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1996 Liouville theorems for nonlinear parabolic equations of second order
G. N. Hile, C. P. Mawata
Differential Integral Equations 9(1): 149-172 (1996). DOI: 10.57262/die/1367969993


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.


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G. N. Hile. C. P. Mawata. "Liouville theorems for nonlinear parabolic equations of second order." Differential Integral Equations 9 (1) 149 - 172, 1996.


Published: 1996
First available in Project Euclid: 7 May 2013

zbMATH: 0840.35043
MathSciNet: MR1364039
Digital Object Identifier: 10.57262/die/1367969993

Primary: 35K55
Secondary: 35B05

Rights: Copyright © 1996 Khayyam Publishing, Inc.


Vol.9 • No. 1 • 1996
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