Large time behavior of reaction–diffusion equations with Bessel generators

We investigate explosion in finite time of one-dimensional semilinear equations of the form ∂ u t ∂ t ( x ) = 1 2 ∂ 2 u t ∂ x 2 ( x ) + φ ′ ( x ) φ ( x ) ∂ u t ∂ x ( x ) − a x 2 u t ( x ) + u t 1 + β ( x ) with initial value ϕ ⩾ 0 , where φ ∈ C 2 ( R ) is positive and a ⩾ 0 , β > 0 are constants. In the free case a = 0 we provide conditions on φ under which any positive nontrivial solution is non-global. In the case a > 0 and φ ( x ) = x μ + 1 / 2 , μ ∈ R , which includes in the special case μ = − 1 / 2 the equation ∂ u t ∂ t ( x ) = 1 2 ∂ 2 u t ∂ x 2 ( x ) − a x 2 u t ( x ) + u t 1 + β ( x ) , we use the Feynman–Kac formula for Bessel processes to give conditions on the equation parameters ensuring finite-time blowup and existence of nontrivial positive global solutions.