Cauchy problem, degenerate operator, fundamental solution, Asymptotic distributions of eigenvalues in context of PDEs, Initial value problems for PDEs with pseudodifferential operators, eigenvalues, Heat and other parabolic equation methods for PDEs on manifolds, Degenerate parabolic equations, principal and sub-principal symbols, rate of its trace, selfadjoint operator, 35S10, compact manifold, Weyl calculus of pseudo- differential operators, parametrix