Lecture 3 Introduction to Nonlinear Partial Differential Equations Iii. Energy Estimates and Existence

In the first lecture I constructed instantaneous point source solution, or so called ZKB-solution of the nonlinear diffusion equation. This solution exhibits finite speed of propagation property which is in sharp contrast to classical linear heat/diffusion equation. In fact, ZKB solution is not a solution in a classical sense, and even first order derivatives are discontinuous along the interfaces or free boundaries separating the regions where solution is positive from the region where it is zero. To justify this important and physically relevant solution, we introduced the notion of the weak solution. Recall the definition of the weak solution to the Dirichlet problem: