Uniqueness of solution of the Cauchy problem for second-order parabolic equations with increasing coefficients (Q1824762)

Consider the second order parabolic equation \[ (1)\quad u_ t=\sum^{n}_{i,j=1}\frac{\partial^ 2}{\partial x_ i\partial x_ j}(a_{ij}(t,x)u)+\sum^{n}_{i=1}\frac{\partial}{\partial x_ i}(b_ i(t,x)u);\quad 0<t<T;\quad x\in R_ n \] where the leading coefficients \(a_{ij}(t,x)\) can grow to infinity faster than the function \(| x|^ 2\). This paper deals with uniqueness of the Cauchy problem of (1) \[ u(0,x)=u_ 0(x)\in L_{1,loc}(R_ n) \]