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

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

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

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

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 $q>1.$