This textbook covers the main graduate themes of partial differential equations, with a clear and rigorous language. It is easy to follow thanks to the examples they use to introduce the several isues, and it contains the proofs of the main theorems or gives the adequate bibliografy. We could say that the book is written with the study of the shockwaves as a common guideline thru the chapters.

The reader needs to know the differential and integral calculus, which is obvious, specially Stokes' Theorem. Also depending on the chapter, he needs to know non-linear autonomous systems of ordinary differential equations which is used on the analysis of traveling waves, or the Lebesgue integral to follow the weak solutions.

About the topics covered in the book, the main objection would be the items which doesn't appear, as for example, in the first order PDE, the general equation is missing. Nonetheless, the linear and quasilinear equation is enough for the rest of the book. Also, the Sturm - Liouville theory is scarcely treated with only theorem 7.8 as a guideline. However, as before, there is a good deal of Fourier series and its properties of convergence. Due to this, the Green function has to do with the PDE instead of the Sturm - Liouville problem, which is the usual in the literature.

Reviewer:

José Luis Guijarro Regalado