Continuity of polynomial solutions with respect to the coefficient of the higher derivative (Q1386814)

Consider the following partial differential equation with constant coefficients \[ \varepsilon D^2_t u+L(D_x)u -D_tu=0, \tag{1} \] where \(\varepsilon \in\mathbb{R}\), and \(L(D_x)\) is any partial differential operator with respect to \(x=(x_1, \dots, x_n)\) having constant coefficients and such that \(L(0)=0\). We are going to investigate polynomial solutions to (1) in order to find such a solution which for \(\varepsilon\to 0\) tends to some polynomial solution of (1) considered for \(\varepsilon =0\). For the method considered below it does not matter at all neither the type of equation (1) nor order of the operator \(L(D_x)\). This method is based on the construction of a 0-normalized system of polynomials.