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Let us consider the heat equation
beginalign*
fracpartialpartial tu(t,x) & = Delta u(t,x), \
u(0,x) & = f(x)
endalign*
on the whole space $mathbbR^d$. If $f in L^p := L^p(mathbbR^d)$ for $p in [1,infty)$ or if $f in C_0(mathbbR^d)$, the long term behaviour of the solution $u(t) := u(t,cdot)$ is well-known:

• If $f in L^p$ and $p in (1,infty)$, then $u(t)$ converges to $0$ with respect to the $L^p$-norm and also with respect to the $L^infty$-norm as $ tto infty$.




• If $f in L^1$, then $|u(t)|_L^1 to |int_mathbbR^d f(x) , dx|$ as $t to infty$, and we do not have convergence of $u(t)$ with respect to the $L^1$-norm, in general. However, we still have $|u(t)|_infty to 0$.




• If $f in C_0(mathbbR^d)$, then also $|u(t)|_infty to 0$ as $t to infty$.

Now, I am interested in the long-term behaviour of $u(t)$ if $f in C_b(mathbbR^d)$, i.e. if $f$ is a bounded continuous function. (Note: In this case, it is maybe less clear what we mean by a solution of the heat equation, but we can still define a "solution" $u(t)$ as the convolution of $f$ with the heat kernel).

Uniform convergence of $u(t)$ as $t to infty$ is probably too much to expect for many functions $f$, but I would be interested in locally uniform convergence:

Question. Is there a characterization of those $f in C_b(mathbbR^d)$ for which $u(t)$ converges uniformly on compact sets to a constant function as $t to infty$?

I would expect that this has been studied somewhere in the literature, so my question is mainly a reference request.

Disclaimer. I discussed this question with a few experts on the matter, but they did not know the solution nor did they know a reference.

First of all, with initial data in $C_b$, classical solution exist, so there is no need for quotation marks. It is easy to see that the convolution of the initial data $f(x)$ with the Gauss–Weierstrass kernel $p_t(x)$ defines the unique bounded classical solution $u(t, x)$.

It is easy to see that $u(t,x) - u(t,y)$ converges to zero as $t to infty$: the functions $p_t(cdot - x) - p_t(cdot - y)$ converge in $L^1$ to zero as $t to infty$. It is only slightly more difficult to see that this convergence is locally uniform with respect to $x$ and $y$. Therefore, your question is equivalent to the following simpler one: for what initial data $f(x)$, the limit of $u(t, 0)$ exists as $t to infty$.

I doubt there is a simpler "if and only if" characterization. One can easily provide sufficient conditions, for example it is sufficient to assume that the mean value of $f$ over the ball $B(0, R)$ has a limit as $R to infty$ (by the same argument that one uses when proving that Cesàro convergence implies Abel convergence). And one can equally easily construct counterexamples.

I doubt that there is a characterization, but one thing I can say is that the complement of the set of such $f$ contains a dense open set.

Let $U(f)$ be the solution with initial condition $u(0,x) = f(x)$, and
$$G = f in C_b: lim_t to infty U(f)(t,cdot) textconverges uniformly on compact sets$$

Take some $f_0 in C_b$ such that $U(f_0)(t,0)$ does not converge as $t to infty$. Let $delta = limsup_t to infty U(f_0)(t,0) - liminf_t to infty U(f_0)(t,0) > 0$. Then for any $f in G$ and any $c ne 0$, if $|g- (f + c f_0)|_infty < |c|delta/2$
then $$limsup_t infty U(g)(t,0) - liminf_t to infty U(g)(t,0) ge |c|delta - 2 |g-(f+c f_0)|_infty > 0 $$