Uniqueness of solutions of diffusion equation with initial condition

Question

In his PDE, Walter A. Strauss claims that the diffusion equation on the whole real line has a unique solution, given an initial condition. However he only proves uniqueness given an initial-boundary condition for solutions on a finite interval (in section 2.3, using the maximum principle). Is this a gap or am I missing something obvious here? The passage I am referring to can be found on page 49 (section 2.4 Diffusion on the whole line) in the second edition.

Answer 1

It seems that he is only claiming here that $u$ is a solution of (1), (2). He does not prove uniqueness in the book, and it should probably be understood that a uniqueness result (with some qualifier) must be supplied by external sources (As a commenter said, there is no uniqueness without qualifiers). Keep in mind that this is not meant to be a rigorous graduate text, and at times you encounter something like this in the book.