On the wellposed Cauchy problem for some dispersive equations
The forward Cauchy problem for the degenerate parabolic operator with real coefficients \[ \partial_t u(t, x)+ a(x) \partial^2_x u(t, x)+ b(x) \partial_x u(t, x)+ c(x) u(t, x)= 0 \] is \(L^2\)- and \(H^\infty\)-well posed if and only if \(a(x)\leq 0\). The question which is studied in the present paper is what happens with the forward Cauchy problem if we add to the operator the dispersive term \(\partial^3_x u(t, x)\). The author shows that the condition \[ \int^y_x a(s) ds\leq K\;\Biggl( \int^y_x a(s) ds= K\log (y- x+ 2)\Biggr),\quad x\leq y \] is necessary and sufficient for the \(L_2\)-well posedness (\(H^\infty\)-well posedness). The necessity will be proved by the construction of asymptotic solutions. For the proof of the sufficiency, the author uses the fact that the Cauchy problem for the equation \[ \partial_t u(t, x)+ \partial^3_x u(t, x)+ a(x) \partial^2_x u(t, x)+ b(x) \partial_x u(t, x)+ c(x) u(t, x)= 0 \] is \(L^2\)-, \(H^\infty\)- and \(S\)-well-posed if \(a(x)\leq 0\). Under the above condition a suitable transformation transfers the original problem to the last one.
scientific article; zbMATH DE number 844446