Some problems for linear partial differential equations with constant coefficients in the entire space and for a~class of degenerate equations in a~halfspace

A. S. Kalashnikov

Sbornik. Mathematics, Tome 14 (1971) no. 2, pp. 186-198

Résumé

In the space $\mathbf R^{n+1}=\mathbf R_t^1\times\mathbf R_x^n$ we consider a linear partial differential equation with constant coefficients which is solvable in the leading derivative with respect to $t$. We prove that two problems with limit conditions as $t\to-\infty$ which are imposed on the Fourier transform $F_{x\to\sigma}[u(t,x)]$ and contain weight factors, are uniquely solvable in the class of functions $u(t,x)$ which for every $t$ belong to $L_2(\mathbf R_x^n)$ along with the derivatives appearing in the equation and which grow at an order no faster that $t$ as $t\to+\infty$ (in $L_2$). We apply these results to a class of equations in a halfspace which degenerate on the boundary hyperplane. Bibliography: 9 titles.