Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 08, pp. 1-17.

Title: Blow-up for parabolic equations in nonlinear divergence form with time-dependent coefficients
       
Authors: Xuhui Shen   (Shanxi Univ., Taiyuan, China)
         Juntang Ding (Shanxi Univ., Taiyuan, China)

Abstract:  
In this article, we study the blow-up of solutions to the nonlinear parabolic equation
in divergence form,
$$\displaylines{
\big(h(u)\big)_t =\sum_{i,j=1}^{n}\big(a^{ij}(x)u_{x_i}\big)_{x_j}-k(t)f(u)
\quad\text{in } \Omega\times(0,t^{*}), \cr
\sum_{i,j=1}^{n}a^{ij}(x)u_{x_i}\nu_j=g(u)
\quad\text{on }  \partial\Omega\times(0,t^{*}),\cr
 u(x,0)=u_0(x)\geq 0 \quad\text{in } \overline{\Omega},
}$$
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.

Submitted January 27, 2021. Published January 25, 2022.
Math Subject Classifications: 35K55, 35B44.
Key Words: Nonlinear parabolic equation; blow-up; upper bound; lower bound.