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Deterministic and stochastic Cauchy problems for a class of weakly hyperbolic operators on \(\mathbb {R}^n\)

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Abstract

We study a class of hyperbolic Cauchy problems, associated with linear operators and systems with polynomially bounded coefficients, variable multiplicities and involutive characteristics, globally defined on \(\mathbb {R}^n\). We prove well-posedness in Sobolev-Kato spaces, with loss of smoothness and decay at infinity. We also obtain results about propagation of singularities, in terms of wave-front sets describing the evolution of both smoothness and decay singularities of temperate distributions. Moreover, we can prove the existence of random-field solutions for the associated stochastic Cauchy problems. To these aims, we first discuss algebraic properties for iterated integrals of suitable parameter-dependent families of Fourier integral operators, associated with the characteristic roots, which are involved in the construction of the fundamental solution. In particular, we show that, also for this operator class, the involutiveness of the characteristics implies commutative properties for such expressions.

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Acknowledgements

We are grateful to an anonymous Referee, for the very careful reading of the paper, the constructive criticism, and the many suggestions, aimed at improving the overall quality of the presentation of our results. The second and third author have been partially supported by their own FFABR 2017 grants by Ministero dell’Istruzione, dell’Università e della Ricerca.

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Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Straße 69, 4040, Linz, Austria

    Ahmed Abdeljawad

  2. Dipartimento di Matematica, Università degli Studi di Ferrara, via Machiavelli 30, 44121, Ferrara, Italy

    Alessia Ascanelli

  3. Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, via Carlo Alberto 10, 10123, Turin, Italy

    Sandro Coriasco

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  1. Ahmed Abdeljawad
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  2. Alessia Ascanelli
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  3. Sandro Coriasco
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Correspondence to Sandro Coriasco.

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Communicated by Joachim Escher.

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Abdeljawad, A., Ascanelli, A. & Coriasco, S. Deterministic and stochastic Cauchy problems for a class of weakly hyperbolic operators on \(\mathbb {R}^n\). Monatsh Math 192, 1–38 (2020). https://doi.org/10.1007/s00605-020-01372-0

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  • DOI: https://doi.org/10.1007/s00605-020-01372-0

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