@article{63311,
abstract = {{AbstractThe Cauchy problem in $$\mathbb {R}^n$$
R
n
, $$n\ge 1$$
n
≥
1
, for the degenerate parabolic equation $$\begin{aligned} u_t=u^p \Delta u \qquad \qquad (\star ) \end{aligned}$$
u
t
=
u
p
Δ
u
(
⋆
)
is considered for $$p\ge 1$$
p
≥
1
. It is shown that given any positive $$f\in C^0([0,\infty ))$$
f
∈
C
0
(
[
0
,
∞
)
)
and $$g\in C^0([0,\infty ))$$
g
∈
C
0
(
[
0
,
∞
)
)
satisfying $$\begin{aligned} f(t)\rightarrow + \infty \quad \text{ and } \quad g(t)\rightarrow 0 \qquad \text{ as } t\rightarrow \infty , \end{aligned}$$
f
(
t
)
→
+
∞
and
g
(
t
)
→
0
as
t
→
∞
,
one can find positive and radially symmetric continuous initial data with the property that the initial value problem for ($$\star $$
⋆
) admits a positive classical solution such that $$\begin{aligned} t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty (\mathbb {R}^n)} \rightarrow \infty \qquad \text{ and } \qquad \Vert u(\cdot ,t)\Vert _{L^\infty (\mathbb {R}^n)} \rightarrow 0 \qquad \text{ as } t\rightarrow \infty , \end{aligned}$$
t
1
p
‖
u
(
·
,
t
)
‖
L
∞
(
R
n
)
→
∞
and
‖
u
(
·
,
t
)
‖
L
∞
(
R
n
)
→
0
as
t
→
∞
,
but that $$\begin{aligned} \liminf _{t\rightarrow \infty } \frac{t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty (\mathbb {R}^n)}}{f(t)} =0 \end{aligned}$$
lim inf
t
→
∞
t
1
p
‖
u
(
·
,
t
)
‖
L
∞
(
R
n
)
f
(
t
)
=
0
and $$\begin{aligned} \limsup _{t\rightarrow \infty } \frac{\Vert u(\cdot ,t)\Vert _{L^\infty (\mathbb {R}^n)}}{g(t)} =\infty . \end{aligned}$$
lim sup
t
→
∞
‖
u
(
·
,
t
)
‖
L
∞
(
R
n
)
g
(
t
)
=
∞
.
}},
author = {{Winkler, Michael}},
issn = {{2662-2963}},
journal = {{Partial Differential Equations and Applications}},
number = {{4}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Oscillatory decay in a degenerate parabolic equation}}},
doi = {{10.1007/s42985-022-00186-z}},
volume = {{3}},
year = {{2022}},
}