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Let $u=u(x,t)$ a solution of $$\begin{cases}\partial _tu=\partial _{xx}u\\ u(x,0)=f(x)\end{cases}$$

I can compute the solution, but I can't interpret this sort of equation. For an ODE $v'(t)=f(v(t))$, I see it as : we look at the mouvement of a particle that has velocity position $v(t)$ and speed $f(v(t))$ at time $t$. But with PDE, I don't see how to interpret it. I know that the solution of such PDE (if I'm not mistaken) can also be see as a density of a Brownian motion. So maybe can some give me a concret interpretation of such equation (avoiding the technical problem of solvability).

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2 Answers 2

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What this equation says is that the time evolution (i.e. the change over time) of the heat distribution $u$ is related to the smoothness of its spatial distribution. $u(x,t)$ tends to change faster in $t$ when $u$ is oscillating rapidly in $x$, because then $u_{xx}$ tends to be larger; and conversely, $u(x,t)$ tends to change more slowly in $t$ when the oscillations of $u$ in $x$ are slow, because then $u_{xx}$ tends to be smaller.

Due to the way the signs work out in this equation, what this tends to do is smooth the solution out over time. Whereever rapid spatial oscillations occur, the large time derivative will kill those oscillations, because rapid spatial oscillations tend to correspond to large spatial fluctuations and you need a large change in your temperature distribution to kill those off. And where rapid spatial oscillations are absent, the temperature distribution does not change much (because $u_t$ is small, since $u_{xx}$ is small), and so the absence of oscillations will persist over time. So eventually all of the rapid spatial oscillations get killed off, leaving you with a very nicely regular temperature distribution. This is the smoothing property of the heat equation: if your initial temperature distribution has sharp corners or edges or rapid fluctuations, it rounds and evens those out over time.

This isn't a proof of the smoothing property; it's only intended to give you an idea of why the smoothing property makes sense from the equation alone.

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I can't see much to add to Gyu's answer, but if you want to look more into the "physics" of it, there is an empirical law called Fourier's law, that says that $\vec{j} = -\lambda\nabla\theta$, where $\vec{j}$ represents the density of the thermal flux, $\lambda$ represents the thermal conductivity and $\theta$ the temperature. :)

So the rate of change of the flux is related to the gradient of the temperature. This can maybe help you see why, after appropriate use of the divergence theorem, you will end up with a result relating $\theta_t$ and $\nabla^2\theta$ (which, in dimension 1, is simply $\theta_{xx}$) :).

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