2021/07/05 by Puskar Mondal, Mondal, Puskar
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc)
paper · pdf · doi:10.48550/arxiv.2107.02323
openalex publication_date 2021/07/05 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
Here we prove a global existence theorem for the solutions of the semi-linear\nwave equation with critical non-linearity admitting a positive definite\nHamiltonian. Formulating a parametrix for the wave equation in a globally\nhyperbolic curved spacetime, we derive an apriori pointwise bound for the\nsolution of the nonlinear wave equation in terms of the initial energy, from\nwhich the global existence follows in a straightforward way. This is\naccomplished in two steps. First, based on Moncrief's light cone formulation we\nderive an expression for the scalar field in terms of integrals over the past\nlight cone from an arbitrary spacetime point to an `initial', Cauchy\nhypersurface and additional integrals over the intersection of this cone with\nthe initial hypersurface. Secondly, we obtain apriori estimates for the energy\nassociated with three quasi-local approximate time-like conformal Killing and\none approximate Killing vector fields. Utilizing these naturally defined\nenergies associated with the physical stress-energy tensor together with the\nintegral equation, we show that the spacetime L\∞ norm of the scalar\nfield remains bounded in terms of the initial data and continues to be so as\nlong as the spacetime remains singularity/Cauchy-horizon free.\n