Self-similar solutions of semilinear wave equation with variable speed of propagation

Abstract

We investigate the issue of existence of the self-similar solutions of the generalized Tricomi equation in the half-space where the equation is hyperbolic. We look for the self-similar solutions via the Cauchy problem. An integral transformation suggested in [K. Yagdjian, A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain, J. Differential Equations 206 (2004) 227-252] is used to represent solutions of the Cauchy problem for the linear Tricomi-type equation in terms of fundamental solutions of the classical wave equation. This representation allows us to prove decay estimates for the linear Tricomi-type equation with a source term. Obtained in [K. Yagdjian, The self-similar solutions of the Tricomi-type equations, Z. Angew. Math. Phys., in press, doi:10.1007/s00033-006-5099-2] estimates for the self-similar solutions of the linear Tricomi-type equation are the key tools to prove existence of the self-similar solutions.

Blow-up and lifespan estimate for generalized Tricomi equations related to Glassey conjecture

“…The Tricomi equation appears in gas dynamic problems, connected to gas flows with nearly sonic speed, describing the transition from subsonic flow (t < 0) to supersonic flow (t > 0), see Frankl [6]. We refer to Yagdjian [36][37][38][39][40] and references therein for more details.…”

The wave equation in the Einstein and de Sitter spacetime

“…Having in mind the scale invariance of the equation and also the applications (see, e.g., [16], [28]), special attention will be given to the self-similar solutions. Results analogous to those presented in this note have already proven to be a good tool in the study of self-similar solutions [35]. This note is organized as follows.…”

“…In particular, and unlike Minkowski or Schwarzschild, the past particle horizons exist. The EdeS spacetime is a good approximation to the large scale structure of the universe during a matter dominated phase, when the averaged (over space and time) energy density evolves adiabatically and pressures are vanishingly small, as, e.g., immediately after inflation [35]. This justifies why such a metric is adopted to model the collapse of overdensity perturbations in the early matter dominated phase that followed inflation.…”

Fundamental Solutions for the Klein-Gordon Equation in de Sitter Spacetime

“…In that way, many properties of the wave equation can be extended to the hyperbolic equations with the time dependent speed of propagation. That approach was successfully applied in [31,32] by the first author to investigate the semilinear Tricomi-type equations.…”