Schauder estimates and existence theory for entire solutions of linear parabolic equations (Q1174806)

The author studies the unique existence of entire solutions vanishing at infinity to the linear parabolic partial differential equation \[ {\mathfrak L}\varphi=\sum^ n_{i,j=1}a_{ij}\partial^ 2\varphi/\partial x_ i\partial x_ j+\sum^ n_{i=1}b_ i\partial\varphi/\partial x_ i+c\varphi-\partial\varphi/\partial t=f. \leqno (NH) \] By an entire solution, the author means a classical solution defined on \(\mathbb{R}^{n+1}\). The 2nd order homogeneous part of \({\mathfrak L}\) is uniformly elliptic, \(c\leq 0\), and \({\mathfrak L}\) approaches the heat operator \({\mathfrak L}_ 0=\Delta-\partial/\partial t\) at infinity. In order to specify the behavior of \(\varphi\) at infinity, Hölder spaces are introduced, normed by an additional weight of some power of \((1+| t|+| x|^ 2)\). The principal technique is the so-called method of continuity applied to the operator \({\mathfrak L}_ r=(1-r){\mathfrak L}_ 0+r{\mathfrak L}\), \(0\leq r\leq 1\), in the Hölder spaces. The Schauder a priori estimates for solutions to \({\mathfrak L}\varphi=f\) are first developed in these spaces. Secondly, the unique existence of solutions to the heat equation \({\mathfrak L}_ 0\varphi=f\) and the well- posedness are investigated by means of the heat kernel. With these preparations, the method of continuity is applied to solve the problem. Under a stronger assumption on the coefficients of \({\mathfrak L}\), an asymptotic relation between the solution of (NH) and the solution to the heat equation is investigated.