Existence of Non-trivial Solutions for the Backward Heat Equation with Boundary Conditions at $x = 0$

Question

Existence of Non-trivial Solutions for the Backward Heat Equation with Boundary Conditions at $x = 0$

I'm working on the backward heat equation and trying to determine whether there exist non-trivial solutions given the following boundary conditions at $x = 0$.

The PDE:

$$ u_t = -k u_{xx} $$ where $k > 0$ is a constant.

Boundary conditions at $x = 0$:

$$ u(0, t) = 0 $$ $$ \frac{\partial u}{\partial x}(0, t) = 0. $$

I'm not imposing any initial conditions. My goal is to find whether there exist non-trivial solutions that satisfy these boundary conditions.

What I've learned so far about the Backward Heat Equation (BHE):

1. Ill-posedness: I understand that BHE is ill-posed, meaning small changes in initial or boundary conditions can lead to significant variations in the solution, or the solution may not exist or be unique. This makes solving it particularly challenging.
2. Sensitivity to initial conditions: BHE is highly sensitive to initial data, but in my case, I am focusing on whether non-trivial solutions can be found without any initial conditions, only relying on the given boundary conditions at $x = 0$.
3. Separation of variables: I've attempted solving the equation using the method of separation of variables, but this approach leads to the trivial solution $u(x, t) = 0$ due to the boundary conditions at $x = 0$.

My Question:

Is it possible to find a non-trivial solution to this equation under these boundary conditions? If so, what methods or approaches should I consider? I am aware of the ill-posed nature of BHE, but I am seeking any non-trivial mathematical solutions regardless of physical interpretation.

I would appreciate any guidance or suggestions on how to proceed, particularly if there are less conventional methods that might help in this case.

Answer 1

Try looking at Tychonoff's construction which is displayed e.g, here:

Tychonoff construction

Journal Reference

Suggestions. Omit the initial term $k=0$ in the series expansion. To solve the backwards heat equation, note that the formal change of variables $x\to i x$ transforms the forward heat equation to its backwards counterpart. Fortunately that formal substitution actually makes sense when you insert it into the Tychonoff construction.

P.S. These weird solutions blow up in the space variables, which is why the uniqueness theory for the heat equation on infinite space requires growth restrictions.

P.P.S. In response to your query about alternative approaches that do not involve infinite sums, I would be surprised if there are indeed any simpler alternative approaches. Why? If for example there were a nice integral formula approach that was stable under perturbations of the initial data, then that would contradict the known instability properties of the backward heat equation. In general I believe that the backwards heat equation is most often studied using power series expansions that are based on the central trick of substituting $x\to ix$ (reducing it to the forward heat equation). If you find an alternative approach, that would, I think be noteworthy.

You may find the following article of interest.

B.Frank Jones, A fundamental solution for the heat equation which is supported in a strip, Journal of Mathematical Analysis and Applications, Volume 60, Issue 2, 1977, Pages 314-324, ISSN 0022-247X, https://doi.org/10.1016/0022-247X(77)90021-X.

In particular see Corollary 3 therein. The behavior at the origin is somewhat similar to what you seek. (Note however that it deals with the forward heat equation. You might ask yourself if the methods can be adapted to the backwards heat equation).

BTW Your usage of the phrase boundary condition is a bit confusing since I assume you are working on the entire real line, which has no boundary. Off hand I wouldn't know what would be a better phrase however.

Here is another article of interest that may perhaps have some methods adaptable to your problem.

JONES, B. FRANK, et al. “On the Existence of Kernel Functions for the Heat Equation.” Indiana University Mathematics Journal, vol. 21, no. 9, 1972, pp. 857–76. JSTOR, http://www.jstor.org/stable/24890353. Accessed 23 Oct. 2024.