Blow-up of semilinear pde's at the critical dimension. A probabilistic approach

We present a probabilistic approach which proves blow-up of solutions of the Fujita equation in the critical dimension . By using the Feynman-Kac representation twice, we construct a subsolution which locally grows to infinity as . In this way, we cover results proved earlier by analytic methods. Our method also applies to extend a blow-up result for systems proved for the Laplacian case by Escobedo and Levine (1995) to the case of -Laplacians with possibly different parameters .

     

Exponential Functionals of Brownian Motion and Explosion Times of a System of Semilinear SPDEs

We investigate lower and upper bounds for the blowup times of a system of semilinear SPDEs. Under certain conditions on the system parameters, we obtain explicit solutions of a related system of random PDEs, which allows us to use a formula due to Yor to obtain the distribution functions of several explosion times. We also give the Laplace transforms at independent exponential times of related exponential functionals of Brownian motion.     

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 , 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.     

Large time behavior of reaction-diffusion equations with Bessel generators

We investigate explosion in finite time of one-dimensional semilinear equations of the form