Asymptotic analysis to a parabolic equation with a weighted localized source

This paper deals with a nonlinear parabolic equation with a more complicated source term, which is a product of localized source uq(0, t), local source up(x, t), and weight function a(x). We investigate how the three factors influence the asymptotic behavior of solutions. It is shown that the blow-up set consists of single point {x = 0} if p > 1. When 0 ⩽ p ⩽ 1 with p + q > 1, the blow-up take place everywhere in B. Moreover, the blow-up rate estimate is established with more precise coefficients determined.