On the Cauchy problem of odd order equation with multiple characteritics (Q2742855)

The following odd order equation \({\partial^{2n+1}U}/{\partial x^{2n+1}}+(-1)^n \frac{\partial U}{\partial y}=0\) in the domain \(D=\{(x,y)\mid-\infty<x<\infty\), \(y>0\}\) is considered. Using the fundamental solution to the above equation NEWLINE\[NEWLINEV(x-\xi,y-\eta)=(y-\eta)^{-\frac 1{2n+1}}\operatorname {Ain}\left((x-\xi)(y-\eta)^{-\frac 1 {2n+1}}\right),NEWLINE\]NEWLINE where \(\operatorname {Ain}(x)\) denotes the Airy function, the continuity of the solution \(U(x,t)= \int_{-\infty}^{\infty}V(x-\xi,t)\varphi(\xi) d\xi\) for \(t\to+0\) is investigated.