Abstract
AbstractIn this paper, we consider a quasi-linear parabolic equation $$u_t=u^p(x_{xx}+u)$$
u
t
=
u
p
(
x
xx
+
u
)
. It is known that there exist blow-up solutions and some of them develop Type II singularity. However, only a few results are known about the precise behavior of Type II blow-up solutions for $$p>2$$
p
>
2
. We investigated the blow-up solutions for the equation with periodic boundary conditions and derived upper estimates of the blow-up rates in the case of $$2<p<3$$
2
<
p
<
3
and in the case of $$p=3$$
p
=
3
, separately. In addition, we assert that if $$2 \le p \le 3$$
2
≤
p
≤
3
then $$\lim _{t \nearrow T}(T-t)^{\frac{1}{p}+\varepsilon }\max u(x,t)=0$$
lim
t
↗
T
(
T
-
t
)
1
p
+
ε
max
u
(
x
,
t
)
=
0
z for any $$\varepsilon >0$$
ε
>
0
under some assumptions.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,General Engineering
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