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

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 ( ∗ ) u t = Δ u + | u | q − 1 u \begin{equation*} u_{t} = \Delta u + |u|^{q-1} u \tag {$\ast $} \end{equation*} with q ∈ ( 1 , 1 + 2 n ] q\in (1, 1+\frac {2}{n}] on the half-space S := ( 0 , + ∞ ) × R n , n ≥ 1 , {\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 S {\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 | u | t ≥ Δ u + | u | q , \begin{equation*} |u|_{t} \geq \Delta u + |u|^{q}, \end{equation*} has no nontrivial solutions on S {\mathbb {S}} when q ∈ ( 1 , 1 + 2 n ] . q\in (1, 1+\frac {2}{n}]. We also show that the inequality u t ≥ Δ u + | u | q − 1 u \begin{equation*} u_{t} \geq \Delta u + |u|^{q-1}u \end{equation*} has no nontrivial nonnegative solutions for q ∈ ( 1 , 1 + 2 n ] q\in (1, 1+\frac {2}{n}] , and it has no solutions on S {\mathbb {S}} bounded below by a positive constant for q > 1. q>1.