Blow-up and stability of semilinear PDEs with gamma generators |
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Authors: | José Alfredo López-Mimbela |
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Institution: | a Centro de Investigación en Matemáticas, Apartado Postal 402, 36000 Guanajuato, Mexico b Département de Mathématiques, Université de La Rochelle, Avenue Michel Crépeau, 17042 La Rochelle cedex 1, France |
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Abstract: | We investigate finite-time blow-up and stability of semilinear partial differential equations of the form , 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 , where x0,c2,a2>0, and a2β>1+σ, then the solution wt is global and satisfies , 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. |
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Keywords: | Semilinear partial differential equations Feynman-Kac representation Blow-up of semilinear systems Gamma processes |
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