# Reference for the parabolic Cauchy problem on $\mathbb{R}^N$ or $\mathbb{T}^N$

I am searching for a reference for the general (uniformly) parabolic Cauchy problem of second order, that is

\begin{align*} \partial_t u - \sum_{1\leq i,j\leq N}\partial_{x_j}(a^{ij}\partial_{x_i}) + \sum_{1\leq i\leq N} b^i \partial_{x_i}u +cu &=f,\\ u(0,\cdot) = u^0, \end{align*} where the coefficients $a^{ij},$$b^i$ and data $f$, $u^0$ are smooth (and the usual uniform elliptic estimate for the $a^{ij}$s), but when this system is considered either on the whole space $\mathbb{R}^N$ or the torus $\mathbb{T}^N$ (so : no boundaries conditions).

This is a purely bibliographic question since I am quite sure (I think ...) that the analysis of this Cauchy problem should mimick exactly the initial-boundary value one on $[0,T]\times\Omega$, only dropping off the assumptions regarding the boundary (well, for $\mathbb{R}^N$ I imagine that one should add an assumption on the behavior of the data at infinity).

There is a bench of references for the boundary-value problem, but nothing for the torus and for the whole space the only one I ran into assumes that the coefficients do not depend on the time variable (Krylov).

Any simple references for this rather academic setting ?

Thanks.

• What kind of results are you looking for? – Nate Eldredge Aug 14 '17 at 13:51
• Existence and uniqueness of a smooth solution when the data is smooth + (weak) maximum principle. – Ayman Moussa Aug 14 '17 at 14:22
• I'm not next to my bookshelf, but I think that lots of standard PDE texts should cover this. Have you checked in Evans, for instance? – Nate Eldredge Aug 14 '17 at 19:10
• Keep in mind that the uniqueness is in general false on unbounded domains. For instance, there are nontrivial smooth solutions to the initial-value problem for the classical heat equation on $\mathbb{R}^n$ with initial values zero. To get uniqueness, you have to impose conditions on the solution (e.g. growth, positivity, integrability, etc), not just on the initial data. – Nate Eldredge Aug 14 '17 at 19:13
• Hi Nate and thanks for your answers. Yes, Evans was one of the first references I checked. In Section 7.1 on parabolic linear problems, he focuses on the boundary-value problem. Yes, for the whole space I expect some growth condition to ensure uniqueness. I think I found references for $\mathbb{R}^n$ (see below), as we say in french: c'est dans les vieux pots qu'on fait les meilleures soupes. – Ayman Moussa Aug 15 '17 at 7:37

I did not manage to find recent references but both books :

O.A. Ladyzhenskaya, N.N. Ural'tseva, "Linear and quasilinear elliptic equations" , Acad. Press (1968) (Translated from Russian)

A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)

state and prove a result of existence in the whole space (via fundamental solution), with uniqueness if the function is assumed not to grow too fast as expected.