Characterization for the solvability of nonlinear partial differential equations

Abstract:

Within the nonlinear theory of generalized functions introduced earlier by the author a number of existence and regularity results have been obtained. One of them has been the first global version of the Cauchy-Kovalevskaia theorem, which proves the existence of generalized solutions on the whole of the domain of analyticity of arbitrary analytic nonlinear $\text {PDEs}$. These generalized solutions are analytic everywhere, except for closed, nowhere dense subsets which can be chosen to have zero Lebesgue measure. This paper gives a certain extension of that result by establishing an algebraic necessary and sufficient condition for the existence of generalized solutions for arbitrary polynomial nonlinear $\text {PDEs}$ with continuous coefficients. This algebraic characterization, given by the so-called neutrix or off diagonal condition, is proved to be equivalent to certain densely vanishing conditions, useful in the study of the solutions of general nonlinear $\text {PDEs}$.