Well‐posedness of heat equations with nonlinearities of arbitrarily rapid growth

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 nontrivial expansion of the classical ‐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 . 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.