Uniqueness of solution to heat equation when initial condition is a generalized function Let $u(t,x)$ be a solution to the heat equation $$\partial_t = \partial_{xx} \quad (t,x) \in [0,T) \times [-1,1]$$ subject to the initial/boundary conditions
$$u(0,x) = f(x), \quad x \in [-1,1], \\
u(t,\pm 1) = g^{\pm}(t), \quad t \in [0,T),$$ with the usual compatibility conditions in corners: $f(\pm 1) = g^{\pm}(0)$. Suppose also that $f$ and $g$ are bounded and continuous. Then one can invoke the maximum principle or the energy method to prove that $u$ is the only solution.
What happens when $f$ or $g$ are unbounded? Say for instance, when $f(x) = \delta_0(x)$ (point mass at zero) and $g^\pm \equiv 0$? This problem has a solution that can be easily represented as a series.
How does one go about proving uniqueness in such a situation?
In fact, come to think of it, how does one prove the uniqueness of the fundamental solution $v(t,x) = \exp \{-x^2 / (4t)\}/ \sqrt{4 \pi t}$?
Is it some kind of weak uniqueness, where you show uniqueness of all classical solutions resulting from mollification of initial/boundary conditions? Is that the best one can do?
Any references would be deeply appreciated.
 A: (Not sure if I understand the question correctly.)
If $p_{t,x}(y) = p(t, x, y)$ is the fundamental solution (a.k.a. the heat kernel), then $p_{t,x}$ converges as $t \to 0^+$ to the Dirac measure $\delta_0$ in the sense of weak* convergence of measures, and hence also in the sense of distributions.
For any initial value given by a distribution $f$ in $(-1,1)$ (say: compactly supported, but this can be extended slightly), $u(t, x) = \langle f, p_{t,x}$ makes sense. Then $u$ can be proved to solve the heat equation, and $u(t, \cdot)$ converges in the space of distributions to $f$ as $t \to 0^+$.
I do not know the literature well, but I would check any book on applications of the distribution theory to PDEs for more on that. For example, Section 16.7 in Vladimirov's Methods of the theory of generalized functions might be a good reference.
A: The fundamental solution of the heat equation is not unique: it is only unique modulo an entire solution of the heat equation, i.e. a solution of the heat equation which is analytic in the whole $\Bbb R_t\times \Bbb R_x \equiv\Bbb R^2$. For example we may consider the class of heat polynomials ([1], §1.4, pp. 17-18)
$$
p_n(t,x)= n! \sum_{k=0}^{[n/2]} \frac{t^k}{k!}\frac{x^{n-2k}}{(n-2k)!}
$$
and see that, if $v(t,x)$ is the standard fundamental solution of the heat equation recalled in the question,
$$
\partial_t \big(p_n(t,x)+v(t,x)\big)-\partial_{xx} \big(p_n(t,x)+v(t,x)\big)=\delta(t,x)=\partial_t v(t,x) -\partial_{xx} v(t,x)
$$
for all $n\in \Bbb N$, by linearity. However, this non uniqueness property is not a unique characteristic of the heat equation: the fundamental solution of every linear partial differential operators (if obviously existing) is always defined modulo a solution of the associated homogeneous partial differential equation. To obtain uniqueness, you should add other conditions which depend on the structure of the given differential operator.
Classically, uniqueness for the solutions to the Cauchy problem for the heat equations can be proved only assuming some restriction on the initial data and/or the non-homogeneous "forcing" term (if present): these restrictions are

*

*measurability of the data and

*spatial growth bounded by $e^{\varepsilon\Vert x\Vert^2}$ for an arbitrary $\varepsilon>0$ in any finite time interval $[0,T]$.

The theory is due to Andrei Tikhonov and is described, for example, in reference [1], §3.6, pp. 40-42 for the one spatial dimensional case and in the wonderful monograph of Vladimirov ([2] §16.7 pp. 228-229) also cited by Mateusz Kwaśnicki in his answer.  
Be it noted that, if the second condition is not fulfilled, then null solutions to the given Cauchy problem may occur ([1], §3.7, p. 42).
Reference
[1] John Rozier Cannon, The one-dimensional heat equation, Foreword by Felix E. Browder, (English) Encyclopedia of Mathematics and Its Applications, Vol. 23. Menlo Park, California etc.: Addison-Wesley Publishing Company; Cambridge etc.: Cambridge University Press, pp. XXV+483 (1984), ISBN: 0-201-13522-1, MR0747979, Zbl 0567.35001.
[2] Vassilij Sergeevič Vladimirov (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, Vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR2012831, Zbl 1078.46029.
A: I'm not a specialist in this area. So for beginning I recommend you to study the theory of equations with $L_1$ data in F.Petitta Not So Long Introduction to the Weak Theory of
Parabolic Problems with Singular Data (Chapter 4). There  is also an extensive literature on nonlinear equations with measure data, for example,

*

*P.Baras, M.Pierre Problems paraboliques semilineaires avec donnees measures. Applicable  Analysis,  1984, V.18, 11l-149,


*L.Boccardo, T.Gallouet Non-linear  Elliptic  and  Parabolic  Equations Involving  Measure  Data. J.  of  Functional  Analysis 1989, V.87,  149-169.
