1. Workshop on Messico, 2010 B. Tirozzi, S.Yu. Dobrokhotov, E. Nazaikinski “Asymptotic solutions of a 2- dimensional wave equations with degenerate velocity and localized initial conditions”

2. Messico 2010 μ is a small parameter is the center of perturbation a=(1,0) Position of the Problem

3. Messico 2010 Operators

4. Messico 2010 Proposition

5. Messico 2010 Proposition Let L be the closure of the operator L 0 Consider the differential expression a and its formal adjoint a* Let A 0 be the operator defined by a in the space D 0 and A be its closure in L 2 (R 2 + ), let A be its adjoint operator

6. Messico 2010 Proposition Then the following identity holds Consider the hyperbolic problem in L 2 (R 2 + ) where L is the self-adjoint operator defined above

7. Messico 2010 Proposition is a strong solution of the problem if it satisfies the equation and the initial conditions and belong to

8. Messico 2010 Theorem

9. Messico 2010 Asymptotic Solution of the Problem Initial Condition Lagrangian manifold Compactification of space, since p 1 goes to infinity in a finite time

10. Messico 2010 Asymptotic Solution of the Problem function with compact support in Λ, regular in ρ and t, polynomial of degree N in h, fast decaying with its derivatives for ρ → ∞ Error Estimate If K h Λ [φρ(t, h)] satisfies the equation with an error of O(h N ), N > 1 the error of u is

11. Messico 2010 Asymptotic Solution of the Problem

12. Messico 2010 Asymptotic Solution of the Problem

13. Messico 2010 Asymptotic Solution of the Problem

14. Messico 2010

15. Asymptotic Solution of the Problem

16. Messico 2010