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On a Liouville-type theorem and the Fujita blow-up phenomenon

Authors: A. G. Kartsatos and V. V. Kurta
Journal: Proc. Amer. Math. Soc. 132 (2004), 807-813
MSC (2000): Primary 35K55, 35R45, 35B40
Published electronically: July 7, 2003
MathSciNet review: 2019959
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Abstract: The main purpose of this paper is to obtain the well-known results of H.Fujita and K.Hayakawa on the nonexistence of nontrivial nonnegative global solutions for the Cauchy problem for the equation

\begin{displaymath}u_{t} = \Delta u + \vert u\vert^{q-1} u \tag{$\ast $ } \end{displaymath}

with $q\in (1, 1+\frac{2}{n}]$ on the half-space ${\mathbb{S}} := (0, +\infty ) \times {\mathbb{R}}^{n},~ n\geq 1,$ as a consequence of a new Liouville theorem of elliptic type for solutions of ($\ast $) on ${\mathbb{S}}$. This new result is in turn a consequence of other new phenomena established for nonlinear evolution problems. In particular, we prove that the inequality

\begin{displaymath}\vert u\vert _{t} \geq \Delta u + \vert u\vert^{q}, \end{displaymath}

has no nontrivial solutions on ${\mathbb{S}}$ when $q\in (1, 1+\frac{2}{n}].$ We also show that the inequality

\begin{displaymath}u_{t} \geq \Delta u + \vert u\vert^{q-1}u \end{displaymath}

has no nontrivial nonnegative solutions for $q\in (1, 1+\frac{2}{n}]$ , and it has no solutions on ${\mathbb{S}}$ bounded below by a positive constant for $q>1.$

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Additional Information

A. G. Kartsatos
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700

V. V. Kurta
Affiliation: Mathematical Reviews, 416 Fourth Street, P.O. Box 8604, Ann Arbor, Michigan 48107-8604

Keywords: Cauchy problem, entire solution, blow-up, Fujita phenomenon, global solution, Liouville theorem
Received by editor(s): October 30, 2002
Published electronically: July 7, 2003
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2003 American Mathematical Society