The critical exponent of doubly singular parabolic equations

In this paper we study the Cauchy problem of doubly singular parabolic equa tions u(t) = div(\delu\(sigma) delu(m)) + t(s)\x\(0)u(p) with non-negative initial data. Here -1 < sigma less than or equal to 0. m > max{0, 1 - sigma - (sigma + 2)/N} satisfying 0 < sigma + m less than or equal to 1, p > 1, and s greater than or equal to 0. We prove that if theta > max{-(sigma + 2) , (1 + s)[N(1 - sigma - m) - (sigma + 2)]}, then p(c) = (sigma + m) + (sigm a + m - 1)s + [(sigma + 2)(1 + s) + theta]/N > 1 is the critical exponent; i.e, if 1 < p less than or equal to p(c) then every non-trivial solution bl ows up in finite time. But for I? s p, a positive global solution exists. ( C) 2001 Academic Press.