Abstract
This paper is devoted to discussing the blow-up problem of the positive solution of the following degenerate parabolic equations:
(
r
(
u
)
)
t
=
div
(
∣
∇
u
∣
p
∇
u
)
+
f
(
x
,
t
,
u
,
∣
∇
u
∣
2
)
,
(
x
,
t
)
∈
D
×
(
0
,
T
∗
)
,
∂
u
∂
ν
+
σ
u
=
0
,
(
x
,
t
)
∈
∂
D
×
(
0
,
T
∗
)
,
u
(
x
,
0
)
=
u
0
(
x
)
,
x
∈
D
¯
.
\left\{\begin{array}{ll}{(r\left(u))}_{t}={\rm{div}}(| \nabla u{| }^{p}\nabla u)+f\left(x,t,u,| \nabla u\hspace{-0.25em}{| }^{2}),& \left(x,t)\in D\times \left(0,{T}^{\ast }),\\ \frac{\partial u}{\partial \nu }+\sigma u=0,& \left(x,t)\in \partial D\times \left(0,{T}^{\ast }),\\ u\left(x,0)={u}_{0}\left(x),& x\in \overline{D}.\end{array}\right.
Here
p
>
0
p\gt 0
, the spatial region
D
⊂
R
n
(
n
≥
2
)
D\subset {{\mathbb{R}}}^{n}\hspace{0.33em}\left(n\ge 2)
is bounded, and its boundary
∂
D
\partial D
is smooth. We give the conditions that cause the positive solution of this degenerate parabolic problem to blow up. At the same time, for the positive blow-up solution of this problem, we also obtain an upper bound of the blow-up time and an upper estimate of the blow-up rate. We mainly carry out our research by means of maximum principles and first-order differential inequality technique.
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