Book recommendation on Fourier analysis technique for PDE

Question

I want a graduate-level textbook which discusses Fourier analysis techniques for solving PDE. To elaborate what I want to study, consider the Laplace operator $-\Delta$. Since the differential operator transforms as a multiplication operator, Fourier analysis techniques can be useful. The following article by Terence Tao https://terrytao.wordpress.com/tag/rage-theorem/?fbclid=IwAR3ovIoURjm8yisDgJp1-Evd1xXcj4Jmfwk1P9kLtVFI1i0THaBGzNuVqrQ illustrates what I want to study. I am interested in "spectral multipliers", which I does not know the definition but it is like a function that is multiplied after Fourier transformation.

I am also interested in fundamental solution of Laplace equations, Helmholtz equations, etc.

Please suggest a book that deals with above topics. I have a strong background in Lebesgue integration and some knowledge in functional analysis.

Answer 1

I would suggest Michael Taylor's first book on PDEs ("Basic Theory"). In particular, the third chapter seems to contain everything that you're looking for.

Answer 2

I would suggest the book

R. Habermann, Applied Partial Differential Equations; with Fourier Series and Boundary Value Problems, 5th ed., Pearson Education Inc., 2013.

It covers the topic of hyperbolic, parabolic and elliptic PDE problems in a quite detailed manner, but learning from it remains simple (many figures and examples). Fourier techniques are one of its main resolution approaches, but Laplace transforms and the method of characteristics are also exposed.