear weakly hyperbolic equations was developed during the last decade in an astonishing way. Today we have a good overview about assumptions which guarantee local well posedness in spaces of smooth functions (C∞, Gevrey). But the situation is completely unclear in the case of Sobolev spaces. Examples from the linear theory show that in opposite to the strictly hyperbolic case we have in general no solutions valued in Sobolev spaces. If so‐called Levi conditions are satisfied, then the situation will be better. Using sharp Levi conditions of C∞‐type leads to an interesting effect. The linear weakly hyperbolic Cauchy problem has a Sobolev solution if the data are sufficiently smooth. The loss of derivatives will be determined in essential by special lower order terms. In the present paper we show that it is even possible to prove the existence of Sobolev solutions in the quasilinear case although one has the finite loss of derivatives for the linear case. Some of the tools are a reduction process to problems with special asymptotical behaviour, a Gronwall type lemma for differential inequalities with a singular coefficient, energy estimates and a fixed point argument.
Highlights
In this paper we want to prove a local existence result in Sobolev spaces with respect to the spatial variables for the weakly hyperbolic Cauchy problem utt − λ2(t) u = f (t, x, u, ut, μi(t)∂xiu), (1.2)
The Levi condition a(t) = O(λ (t)) is sharp in the following sense: If we weaken it to a(t) = o(λ (t)s), s ∈ (0, 1), there doesn’t exist a distributional solution
Allowing this kind of sharp Levi condition we have to take into consideration the following question
Summary
In this paper we want to prove a local existence result in Sobolev spaces with respect to the spatial variables for the weakly hyperbolic Cauchy problem (1.1). Quasilinear weakly hyperbolic equations, time degeneracy, local existence, Levi conditions, Sobolev spaces, energy method. The Levi condition a(t) = O(λ (t)) is sharp in the following sense: If we weaken it to a(t) = o(λ (t)s), s ∈ (0, 1), there doesn’t exist a distributional solution (see [5]) If we sharpen it to a(t) = o(λ (t)), the term of lower order has no influence on the loss of Sobolev regularity. Allowing this kind of sharp Levi condition we have to take into consideration the following question Is it possible to connect the quasilinear structure of our problem (1.1), (1.2) with the loss of derivatives of the solution which appears even in the linear case?. There is no nonlinear dependence on ∇xu or the Levi condition is not sharp for time degeneracies of infinite order
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