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Wave equations on asymptotically de Sitter spaces
Asymptotically de Sitter spaces are Lorentzian manifolds modeled on the de Sitter space of general relativity. In this dissertation, we construct the forward fundamental solution for the wave and Klein-Gordon equations on asymptotically de Sitter spaces. We adapt classes of conormal and paired Lagrangian distributions to this setting and show that the lift of the kernel of the forward fundamental solution to a blown-up space is a sum of distributions in these classes. We use the structure of the kernel of the fundamental solution to study its mapping properties. We show that Strichartz estimates with loss hold for the positive mass Klein-Gordon equation on asymptotically de Sitter spaces. When the mass parameter is the conformal value, Strichartz estimates hold without loss. As an application of these estimates, we prove a small-data global existence result for a defocusing Klein-Gordon equation.
- "Asymptotically de Sitter spaces are Lorentzian manifolds modeled on the de Sitter space of general relativity. In this dissertation, we construct the forward fundamental solution for the wave and Klein-Gordon equations on asymptotically de Sitter spaces. We adapt classes of conormal and paired Lagrangian distributions to this setting and show that the lift of the kernel of the forward fundamental solution to a blown-up space is a sum of distributions in these classes. We use the structure of the kernel of the fundamental solution to study its mapping properties. We show that Strichartz estimates with loss hold for the positive mass Klein-Gordon equation on asymptotically de Sitter spaces. When the mass parameter is the conformal value, Strichartz estimates hold without loss. As an application of these estimates, we prove a small-data global existence result for a defocusing Klein-Gordon equation."@en
- "Wave equations on asymptotically de Sitter spaces"@en