Two-sided estimates of fundamental solutions of second-order parabolic equations, and some applications
F O Porper, Samuil D. Eidelman · Russian Mathematical Surveys · 1984
CONTENTS Introduction Chapter I. (Subsidiary). Fundamental solutions of second-order parabolic equations ??1. Fundamental solutions ??2. The Cauchy problem Chapter II. Harnack's inequality ??1. Proof of Harnack's inequality ??2. Consequences of Harnack's inequality Chapter III. Two-sided estimates of classical fundamental solutions ??1. Preliminary estimates ??2. An upper estimate of the fundamental solution ??3. A lower estimate of the fundamental solution Appendix Chapter IV. Estimates of the derivatives of classical fundamental solutions of stationary equations ??1. Estimates of the time derivatives of fundamental solutions ??2. Integral estimates of special differential expressions of fundamental solutions Chapter V. Weak fundamental solutions of parabolic and elliptic equations with measurable coefficients ??1. Definitions. Energy inequalities ??2. Weak fundamental solutions of the Cauchy problem ??3. Weak fundamental solutions of stationary parabolic and elliptic equations Chapter VI. Generalizations. Applications ??1. Generalization to the case of equations containing lower-order derivatives ??2. The asymptotic proximity of solutions of the Cauchy problem. A stabilization theorem Comments References