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

Abstract

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}) .