The monograph by Ladyzhenskaya is impossible for me to penetrate... I feel frustrated that such important and widely cited book has not got re-published with modern typography.

Thank you very much for your enlightening explanation.

Let $f(x)=\frac{\ln2}{x\ln^2 x}\,1(0<x<1/2)$. Could you explain how $\int_0^\varepsilon \varphi \circ f = \infty$ for sufficiently small $\varepsilon >0$?

You meant if well-posedness still holds if $b$ is locally integrable in space-time, right?

I am still not clear about what your question is.

I don't understand the sentence "integeability of the Lipschitz constant can replace in some cases uniformity of the Lipschitz condition...". Could you be more precise?

You can have a look at the works by Nicola Gigli on second-order differential calculus on some special metric spaces.

For interested readers, this question has got an answer at math.stackexchange.com/questions/3654495/….

The book is a classic and I guess most libraries (and researchers using PDE) would love to have its re-edited edition. Even just typesetting the book in LaTeX is already very beneficial very the communities. I guess just typesetting the book is not expensive at all, no?

The book [1] is importantc and widely cited. The biggest problem with this book is that its notations are old and very hard to read. I don't get why AMS has not released it with modern typesetting