On a class of Cauchy problems with applications in nonlinear partial differential equations

In this paper, we investigate the existence and nonexistence of entire solutions to a general class of Cauchy problems in the positive half line. Our results provide a unified approach to proving sharp local and entire solvability of nonlinear partial differential equations in n-dimensional Euclidean space. As applications of the general framework, we present such results for two series of nonlinear equations: a series of k-Hessian type equations and a new series of P-k-Hessian type equations. These results are also proved for the more general p-Hessian matrices, with p > 1, and degenerate non homogeneous terms.