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 \begin{equation*} u_{t} = \Delta u + |u|^{q-1} u \tag {$\ast $} \end{equation*} 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{equation*} |u|_{t} \geq \Delta u + |u|^{q}, \end{equation*} has no nontrivial solutions on ${\mathbb {S}}$ when $q\in (1, 1+\frac {2}{n}].$ We also show that the inequality \begin{equation*} u_{t} \geq \Delta u + |u|^{q-1}u \end{equation*} 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 $q>1.$