Consider the initial value problem
$$ \partial_t u = \partial_{xx} u$$
$$ u(0,x) = u_0(x)$$
for the heat equation in one dimension, where $u_0: {\bf R} \to {\bf R}$ is a smooth initial datum and $u: [0,+\infty) \times {\bf R} \to {\bf R}$ is the smooth solution. Under reasonable growth conditions on $u_0$ and $u$ (e.g. $u_0$ and $u$ are at most polynomial growth), there is a unique solution $u$ to this problem given by the classical formula
$$ u(t,x) = \frac{1}{\sqrt{4\pi t}} \int_{\bf R} e^{-|x-y|^2/4t} u_0(y)\ dy.$$
However, as is well known, once one allows $u_0$ or $u$ to grow sufficiently rapidly at infinity, then smooth solutions to the heat equation are no longer unique, as demonstrated first by Tychonoff in 1935.
My question then concerns the corresponding question for existence : does there exist smooth initial data $u_0$ for which there are no global smooth solutions $u$ to the initial value problem for the heat equation?
One obvious candidate for such "bad" data would be a backwards heat kernel, such as
$$ u_0(x) = e^{|x|^2/4}.$$
One can verify that
$$ u(t,x) = \frac{1}{(1-t)^{1/2}} e^{|x|^2/4(1-t)}$$
is a smooth solution to the initial value problem for the heat equation with initial datum $u_0$ up to time $t=1$, at which point it blows up (rather dramatically). However, this does not fully solve the problem due to the aforementioned lack of uniqueness; just because this particular solution $u$ blows up, there could be some other more exotic smooth solution with the same data which somehow manages to retain its smoothness beyond the time $t=1$. This seems highly unlikely to me, but I was not able to demonstrate such an "unconditional non-existence" result – the absence of any growth hypotheses at infinity seems to destroy most methods of controlling solutions, and could potentially create some strange scenario in which one could continually keep singularities from forming by pumping in infinite quantities of energy from spatial infinity in just the right manner. But perhaps there is some literature on this problem?
Best Answer
It is true that for any initial datum $u_0\in C^\infty(\mathbb{R})$ there exists a solution $u\in C^\infty(\mathbb{R}^+\times\mathbb{R})$ to the heat equation with initial condition $u(0,x)=u_0(x)$. As you point out, this will not be unique.
I can give a method of constructing such solutions now. The idea is to show that we can write $u=\sum_{n=1}^\infty f_n$ where $f_n$ are carefully constructed solutions chosen such that the partial sums $\sum_{n=0}^mf_n(0,x)$ eventually agree with $u_0(x)$ any bounded subset of the reals, and for which $f_n$ tends to zero arbitrarily quickly in the compact-open topology. First a bit of notation. I use $\mathbb{R}^+=[0,\infty)$ for the nonnegative reals. For a space $X$ then $C_0(X)$, $C^\infty(X)$, $C^\infty_0(X)$, and $C^\infty_K(X)$, represent the continuous real-valued functions on $X$ which are respectively vanishing at infinity, smooth, smooth and vanishing at infinity, and smooth with compact support. Let $(P_t)_{t\geq 0}$ be the kernels $$ \begin{align} &P_t\colon C_0(\mathbb{R})\to C_0(\mathbb{R}),\\\\ &P_tu(x)=\frac{1}{\sqrt{4\pi t}}\int_\mathbb{R}e^{-(x-y)^2/4t}u(y)\,dy \end{align} $$ for $t > 0$, and $P_0u=u$. This is the Markov transition function for Brownian motion (more precisely, for standard Brownian motion scaled by $\sqrt{2}$, because of the normalization used here). For $u\in C_K^\infty(\mathbb{R})$, then $f(t,x)=P_tu(x)$ is in $C_0^\infty(\mathbb{R}^+\times\mathbb{R})$, and is a solution to the heat equation with initial condition $f(0,x)=u(x)$, agreeing with the classical solution stated in the question. I'll also consider initial conditions $u\in C^\infty_K((a,\infty))$ (for $a\in\mathbb{R}$) by setting $u(x)\equiv0$ for all $x\le a$. The first step in the construction is to find initial conditions supported in $(a,\infty)$ so that $P_tu(0)$ approximates any given continuous function of time that we like.
Consider the closure, $V$, in $C_0((0,\infty])$ (under the uniform norm) of the space of functions $t\mapsto\sqrt{4\pi t}P_tu(0)$ for $u\in C^\infty_K((a,\infty))$. Note that the limit as $t\to\infty$ does always exist and is just the integral of $u$. Then $V$ is a closed linear subspace of $C_0((0,\infty])$. Consider a sequence $u_n\in C_0((a,\infty))$ tending to the delta function $\delta_b$ at a point $b > a$, in the sense that $u_n$ all have support in the same compact set, and converge in distribution to $\delta_b$. Then, from the expression defining $P_t$, $\sqrt{4\pi t}P_tu_n(0)$ converges uniformly over $t > 0$ to $\exp(-b^2/4t)$. So, the function $t\mapsto\exp(-b^2/4t)$ is in $V$. As the set of functions of the form $t\mapsto\exp(-b^2/4t)$ for $b > a$ is closed under multiplication and separates points, the locally compact version of the Stone-Weierstrass theorem says that $V=C_0((0,\infty])$. Then, (1) follows.
Choosing $0 < \epsilon < a$ so that the support of $u$ is contained in $(\epsilon,\infty)$, (1) implies that we can choose $u_n\in C^\infty_K((a,\infty)$ so that $\sqrt{t}P_tu_n(\epsilon)$ converges uniformly to $-\sqrt{t}P_tu(\epsilon)$ over $t\ge0$ as $n\to\infty$. Then, $f_n(t,x)\equiv P_t(u+u_n)(x)$ is a bounded solution to the heat equation with boundary conditions $f_n(0,x)=0$ for $x\le\epsilon$ and $f_n(t,\epsilon)=P_t(u+u_n)(\epsilon)$. It is then standard that the solution is given by an integral over the boundary, $$ f_n(t,x)=\int_0^t\frac{\epsilon-x}{\sqrt{4\pi (t-s)^3}}e^{-(\epsilon-x)^2/4(t-s)}\sqrt{s}P_s(u_n+u)(\epsilon)\frac{ds}{\sqrt{s}} $$ for $x\le0$. Differentiating this wrt $x$ and $t$ an arbitrary number of times, and using dominated convergence as $n\to\infty$, it follows that $f_n(t,x)$ (and all its partial derivatives) converge uniformly to zero over $x\le0$ and $t\in[0,T]$ as $n\to\infty$.
Set $u^+(x)=1_{\{x > 0\}}u(x)$ and $u^-(x)=1_{\{x < 0\}}u(x)$. Applying (2) to $u^+(x+a)$, there exists a sequence $u^+_n\in C^\infty_K(\mathbb{R})$ with supports in $(b,\infty)$ such that $P_t(u^++u^+_n)(x)$ tends to 0 uniformly over $x\le a$ and $t\in[0,T]$. Applying the same argument to $u^-(-x-a)$, there exists $u^-_n\in C^\infty_K(\mathbb{R})$ with support in $(-\infty,-b)$ such that $P_t(u^-+u^-_n)(x)$ tends to zero uniformly over $x\ge -a$ and $t\in[0,T]$. A sequence satisfying the statement of (3) is $u_n=u^+_n+u^-_n$.
We can inductively choose a sequence $u_n\in C^\infty_K(\mathbb{R})$ such that $\sum_{m=1}^nu_m(x)=u(x)$ for $\vert x\vert\le n+1$ and $n\ge1$. Let $u_1=u$ on $[-2,2]$ and then, for each $n\ge2$, apply the following step.
Setting $f_n(t,x)=P_tu_n(x)$, then $f_n$ are smooth functions satisfying the heat equation and the initial conditions $\sum_{m=1}^nf_m(0,x)=u(x)$ for $\vert x\vert\le n+1$. Also, by the choice of $u_n$, for $n > 1$ then $f_n$ together with all its derivatives up to order $n$ is bounded by $2^{-n}$ on $[0,n]\times[1-n,n-1]$. As $\sum_n2^{-n} < \infty$, the limit $f=\sum_nf_n$ exists and is smooth with all partial derivatives commuting with the summation. Then $f$ has the required properties.