We consider the Cauchy problem (f)
\[
{
u
t
−
div
(
|
D
u
|
p
−
2
D
u
)
=
0
a
m
p
;
in
R
N
×
(
0
,
∞
)
,
p
>
2
,
u
(
x
,
0
)
=
u
0
(
x
)
,
a
m
p
;
x
∈
R
N
,
\left \{ {\begin {array}{*{20}{c}} {{u_t} - \operatorname {div}(|Du{|^{p - 2}}Du) = 0} \hfill & {{\text {in}}\;{{\mathbf {R}}^N} \times (0,\infty ),p > 2,} \hfill \\ {u(x,0) = {u_0}(x),} \hfill & {x \in {{\mathbf {R}}^N},} \hfill \\ \end {array} } \right .
\]
and discuss existence of solutions in some strip
S
T
≡
R
N
×
(
0
,
T
)
{S_T} \equiv {{\mathbf {R}}^N} \times (0,T)
,
0
>
T
≤
∞
0 > T \leq \infty
, in terms of the behavior of
x
→
u
0
(
x
)
x \to {u_0}(x)
as
|
x
|
→
∞
|x| \to \infty
. The results obtained are optimal in the class of nonnegative locally bounded solutions, for which a Harnack-type inequality holds. Uniqueness is shown under the assumption that the initial values are taken in the sense of
L
loc
1
(
R
N
)
L_{{\text {loc}}}^1({{\mathbf {R}}^N})
.