Well-Posedness of the Cauchy Problem for Some Evolution Equations

Problem. Under what conditions is the above Cauchy problem wellposed ? This problem was studied by many authors. Some of them studied this problem for equations of more general form, namely for p-evolution equations. In the case where the coefficients are constant, moreover in the case where the coefficients are functions of only t as well, we know the necessary and sufficient condition for (1.1) to be well-posed. Namely Hadamard's condition, Petrowsky's theorem and so on are well-known. However, in the case where the coefficients are functions of x and t, the situation is much more complicated. One of the most important results already known is the theorem of