On the Cauchy problem for Dt2–Dxa(t,x)nDx

Abstract

The well posedness of the Cauchy problem for the operator P=D t 2–D x a(t,x) n D x , $t,x \in \mathbb{R}$ with data on t=0 is studied assuming a ∈ C N ( $[0,T];\gamma^{(s_0)}$ (R)), s0>1 and sufficiently close to 1, a(t,x)≥ 0. Well posedness is proved in Gevrey classes γ(s), for $1\leq s< \frac{Nn^2}{2(N+2n)}$ , n≥ n0. Keywords: Partial differential equations, Cauchy problem, Well posedness