Liouville theorems for nonlinear parabolic equations of second order

Abstract

We study entire solutions (i.e., solutions defined in all of $\mathcal {R}^{n+1}$) of second-order nonlinear parabolic equations with linear principal part. Under appropriate hypotheses, we establish existence and uniqueness of an entire solution vanishing at infinity. More generally, we discuss existence and uniqueness of an entire solution approaching a given heat polynomial at infinity. When specialized to the linear homogeneous parabolic equation, containing no zero-order term, our results yield a Liouville theorem stating that an entire and bounded solution must be constant; certain asymptotic behavior of the coefficients at infinity however is required. Our methods involve the establishment of a priori bounds on entire solutions, first for the nonhomogeneous heat equation and then for a more general linear parabolic equation; we use these bounds with a Schauder continuation technique to study the nonlinear equation.