Barros-Neto and Gelfand (Duke Math. J. 98 (3) (1999) 465; Duke Math. J. 117 (2) (2003) 561) constructed for the Tricomi operator y* 2 x + * 2 y on the plane the fundamental solutions with the supports in the regions related to the geometry of the characteristics of the Tricomi operator.We give for the Tricomi-type operator * 2 t −t m $ x a fundamental solution relative to an arbitrary point of R n+1 with the support in the region t 0, where the operator is hyperbolic. Our key observation is that the fundamental solution for the Tricomi-type operator can be written like an integral of the distributions generated by the fundamental solution of the Cauchy problem for the wave equation. The application of that fundamental solution to the L p − L q estimate for the forced Tricomi-type equation is given as well.
In this article we construct the fundamental solutions for the Klein-Gordon equation in de Sitter spacetime. We use these fundamental solutions to represent solutions of the Cauchy problem and to prove L p − L q estimates for the solutions of the equation with and without a source term.
In this article we study the blow-up phenomena for the solutions of the semilinear Klein-Gordon equation ✷gφ − m 2 φ = −|φ| p with the small mass m ≤ n/2 in de Sitter space-time with the metric g. We prove that for every p > 1 the large energy solution blows up, while for the small energy solutions we give a borderline p = p(m, n) for the global in time existence. The consideration is based on the representation formulas for the solution of the Cauchy problem and on some generalizations of the Kato's lemma.3 √ Λ , is considered by Dafermos and Rodnianski [3]. They proved that in the region bounded by a set of black/white hole horizons and cosmological horizons, solutions converge pointwise to a constant faster than any given polynomial rate, where the decay is measured with respect to natural future-directed advanced and retarded time coordinates. The bounds on decay rates for solutions to the wave equation in the Schwarzschild -de Sitter spacetime is a first step to a mathematical understanding of non-linear stability problems for spacetimes containing black holes.Catania and Georgiev [2] studied the Cauchy problem for the semilinear wave equation ✷ g φ = |φ| p in the Schwarzschild metric (3 + 1)-dimensional space-time, that is the case of Λ = 0 in 0 < M bh < 1 3 √ Λ .
L q estimates Higgs boson equation a b s t r a c tIn this article we prove the global existence of the small data solutions of the Cauchy problem for the semilinear Klein-Gordon equation in the de Sitter spacetime. Unlike the same problem in the Minkowski spacetime, we have no restriction on the order of nonlinearity and structure of the nonlinear term, provided that a physical mass of the field is outside of some bounded interval.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.