On the solution of the heat equation using distribution theory

Question

The Green's function $$ \tag{1} \displaystyle K(t,x,y)={\frac {1}{(4\pi t)^{d/2}}}e^{-\|x-y\|^{2}/4t}$$ solves the heat equation $$ {\frac {\partial K}{\partial t}}(t,x,y)=\Delta _{x}K(t,x,y)\ $$ for all $t > 0$ and $x,y ∈ \mathbb{R}^d$, where $Δ$ is the Laplace operator, with the initial condition $$ \lim _{t\to 0}K(t,x,y)=\delta (x-y)=\delta _{x}(y) $$

Question. Is the heat kernel $(1)$ still of some use for different initial conditions? For instance, for a Gaussian initial condition or negative exponential is the solution still traceable to the heat kernel?

Answer 1

As I said in comments, the standard answer to this question is just to look up on a PDE textbook. For example, L.Evans "Partial Differential Equations", 2nd ed, treats this in detail in §2.3.1.

However, comments revealed that the actual question is another one, and it is more interesting. I am spelling this question out now.

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In literature, "fundamental solution of the heat equation" refers to two seemingly different objects.

1. a solution to the initial value problem, for $t\ge 0, x\in\mathbb R^d$, $$\begin{cases} \partial_t K = \Delta K, \\ K(0, x)=\delta(x).\end{cases}$$
2. a solution to the inhomogeneous equation, again for $t\ge 0, x\in\mathbb R^d$, $$\partial_t E - \Delta E =\delta(t)\delta(x).$$

What is the relation between $K$ and $E$?

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Let us discuss this at a formal level, without going into analytical details (function spaces, uniqueness, etc...).

Remark. These two fundamental solutions are used in solving different problems. Indeed, formally at least, the solution to $$ \begin{cases} \partial_t u=\Delta u, \\ u(0, x)=f(x), \end{cases} $$ is given by $u(t, x)=K(t, \cdot)\ast_x f$, whereas $$ \partial_t v - \Delta v = F$$ is given by $v(t, x)=E\ast_{(t,x)} F$. (Here, $\ast_y$ denotes the convolution product in the vector variable $y$).

An answer. Suppose you already know a fundamental solution $K$ in the sense of 1. above. Then define $$ E(t, x):=\mathbf{1}_{\{t>0\}} K(t, x).$$ This is a fundamental solution in the sense of 2. above. Indeed, $$ \begin{split} \partial_t E(t, x)&=\delta(t)K(0, x) +\mathbf{1}_{\{t>0\}} \partial_t K(t, x)\\ &= \delta(t)\delta(x) + \mathbf{1}_{\{t>0\}}\Delta K(t, x) \\ &=\delta(t)\delta(x) + \Delta E(t, x). \end{split} $$ The upshot of all this is the following. Suppose we know a fundamental solution like in 1. above. (This is what is commonly called "heat kernel"). We can then solve the general problem $$ \begin{cases} \partial_t w -\Delta w = F(t, x),& t\ge 0, x\in\mathbb R^d,\\ w(0, x)=f(x), \end{cases}$$ as $$ \begin{split} w(t, x)&= E\ast_{(t, x)} F + K\ast_x f \\ &=\int_{0}^\infty \int_{\mathbb R^d} \mathbf{1}_{\{t-\tau>0\}} K(t-\tau, x-\xi)F(\tau, \xi)\, d\tau d\xi + \int_{\mathbb R^d} K(t, x-\xi)f(\xi)\, d\xi, \end{split} $$ which indeed is exactly the formula we find on the aforementioned Evans, eq. (13) pg.49.