Two-dimensional wave equation with degeneration on the curvilinear boundary of the domain and asymptotic solutions with localized initial data

In a two-dimensional domain Ω ⊂ R2, we consider the wave equation with variable velocity c(x1, x2) degenerating on the boundary Γ = ∂Ω as the square root of the distance to the boundary, and construct an asymptotic solution of the Cauchy problem with localized initial data. This problem is related to the so-called “run-up problem” in tsunami wave theory. One main idea (also used by the authors in earlier papers in the one-dimensional case and the two-dimensional case with c2(x1, x2) = x1) is that the (singular) curve Γ is a caustic of special type. We use this idea to introduce a generalization of the Maslov canonical operator covering the problem with degeneration and obtain efficient formulas for the asymptotic solutions.