As mentioned on Wikipedia, a fundamental solution for a linear differential operator $L$ is a function (or distribution) $G$ such that $$LG = \delta$$ which by linearity of $L$ gives the following property: to solve $$Lu = f$$ we can use a convolution, $$u = G \star f.$$
Take $L$ to be the differential operator defining the heat equation and consider the inhomogeneous problem: $$Lu = (\partial_t - \Delta)u = f. \tag{1}$$
From the above definition given on Wikipedia, one would expect that if $K$ is the heat kernel/fundamental solution to the heat equation, (1) can be solved as $$u = K \star f. \tag{2}$$.
But this is of course not true and one would need to use Duhamel's principle.
Since this property is not true for the heat equation, what justifies the heat kernel being called a fundamental solution?
