Differential and Integral Equations

Liouville theorems for nonlinear parabolic equations of second order

G. N. Hile and C. P. Mawata

Full-text: Open access


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.

Article information

Differential Integral Equations, Volume 9, Number 1 (1996), 149-172.

First available in Project Euclid: 7 May 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc.


Hile, G. N.; Mawata, C. P. Liouville theorems for nonlinear parabolic equations of second order. Differential Integral Equations 9 (1996), no. 1, 149--172. https://projecteuclid.org/euclid.die/1367969993

Export citation