Abstract
In this article, we study the blow-up of solutions to the nonlinear parabolic equation in divergence form, $$\displaylines{\big(h(u)\big)_t =\sum_{i,j=1}^{n}\big(a^{ij}(x)u_{x_i}\big)_{x_j}-k(t)f(u)\quad\hbox{in } \Omega\times(0,t^{*}), \cr \sum_{i,j=1}^{n}a^{ij}(x)u_{x_i}\nu_j=g(u) \quad\hbox{on } \partial\Omega\times(0,t^{*}),\cr u(x,0)=u_0(x)\geq 0 \quad\hbox{in } \overline{\Omega},}$$ where \(\Omega\) is a bounded convex domain in \(\mathbb{R}^n\) \((n\geq2)\) with smooth boundary \(\partial\Omega\). By constructing suitable auxiliary functions and using a differential inequality technique, when \(\Omega\subset\mathbb{R}^n\) \((n\geq2)\), 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 \(\Omega\subset \mathbb{R}^n\) with \((n\geq3)\), we use a Sobolev inequality to obtain a lower bound for the blow-up time.