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The Cauchy problem for properly hyperbolic equations in one space variable

Sergio Spagnolo

Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

E-mail Address: sergio.spagnolo@unipi.it

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and

Giovanni Taglialatela

Dipartimento di Economia e Finanza, Università di Bari “Aldo Moro”, Largo Abbazia S. Scolastica, 70124 Bari, Italy

E-mail Address: giovanni.taglialatela@uniba.it

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https://doi.org/10.1142/S0219891622500138Cited by:2 (Source: Crossref)

Abstract

In this paper, we consider the Cauchy problem for higher-order weakly hyperbolic equations assuming that the principal symbol depends only on one space variable and the characteristic roots $τ_{j}$ verify an inequality like

τ 2 j (x) + τ 2 k (x) \leq M (τ j (x) - τ k (x)) 2 . <math display="block" altimg="eq-00002.gif"><mrow><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>j</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mo stretchy="false" class="MathClass-bin">+</mo><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">\leq</mo><mi>M</mi><msup><mrow><mo class="MathClass-open" stretchy="false">(</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mo stretchy="false" class="MathClass-bin">-</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-close" stretchy="false">)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo class="MathClass-punc">.</mo></mrow></math>

We prove that the Cauchy problem is well-posed in

$𝒞^{\infty}$ if the operators with frozen coefficients are uniformly hyperbolic in the sense of Gårding.

Dedicated to our friend Enrico Jannelli

Communicated by T. Nishitani

Keywords:

AMSC: 35L30

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