On the Cauchy problem and initial traces for a degenerate parabolic equation

Abstract:

We consider the Cauchy problem (f) \[ \left \{ {\begin {array}{*{20}{c}} {{u_t} - \operatorname {div}(|Du{|^{p - 2}}Du) = 0} \hfill & {{\text {in}}\;{{\mathbf {R}}^N} \times (0,\infty ),p > 2,} \hfill \\ {u(x,0) = {u_0}(x),} \hfill & {x \in {{\mathbf {R}}^N},} \hfill \\ \end {array} } \right .\] and discuss existence of solutions in some strip ${S_T} \equiv {{\mathbf {R}}^N} \times (0,T)$, $0 < T \leq \infty$, in terms of the behavior of $x \to {u_0}(x)$ as $|x| \to \infty$. 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 $L_{{\text {loc}}}^1({{\mathbf {R}}^N})$.