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 (*) u(t)=Deltau+\u\(q-1)u with q is an element of (1, 1+2/n] on the half- space S := (0,+ infinity) x R-n, ngreater than or equal to1; as a consequence of a new Liouville theorem of elliptic type for solutions of (*) on 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)greater than or equal toDeltau+\u\(q), has no nontrivial solutions on S when q is an element of (1, 1+2/n]. We also show that the inequality u(t)greater than or equal toDeltau+\u\(q-1)u has no nontrivial nonnegative solutions for q is an element of (1, 1+2/n], and it has no solutions on S bounded below by a positive constant for q>1.