This paper presents a probabilistic approach to proving blow-up of solutions to the Fujita equation in the critical dimension $ d = \alpha/\beta $. The equation is given by $ \partial w/\partial t = -(-\Delta)^{\alpha/2}w + w^{1+\beta} $. The authors use the Feynman-Kac representation twice to construct a subsolution that grows to infinity as $ t \to \infty $, thereby proving blow-up. This method extends results previously obtained by analytic methods and applies to systems involving $ \alpha $-Laplacians with different parameters. The paper begins by introducing the semilinear equation $ \partial w_t/\partial t = \Delta_\alpha w_t + \gamma w_t^{1+\beta} $, where $ \Delta_\alpha $ is the $ \alpha $-Laplacian. It discusses the critical dimension $ d = \alpha/\beta $, where solutions blow up. The authors show that for $ d < \alpha/\beta $, the solution grows to infinity, and for $ d = \alpha/\beta $, the solution also blows up. They use probabilistic methods, including the Feynman-Kac formula, to construct subsolutions that grow to infinity, proving blow-up. The paper also extends results to systems and shows that blow-up occurs at the critical dimension for certain systems. It discusses the critical dimension for blow-up in systems and provides conditions under which blow-up occurs. The authors also mention that their method can be applied to time-dependent nonlinearities and that it can be used to prove blow-up for systems analyzed in previous works. The paper concludes with a discussion of the critical dimension for blow-up in systems and the conditions under which blow-up occurs.This paper presents a probabilistic approach to proving blow-up of solutions to the Fujita equation in the critical dimension $ d = \alpha/\beta $. The equation is given by $ \partial w/\partial t = -(-\Delta)^{\alpha/2}w + w^{1+\beta} $. The authors use the Feynman-Kac representation twice to construct a subsolution that grows to infinity as $ t \to \infty $, thereby proving blow-up. This method extends results previously obtained by analytic methods and applies to systems involving $ \alpha $-Laplacians with different parameters. The paper begins by introducing the semilinear equation $ \partial w_t/\partial t = \Delta_\alpha w_t + \gamma w_t^{1+\beta} $, where $ \Delta_\alpha $ is the $ \alpha $-Laplacian. It discusses the critical dimension $ d = \alpha/\beta $, where solutions blow up. The authors show that for $ d < \alpha/\beta $, the solution grows to infinity, and for $ d = \alpha/\beta $, the solution also blows up. They use probabilistic methods, including the Feynman-Kac formula, to construct subsolutions that grow to infinity, proving blow-up. The paper also extends results to systems and shows that blow-up occurs at the critical dimension for certain systems. It discusses the critical dimension for blow-up in systems and provides conditions under which blow-up occurs. The authors also mention that their method can be applied to time-dependent nonlinearities and that it can be used to prove blow-up for systems analyzed in previous works. The paper concludes with a discussion of the critical dimension for blow-up in systems and the conditions under which blow-up occurs.