Large time behavior of reaction–diffusion equations with Bessel generators

We investigate explosion in finite time of one-dimensional semilinear equations of the form∂ut∂t(x)=12∂2ut∂x2(x)+φ′(x)φ(x)∂ut∂x(x)−ax2ut(x)+ut1+β(x) with initial value ϕ⩾0ϕ⩾0, where φ∈C2(R)φ∈C2(R) is positive and a⩾0a⩾0, β>0β>0 are constants. In the free case a=0a=0 we provide conditions on φ   under which any positive nontrivial solution is non-global. In the case a>0a>0 and φ(x)=xμ+1/2φ(x)=xμ+1/2, μ∈Rμ∈R, which includes in the special case μ=−1/2μ=−1/2 the equation∂ut∂t(x)=12∂2ut∂x2(x)−ax2ut(x)+ut1+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.