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A Note on \(L^{p}-L^{q}\) Estimates for Semilinear Critical Dissipative Klein–Gordon Equations

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Abstract

In this note, we consider a semilinear wave equation with scale-invariant mass and dissipation. The scale of the dissipation is the same of the mass term and this creates an interplay in which a relation between the coefficients comes into play to determine if the dissipation is dominant with respect to the mass, or viceversa. We prove global existence of small data solutions for supercritical power nonlinearities when the mass term is dominant with respect to the dissipative term (“Klein–Gordon type” case). The critical exponent is a modified Strauss exponent for global small data solutions to semilinear Klein–Gordon equations.

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Acknowledgements

Marcello D’Abbicco has been supported by INdAM - GNAMPA Project 2017 Equazioni di tipo dispersivo e proprietà asintotiche. Alessandro Palmieri is supported by the University of Pisa, Project PRA 2018 49.

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Authors and Affiliations

  1. Department of Mathematics, University of Bari, Via E. Orabona 4, 70125, Bari, Italy

    Marcello D’Abbicco

  2. Department of Mathematics, University of Pisa, Largo B. Pontecorvo 5, 56127, Pisa, Italy

    Alessandro Palmieri

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  1. Marcello D’Abbicco
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  2. Alessandro Palmieri
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Correspondence to Marcello D’Abbicco.

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D’Abbicco, M., Palmieri, A. A Note on \(L^{p}-L^{q}\) Estimates for Semilinear Critical Dissipative Klein–Gordon Equations. J Dyn Diff Equat 33, 63–74 (2021). https://doi.org/10.1007/s10884-019-09818-2

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  • DOI: https://doi.org/10.1007/s10884-019-09818-2

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