Breaking symmetry in focusing nonlinear Klein-Gordon equations with potential

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    We study the dynamics for the focusing nonlinear Klein-Gordon equation, utt - Δu + m2u = V (x)|u|p-1u, with positive radial potential V and initial data in the energy space. Under suitable assumption on the potential, we establish the existence and uniqueness of the ground state solution. This enables us to define a threshold size for the initial data that separates global existence and blow-up. An appropriate Gagliardo-Nirenberg inequality gives a critical exponent depending on V. For subcritical exponent and subcritical energy global existence vs blow-up conditions are determined by a comparison between the nonlinear term of the energy solution and the nonlinear term of the ground state energy. For subcritical exponents and critical energy some solutions blow-up, other solutions exist for all time due to the decomposition of the energy space of the initial data into two complementary domains.

    Original languageEnglish
    Pages (from-to)755-788
    Number of pages34
    JournalJournal of Hyperbolic Differential Equations
    Issue number4
    Publication statusPublished - 2018 Dec 1


    • critical energy
    • global existence/blow up
    • Ground state

    ASJC Scopus subject areas

    • Analysis
    • Mathematics(all)

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