Nonlinear Heat Flow

In this chapter, we discuss fundamental properties of the nonlinear heat equation ∂tu = Δu associated with the nonlinear Laplacian Δ defined in Chap. 11. In particular, we establish the existence and the regularity of global solutions to the heat equation. Coupled with the Bochner inequalities in the previous chapter, the analysis of heat flow leads to various analytic and geometric applications as we will see in the following chapters. We remark that, due to the nonlinearity, there is no heat kernel.