Existence of solutions to a nonhomogeneous partial differential equation in the class of functions not growing faster than a polynomial (Q1819282)

Let \(M_{\gamma}\), \(\gamma\geq 0\), denote a class of functions \(u: R^ 2\to R\) such that \[ | D^ k_ x D^ m_ t u(x,t)| \leq c_{km}(1+| x| +| t|)^{\gamma},\quad m,k=0,1,2,... \] M\(=\cup_{\gamma \geq 0}M_{\gamma}\). The results of the paper are connected with the equation \[ (1)\quad \partial^ nu/\partial t^ n+\sum^{n}_{k=1}p_ k(i \partial /\partial x)\partial^{n- k}u/\partial t^{n-k}=f(t,x), \] \(p_ k\) being given polynomials. It is proved that if \(f\in M\) then (1) has a solution \(u\in M\). If the complete symbol \[ p(x,t)\equiv (-it)^ n+\sum^{n}_{k=1}p_ k(x)(-it)^{n- k}\neq 0\quad \forall (x,t), \] then the homogeneous equation (1) has no nontrivial solution in M, elsewise the solution space in M is infinite- dimensional. The solution space of the homgeneous equation in \(M_{\gamma}\) is finite-dimensional if and only if \(p(x,t)=0\) for a finite number of points \((x,t)\in R^ 2\).