Blow-up and stability of semilinear PDEs with gamma generators

We investigate finite-time blow-up and stability of semilinear partial differential equations of the form ∂wt /∂t = Γwt + νtσw 1+β t , w0(x) = ϕ(x) 0, x ∈ R+, where Γ is the generator of the standard gamma process and ν > 0, σ ∈ R, β > 0 are constants. We show that any initial value satisfying c1x−a1 ϕ(x), x >x0, for some positive constants x0, c1, a1, yields a non-global solution if a1β < 1 + σ. If ϕ(x) c2x−a2, x>x0, where x0, c2, a2 > 0, and a2β > 1 + σ, then the solution wt is global and satisfies 0 wt (x) Ct−a2, x 0, for some constant C >0. This complements the results previously obtained in [M. Birkner et al., Proc. Amer. Math. Soc. 130 (2002) 2431; M. Guedda, M. Kirane, Bull. Belg. Math. Soc. Simon Stevin 6 (1999) 491; S. Sugitani, Osaka J. Math. 12 (1975) 45] for symmetric α-stable generators. Systems of semilinear PDEs with gamma generators are also considered.  2004 Elsevier Inc. All rights reserved.