The paper presents a probabilistic approach to prove blow-up of solutions to the Fujita equation \(\partial w/\partial t = -(-\Delta)^{\alpha/2} w + w^{1+\beta}\) in the critical dimension \(d = \alpha/\beta\). By using the Feynman-Kac representation twice, the authors construct a subsolution that locally grows to infinity as \(t \to \infty\). This method covers earlier results proved by analytic methods and extends a blow-up result for systems with the Laplacian case to the case of \(\alpha\)-Laplacians with possibly different parameters \(\alpha\). The paper also discusses blow-up in subcritical dimensions and provides conditions for blow-up of a class of semilinear systems at the critical dimension. Additionally, it addresses a time-dependent nonlinearity and extends the blow-up result to a more general system.The paper presents a probabilistic approach to prove blow-up of solutions to the Fujita equation \(\partial w/\partial t = -(-\Delta)^{\alpha/2} w + w^{1+\beta}\) in the critical dimension \(d = \alpha/\beta\). By using the Feynman-Kac representation twice, the authors construct a subsolution that locally grows to infinity as \(t \to \infty\). This method covers earlier results proved by analytic methods and extends a blow-up result for systems with the Laplacian case to the case of \(\alpha\)-Laplacians with possibly different parameters \(\alpha\). The paper also discusses blow-up in subcritical dimensions and provides conditions for blow-up of a class of semilinear systems at the critical dimension. Additionally, it addresses a time-dependent nonlinearity and extends the blow-up result to a more general system.