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+νtσwt1+β, 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.