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

TL;DR

This paper studies the existence and nonexistence of solutions to a broad class of Cauchy problems, offering a unified approach applicable to various nonlinear PDEs including k-Hessian and P-k-Hessian equations.

Contribution

It introduces a general framework for analyzing solvability of nonlinear PDEs, covering new classes of equations and more general conditions than previous methods.

Findings

Established sharp conditions for solution existence and nonexistence.

Unified approach applicable to multiple classes of nonlinear PDEs.

Extended results to p-Hessian matrices with p > 1.

Abstract

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.