Energy estimate and fundamental solution for degenerate hyperbolic Cauchy problems

Abstract

The aim of this paper is to give an uniform approach to different kinds of degenerate hyperbolic Cauchy problems. We prove that a weakly hyperbolic equation, satisfying an intermediate condition between effective hyperbolicity and the C ∞ Levi condition, and a strictly hyperbolic equation with non-regular coefficients with respect to the time variable can be reduced to firstorder systems of the same type. For such a kind of systems, we prove an energy estimate in Sobolev spaces (with a loss of derivatives) which gives the well-posedness of the Cauchy problem in C ∞ . In the strictly hyperbolic case, we also construct the fundamental solution and we describe the propagation of the space singularities of the solution which is influenced by the non-regularity of the coefficients with respect to the time variable.

A degenerate hyperbolic equation under Levi conditions

“…Well-posedness in H ±∞ hence follows from (2.3). The proof of Proposition 2.1 is given in [2]. Now we are ready to prove Theorem 1.1.…”

“…Dependence on the space variable x ∈ R n , n ≥ 1, of the lower order terms b, c is allowed by Colombini and Nishitani, [10]; there C ∞ well posedness is proved for a larger, for k > 2, value of γ. In [2] C ∞ well posedness with the optimal γ is reobtained for a second order equation (1.1) where a has the particular structure a(t, x, ξ ) = α(t)Q(t, x, ξ ), for a non negative function α ∈ C ∞ [0, T ] and for an elliptic symbol of the second order Q.…”

Random-field solutions to linear hyperbolic stochastic partial differential equations with variable coefficients

“…In this section we want to construct a representation of the solution of (3.12) by producing its fundamental solution. The construction presented here follows the procedure in [2,3,4] and [18,Section 10.7] and goes in three steps: first we reduce the higher-order hyperbolic equation to a first-order system, then we compute the fundamental solution to the resulting first-order system and finally, we obtain a representation formula for the fundamental solution to the higher-order equation. First step: Reduction to a first-order system.…”

“…However, results of "well-posedness with loss of derivatives" in Sobolev spaces can be obtained under suitable assumptions in the case of weakly hyperbolic equations, i.e. equations having real characteristic roots which are not necessarily distinct and separate at every time, see for example [2] and the references therein. Here well-posedness with loss of derivatives means that in this case the fundamental solution E(t, s) results to be a Fourier integral operator of order δ > 0, and so by Duhamel's formula (3.11) one obtains a solution U which is less regular then the data, i.e.…”

A dyadic decomposition approach to a finitely degenerate hyperbolic problem

“…In particular in [6] the technique is very similar to that one in [7]. In [5] and [9] some convolution weights are introduced to treat the coefficients depending on x, while in [1] the approximate energies method is used together with some pseudodifferential theory arguments.…”