Existence theory for the exterior fractional heat problem

Establish a complete existence theory for solutions of the time-fractional heat equation on a half-line with prescribed boundary trace, including a sharply delineated admissible class of boundary data and verification of the boundedness regularity hypothesis.

Background

The paper proves uniqueness for the exterior half-line problem under an assumed Laplace-transform and boundedness condition, but it does not construct solutions for a precisely specified class of boundary traces. Although an inverse Laplace representation involving the M-Wright function is given, the authors do not verify that the resulting solution satisfies the standing regularity assumption. They indicate that a complete proof would presumably require Hölder-continuous boundary data satisfying g(0)=0.

References

We do not carry out this construction in detail here (in particular, we do not verify that the resulting $u$ satisfies Assumption~\ref{ass:regularity} for a specific, sharply delineated class of boundary data $g$). A complete existence proof, presumably requiring $g$ Hölder continuous with $g(0)=0$ along the lines of, is left for future work together with the existence question raised after Proposition~\ref{prop:wellposed}.

— Transparent Boundary Conditions for the Time-Fractional Heat Equation on Metric Graphs  (2610.06631 - Hassin et al., 5 Oct 2026) in Section 2, immediately after Proposition 2.2