My textbook has this image of the fundamental solution of the heat equation $u_t - u_{xx} = 0$ for $0.25 \le t \le 10$:
It says that, when we deduce from the PDE that at locations where $u_{xx} < 0$ (so $u$ is convex in $x$) the solution decreases in time at fixed $x$ whereas, when $u_{xx} > 0$ (so $u$ is concave) the solution increases in time. The points of inflection, where $u_{xx}$ changes sign, separate regions where the solution is increasing in time from those where it is decreasing.
Convex vs. Concave:
Am I thinking about this incorrectly, or does this description not match the figure provided? For constant $x$ "beneath" the line (when $u$ is convex in $x$), we see that the solution decreases till about $t = 1$ and then actually increases. And, for constant $x$ "above" the line (when $u$ is concave in $x$), the solution actually decreases in time till about $t = 1$ and only then seems to increase.
Hmm, I'm not sure if I'm interpreting this incorrectly, or if this is an error by the author?
The author has accidentally switched the words "concave" and "convex", but if you ignore those two words and read the maths(which is what I did originally), he is correct :) Its nothing more than combining the PDE $u_t = u_{xx}$ with the fact that $u_t < 0$ means $u$ is decreasing (and similarly for increasing.)
I remember which way round things are by remembering that $x^2$ is convex, and $(x^2)'' > 0$.
I've also attempted to recreate the graph. Note the contour lines, showing the decrease in height at $x=0$.
Not quite the same, but hopefully this helps. From different angles-
Edit - I just found out that Geogebra has an online 3D grapher. It even has a mode for the oldschool red-blue 3D glasses. I recreated the graph again here, and this time its interactive.