A boundary value problem in a strip for partial differential equations in classes of tempered functions (Q5942092)

Let \(M_\gamma(D)\) be class of functions \(u(x,t)\in C^\infty(\overline D)\) defined in the strip \(D=\{(x,t):x\in \mathbb{R}\), \(0<t<1\}\) and satisfying in \(\overline D\) the following estimates \(|\partial^{j+k}u(x,t)/\partial x^j\partial t^k|\leq c_{jk}(1+|x|)^\gamma\). The differential equation \[ P_n(i\partial/\partial x,\partial/\partial t)u(x,t)=\partial^nu(x,t)/ \partial t^n+\sum_{j=1}^na_j(i\partial/\partial x)\partial^{n-j}u(x,t)/ \partial t^{n-j}=0,\tag{*} \] where \(u\in M(D)\equiv\bigcup_{\gamma\in \mathbb{R}}M_\gamma(D)\) and coefficients \(a_j(\xi)\) are polynomials in \(\xi\in \mathbb{R}\) with constant coefficients, is considered. The main condition on the operator \(P_n(i\partial/ \partial x,\partial/\partial t)\) is: the roots \(\lambda_1(\xi),\dotsc, \lambda_n(\xi)\) split in two groups \(\lambda_1(\xi),\dotsc,\lambda_m(\xi)\) and \(\lambda_{m+1}(\xi),\dotsc,\lambda_n(\xi)\) such that (i) \(\text{Re}\lambda_j(\xi)\leq\alpha\) for some \(\alpha\in \mathbb{R}\) and \(1\leq j\leq m\), \(\xi\in \mathbb{R}\), and (ii) \(\text{Re }\lambda_j(\xi)\to\infty\) as \(|\xi|\to\infty\) for \(m+1\leq j\leq n\). Problem A: Find a solution \(u\in M(D)\) to the equation (*) with the boundary conditions \(\partial^ju(x,0)/\partial t^j=f_j(x)\), for \(0\leq j\leq m-1\) and \(\partial^ju(x,1)/\partial t^j=f_{j+m}(x)\), for \(0\leq j\leq n-m-1\), where \(f_j\in M(\mathbb{R})\). It is proved that the homogeneous problem A has at most finitely many linearly independent solutions, and the nonhomogeneous problem A is solvable for any \(f_j\in M(\mathbb{R})\). The uniqueness theorem for the problem A is given, too. Some illustrative examples are considered.