Backward uniqueness vs well-posedness of the backward heat equation vs time reversibility for the heat equation

Question

I am trying to understand the differences between the following three concepts related to the heat equation:

1. Backward Uniqueness: The heat equation has backward uniqueness, meaning that if the solution at some final time $t = T$ is known, the solution is uniquely determined backward in time.

2. Well-posedness of the Backward Heat Equation: The backward heat equation is not well-posed due to instability, even though the solution might exist in a weak sense.

3. Time Reversibility: The heat equation is not time-reversible, meaning that running the solution backward in time does not necessarily recover the original solution, due to the dissipative nature of the equation.

All of these concepts seem to concern the "backward nature" of the solution in some way, yet they are distinct. Could someone explain the subtle difference between these three statements? Specifically:

• Why does backward uniqueness hold for the heat equation, but the backward heat equation is not well-posed?
• How does time reversibility differ from backward uniqueness and well-posedness?

I am interested in understanding the mathematical distinctions and the intuition behind why these concepts behave differently.

An example of a PDE where all three of these hold would be greatly appreciated.

Thanks in advance...

Answer 1

For instance, consider the heat equation on the real line, with initial condition $f(0,x)=\delta_0(x)$, then its solution is a gaussian $f(t,x)=\frac1{\sqrt{2\pi t}}\exp(-x^2/(2t))$ which is smooth in $x$ for any $t>0$. If you consider the state at $t=1$, you can predict the evolution back in time (so you have backward uniqueness), but you will get a delta function at the past time $t=0$ (so you don't have well posedness of the backward equation). Finally it is clear that you don't have time reversibility because in the future direction the variance increases. An example of a pde with all the properties you mention is $\partial_t f(t,x)=0$ or for a less trivial example, the wave equation $(\partial_t^2-\partial_x^2)f(t,x)=0$ should work.