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) indicated by Mateusz Kwaśnicki in his answer above.

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.

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