Shen, XuhuiDing, Juntang2023-03-302023-03-302022-01-25Shen, X., & Ding, J. (2022). Blow-up for parabolic equations in nonlinear divergence form with time-dependent coefficients. <i>Electronic Journal of Differential Equations, 2022</i>(08), pp. 1-17.1072-6691https://hdl.handle.net/10877/16512In this article, we study the blow-up of solutions to the nonlinear parabolic equation in divergence form, (h(u))t = n∑i,j=1 (ɑ ij(x)uxi)xj - k(t)ƒ(u) in Ω x (0, t*), n∑i,j=1 ɑ ij(x)uxi vj = g(u) on ∂Ω x (0, t*), u(x, 0) = u0(x) ≥ 0 in Ω̅, where Ω is a bounded convex domain in ℝn (n ≥ 2) with smooth boundary ∂Ω. By constructing suitable auxiliary functions and using a differential inequality technique, when Ω ⊂ ℝn (n ≥ 2), we establish conditions for the solution blow up at a finite time, and conditions for the solution to exist for all time. Also, we find an upper bound for the blow-up time. In addition, when Ω ⊂ ℝn with (n ≥ 3), we use a Sobolev inequality to obtain a lower bound for the blow-up time.Text17 pages1 file (.pdf)enAttribution 4.0 InternationalNonlinear parabolic equationBlow-upUpper boundLower boundArticle