Well-posedness of Heat Equations with Nonlinearities of Arbitrarily Rapid Growth
Yohei Fujishima, Kotaro Hisa, Robert Laister
TL;DR
The paper develops a comprehensive Orlicz-space framework to address well-posedness for the nonlinear heat equation $u_t=\Delta u+f(u)$ without priori growth restrictions on $f$. It introduces sharp smoothing estimates for the heat semigroup between general Orlicz spaces (Theorem A), enabling a contraction-based local theory and, for small data, global well-posedness in Morse-Transue spaces (Theorem B). A central contribution is the identification of a critical Young function $\Phi_c$ with $\Phi_c \sim f$ at infinity, which sharply delineates well-posedness from nonexistence for FP-type nonlinearities (Theorem C), and extends the theory to nonlinearities of arbitrarily rapid growth. The results are demonstrated on a broad canonical family of nonlinearities, including exponential and doubly exponential growth, and are supported by an extensive set of applications, illustrating significant progress in the qualitative theory of nonlinear heat equations beyond polynomial and exponential growth regimes.
Abstract
We address local- and global-in-time well-posedness of the Cauchy problem for nonlinear heat equations without imposing growth rate restrictions on the nonlinearity a priori. Our results constitute a non-trivial expansion of the classical $L^q$-theory for nonlinearities dominated by polynomial growth and the exponential-Orlicz space theory for nonlinearities of exponential growth, to one dealing with nonlinearities of arbitrarily large growth rate. A key ingredient is a new smoothing estimate for the action of the heat semigroup between two arbitrary Orlicz spaces, and in particular into $L^{\infty}$. For nonlinearities growing at least exponentially we are able to identify explicitly a critical space for local well-posedness and for small initial data global well-posedness.