Boundary behavior for the heat equation on the half‐line

The initial‐boundary value problem for the heat equation on x>0,t>0$$ \left\{x>0,t>0\right\} $$ with nonzero Dirichlet boundary data is studied rigorously, with emphasis on the behavior of the solution along the boundary of the spatiotemporal domain, that is, in the limits x→0+$$ x\to {0}^{+} $$ or t→0+$$ t\to {0}^{+} $$, including the case (x,t)→(0,0)$$ \left(x,t\right)\to \left(0,0\right) $$. Our results specify how the regularity of the solution at the boundary is dictated by the level of smoothness of the relevant initial and boundary data. To the best of our knowledge, for unbounded domains like the half‐line {x>0}$$ \left\{x>0\right\} $$, such results are not available in the literature. The corresponding analysis for the derivatives of the solution is also performed and yields additional compatibility conditions on the initial and boundary data which guarantee the extension of the solution to a C∞$$ {C}^{\infty } $$ function up to the boundary, that is, on x⩾0,t⩾0$$ \left\{x\geqslant 0,t\geqslant 0\right\} $$. The analysis relies on the explicit solution formula provided by the unified transform, also known as the Fokas method, whose rigorous derivation is presented. The main advantage of this formula, namely, its uniform convergence at the boundary, largely reduces the proof of our results to fundamental tools from real and complex analysis such as Lebesgue's dominated convergence theorem and Cauchy's theorem. Moreover, thanks to uniform convergence, the unified transform solution formula can actually be directly evaluated at the boundary to yield the prescribed initial and boundary data.