Linear parabolic equations with singular lower order coefficients

In chapter one: we obtain the existence of the weak Green\u27s functions of parabolic equations with lower order coefficients in the so called parabolic Kato class which is being proposed as a natural generalization of the Kato class in the study of elliptic equations. As a consequence we are able to prove the existence of some initial boundary value problems. Moreover based on a lower and an upper bound of the Green\u27s function we prove a Harnack inequality for the non-negative weak solutions. In chapter two: we establish a lower and an off-diagonal upper bounds for the heat kernel of sum of square operators plus a potential with a minimum integrability assumption. We use these bounds to prove the Harnack inequality and continuity of weak solutions. Our main result also implies the continuity of weak solutions of parabolic equations with lower order coefficients in a suitable parabolic Morrey-Campanato space. In chapter 3: we study the parabolic equation div$(A\bigtriangledown u)+V\ u-u\sb{t}=0.$ The potential $V=V(x, t)$ is assumed to be in a parabolic Kato class characterized by an integrability condition involving heat kernels. Under this assumption and using a new method we prove the existence of Gaussian bounds for the fundamental solutions of the equation. In addition we give a sufficient condition for V so that the Gaussian upper bound is global in time. We also establish similar results for the heat kernels of some subelliptic operators