Structure of positive solutions to second order elliptic and parabolic partial differential equations

M.Murata和K.Ishige研究了二阶抛物型方程Cauchy问题非负解的唯一性，并通过方程的内在度量给出了方程系数在无穷远处的最优增长率。他们进一步深入地研究了唯一性问题，建立了一个统一了以往所有唯一性结果的尖锐而一般的唯一性定理。这一结果得到了a.a ancona、L.Salloff-Coste、k.t。m . murata在二阶椭圆方程正解的Martin理论中引入了半小摄动的概念，建立了半小摄动下正解结构的稳定性，并给出了摄动为半小的充分条件。该结果与二阶抛物型方程Cauchy问题非负解的非唯一性、概率论中扩散过程的寿命估计以及椭圆型方程正解的结构有关。它得到了Y.Pinchover和R.O.Pinsky等专家的高度评价，最近H.Aikawa和Y.Pinchover也给出了相关的结果。M.Murata和K.Kurata还出版了一本关于椭圆型和抛物型偏微分方程理论的书，这是上述和其他偏微分方程研究的基础。kurata也给出了关于椭圆方程的几个结果。

M.Murata and K.Ishige studied uniqueness of nonnegative solutions of the Cauchy problem to second order parabolic equations, and gave optimal growth rates at infinity of the coefficients of equations via intrinsic metrics for equations. They further investigated the uniqueness problem deeply, and established a sharp and general uniqueness theorem which unifies all previous results on the uniqueness. This result was reviewed by such experts as A.Ancona, L.Salloff-Coste, K.Th. Sturm, E.B.Davies, A.Grigor'yan, and has been submitted.M.Murata introduced a notion of semismall perturbation in the Martin theory for positive solutions of second order elliptic equations, established stably of the structure of positive solutions under semismall perturbations, and gave sufficient conditions for perturbations to be semismall. This result is related to non-uniqueness of nonnegative solutions of the Cauchy problem to second order parabolic equations, lifetime estimates of diffusion processes in probability theory, and the structure of positive solutions to elliptic equations. It is highly estimated by such experts as Y.Pinchover and R.O.Pinsky, and results related to it have been given recently by H.Aikawa and Y.Pinchover. M.Murata also published a survey on the structure of positive solutions to stationary Schrdinger equations.M.Murata and K.Kurata published a book on the theory of elliptic and parabolic partial differential equations which is fundamental for the above and other investigations on partial differential equations . K.Kurata also gave several results on elliptic equations.