Solutions of Nonlinear Parabolic Equations without Growth Restrictions on the Data

Abstract

The purpose of this paper is to prove the existence of solutions for certain types of nonlinear parabolic partial differential equations posed in the whole space, when the data are assumed to be merely locally integrable functions, without any control of their behaviour at infinity. A simple representative example of such an equation is ut - ∆u + |u|s-1 u = f, which admits a unique globally defined weak solution u(x, t) if the initial function u(x, 0) is a locally integrable function in ℝN, N ≥ 1, and the second member ƒ is a locally integrable function of x ∈ ℝN and t ∈ [0, T] whenever the exponent s is larger than 1. The results extend to parabolic equations results obtained by Brezis and by the authors for elliptical equations. They have no equivalent for linear or sub-linear zero-order nonlinearities.