On Regularity for Solutions of Non-linear Equation with Constant Multiple Characteristic

In this paper we consider the propagation of microlocal regularity near constant multiple characteristic or a real solution u ∈ H^s (s > m + max{μ, 2} + \frac{n}{2})or non-linear partial differential equation F(x, u,…, ∂^βu,…)_{(|β|≤m)} = 0 We will prove that the microlocal regularity ncar constant multiple characteristic of the solution u will propagate along bicharacteristic with constant multiplicity μ and have loss of smoothness up to order μ - 1 under Levi condition.