Uniqueness of solutions of the Cauchy problem for parabolic equations degenerating at infinity

Abstract

This paper is devoted to the investigation of classes of uniqueness for classical solutions of the Cauchy problem $\begin{eqnarray} && \rho (t,x)\curpartial_{t}u=\sum\limits^{n}_{i,j=1}a_{ij}(t,x)\curpartial_{x_{i}} \curpartial_{x_{j}}u +\sum\limits^{n}_{i=1}b_{i}(t,x)\curpartial_{x_{i}}u+c(t,x)u,\\ && u\vert_{t=0}=u_{0}(x),\quad x\in R^{n} . \end{eqnarray}$ We study how the behaviour of the function $\rho(t,x)$ as $\vert x\vert \to \infty$ influences the classes of uniqueness of the Cauchy problem.

In particular the dependence of such classes on the number of the space coordinates is clarified. We also discuss the sharpness of the obtained results.