We call $a^x$ as an exponential function. Many things like compound interest, bacteria growth etc follow this model of $ba^x$.
It's behaviour is $f(x)$ and all order derivatives will keep on increasing x increases and at any point proportional to $a^x$. To put it in another way, it will be a straight line with slope $b$ in a logarithmic scale.
Now let's take this function $f(x) = xa^x$. This won't be a straight line in log scale. It will concaving upward.
Is there any name for this particular curve?
I think the most apt name for such a function would be a "product exponential", as its famous inverse is sometimes called a product logarithm.