Partial Differential Equations Example sheet 4

for some positive constants m,M and all x, t and ξ. The basic example is the heat, or diffusion, equation ut − ∆u = 0 , which we start by solving, first for x in an interval and then in R. We then show that in both situations the solutions fit into an abstract framework of what is called a semi-group of contraction operators. We then discuss some properties of solutions of general parabolic equations (maximum principles and regularity theory).