Monge–Ampère type equations and their applications and heat equations driven by irregular terms

This work is presented in two parts. In Part I, we focus on the Monge-Ampere type equations and their applications. In Part II, we achieve regularity results for the Cauchy problems of heat equations driven by a separable inhomogeneous term or a nonseparable general term. As an application, the existence and uniqueness of solutions to the Cauchy problems are arrived. In addition, the results in Part II are applied to obtain pathwise estimates for the heat equation driven by a fractional Brownian sheet.