Motivation for the fractional Sobolev spaces I am strying studying on my own parabolic PDEs throughout the book "Linear and Quasi-linear Equations of Parabolic Type" by  Vsevolod A. Solonnikov, Nina Uraltseva, Olga Ladyzhenskaya, but there is one thing that I do not understand: why we are looking for the solutions of linear and quasi-linear parabolic PDEs in fractional Sobolev spaces instead of the classical Sobolev spaces. I believe that there is some problem that we find when we try looking for these solutions in the classical Sobolev spaces, but I could not realize what is this problem and the only thing that I could find about the motivation for fractional Sobolev spaces is this. I will be grateful if someone can explain why we work in fractional Sobolev spaces instead of the classical Sobolev spaces.
 A: The simple answer is that  you can find better and sharper estimates using fractional spaces, or interpolation spaces. Let me give an example, our favorite parabolic pde:
\begin{align}
u_t=u_{xx}. 
\end{align}
When we denote by $S(t)$ the semigroup generated by the laplacian, we can solve the equation as
\begin{align}
u(t)=S(t)u_0.
\end{align}
Suppose we want to measure $u(t)$ in some Hilbert space $X$, and the initial condition is from a space $Y$, then we find
\begin{align}
||u(t)||_X\leq ||S(t)||_{L(Y,X)}||u_0||_Y.
\end{align}
The key question is now how the operator norm depends on time. For $X=H^2$ and $Y=L^2$, we know that the operator norm has a singularity of $t^{-1}$, but when we take $Y=H^2$ there is no singularity. Now what if we take an initial condition that is smoother then $L^2$, but not as smooth as $H^2$? How strong will the singularity be? In order to answer these questions, you need interpolation spaces between $H^2$ and $L^2$, in other words, you want to construct a family of spaces $H^\alpha$ in between $H^2$ and $L^2$, and fractional Sobolev spaces are a nice explicit way to construct these spaces. I do recommend to study these lecture notes, http://people.dmi.unipr.it/alessandra.lunardi/. Corollary 4.1.11 is the famous Ladyzhenskaja – Solonnikov – Ural’ceva-theorem, and the use of interpolation spaces becomes very clear here. 
