On the \(C^\infty\)-Goursat problem for some second order equations with variable coefficients (Q1130093)

We study the following \(C^\infty\)-Goursat problem: \[ \partial_t\partial_xu- a(t,x)\partial^2_y u- b(t,x)\partial_yu- c(t,x)u= f, u(0,x,y)= g(x,y), u(t,0,y)= h(t,y),\tag{P} \] where the coefficients \(a(t,x)\), \(b(t,x)\), \(c(t,x)\) and data \(f(t,x,y)\), \(g(x,y)\), \(h(t,y)\) are \(C^\infty\) functions. Under some assumptions, we obtained a necessary condition and a sufficient condition for (P) to be \({\mathcal E}\)-wellposed.