P.R. Popivanov LOCAL SOLVABILITY OF SOME CLASSES OF LINEAR AND NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

The paper deals with the local nonsolvability of several examples of linear and nonlinear partial differential equations. In the linear case we prove nonsolvability in Schwartz distribution space while in the nonlinear case we prove the nonexistence of classical solutions as well as the nonexistence of L∞ ∩ H s , s > 0, solutions. 1. This paper deals with the local nonsolvability of several examples of linear and nonlinear partial differential equations (PDE). In the linear case we prove nonsolvability in Schwartz distribution space D′ while in the nonlinear case we prove the nonexistence of classical solutions as well as the nonexistence of L∞ ∩ H s , s > 0 solutions. We hope that some illustrative examples in the nonlinear case could be useful in a further development of the theory of the local nonsolvability. Y.V. Egorov stated the problem of finding necessary conditions for the local solvability of nonlinear PDE having in mind the well known Hormander’s necessary condition for the local solvability of linear PDE in D′ [2]. We analyse in this paper several examples in order to stress some difficulties arising in the nonlinear situation. 2. We shall propose at first some results on nonsolvability (nonhypoellipticity) of several examples of linear PDE in D′. So consider the following class of PDE with C∞ coefficients (1) P(x, D) = ∑ |α|≤m aα(x)D , aα(x) ∈ C∞ ( R n) . DEFINITION 2.1. The operator (1) is quasihomogeneous if and only if P ( tμx, t−μξ ) = tγ P(x, ξ), ∀t > 0, ∀(x, ξ) ∈ R2n , γ = const . As usual, μ = (μ1, . . . , μn), μ j > 0, 1 ≤ j ≤ n, tμx = ( tμ1 x1, . . . , t μn xn ) . Without loss of generality we assume that 0 < μ1 ≤ μ2 ≤ μ2 ≤ . . . ≤ μn . REMARK 2.1. P quasihomogeneous implies that its formal adjoint operator t P is quasihomogeneous too. Assume that (i) K er t P∩S (