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LECTURE 1 INTRODUCTION TO NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS I. NONLINEAR DIFFUSION EQUATION

I am going to start a series of lectures on nonlinear partial differential equations. Advancements made in the theory of nonlinear PDEs is one of the main achievements in XX century mathematics. Let me first make a remark on linear PDEs. In some sense one can say that there is a complete theory of linear PDEs, and perhaps the best source would be the four volumes of "The Analysis of Linear Partial Differential Operators" by Lars Hörmander [3]. Importance of the nonlinear PDEs is associated with the fact that they reflect more precise nonlinear laws of the nature, while linear PDEs were derived from the linearized versions of the nonlinear laws. Despite the great advances in the theory of nonlinear PDEs it is far from being complete. It is remarkable that almost every major nonlinear PDE has its own personality and requires unique approach. However, looking to current state of art in the theory of nonlinear PDEs, one can observe a similarity with the classical linear theory in the following sense: there are some major individual nonlinear PDEs, and analysis and understanding of those PDEs is a key towards the general theory of a broad class of nonlinear PDEs. I am going to concentrate on one of those key nonlinear PDEs called the nonlinear diffusion equation, which is a generalization of the classical heat/diffusion equation. I will use the following notation