EXISTENCE AND NONEXISTENCE OF GLOBAL-SOLUTIONS FOR U(T)=DELTA-U+A(X)U(P) IN R(D)

We study nonnegative solutions of the equation u(t) = Delta u + a(x) u (p) in R(d), t > 0, under the assumption that a(x) not greater than or equal to 0 is on the order \x\(m), for m is an element of (-2,infinit y), or that 0 not less than or equal to a(x) less than or equal to C\x \(-2). Extending the classical result of Fujita and more recent result s of Bandle and Levine and of Levine and Meier, we find a critical exp onent p = p*(m,d) such that if 1 < p less than or equal to p*, then t here exist no solutions that are global in time, while if p > p, then there exist both global and nonglobal solutions. (C) 1997 Academic Pr ess