Blow-up and stability of semilinear PDEs with gamma generators

Abstract: We investigate finite-time blow-up and stability of semilinear partial differential equations of the form , , , where Γ is the generator of the standard gamma process and , , are constants. We show that any initial value satisfying , , for some positive constants , , , yields a non-global solution if . If , where , and , then the solution is global and satisfies , for some constant . 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. [Copyright &y& Elsevier]