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Incompatible initial and boundary values prevent corner continuity

Statement refuted

The claim refuted is that the interval Dirichlet heat problem on $(0, π) \times (0, T]$ with initial data $u (x, 0) = 1$ for $0 < x < π$ and lateral data $u (0, t) = u (π, t) = 0$ for $t > 0$ can be solved by a function $u$ that is continuous on $[0, π] \times [0, T]$ and attains its data: no such continuous function exists, because the two prescriptions disagree at the corners.

Facts & Assumptions

Given: $T > 0$ , the rectangle $[0, π] \times [0, T]$ , and the prescribed data $u (x, 0) = 1$ on $(0, π)$ , $u (0, t) = u (π, t) = 0$ on $(0, T]$ .

[F1]

Continuity at a point means the $ε$ - $δ$ condition of Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form; in particular, if $p_{k} \to p$ in the domain then $u (p_{k}) \to u (p)$ .

[F2]

In the parabolic-cylinder vocabulary the initial face is $\overline{Ω} \times {0}$ , the lateral face is $\partial Ω \times [0, T]$ , and the closure of the cylinder contains the corners $(\partial Ω \times {0})$ (Parabolic cylinder and parabolic boundary).

Counterexample

Given: $T > 0$ and the data $u (x, 0) = 1$ on $(0, π)$ , $u (0, t) = u (π, t) = 0$ on $(0, T]$ .

1.1F1F2given

If $u$ is continuous on $[0, π] \times [0, T]$ , then at the corner $(0, 0)$ the sequence $(1 / (k + 1), 0)$ lies in the initial face, converges to $(0, 0)$ , and $u (1 / (k + 1), 0) = 1$ for every $k \geq 0$ ; [F1] therefore forces $u (0, 0) = \lim_{k} u (1 / (k + 1), 0) = 1$ .

1.2F1F2given

Along the lateral face, the sequence $(0, T / (k + 1))$ converges to $(0, 0)$ with $u (0, T / (k + 1)) = 0$ for every $k \geq 0$ , so the same continuity forces $u (0, 0) = \lim_{k} u (0, T / (k + 1)) = 0$ .

2.1step 1.1step 1.2given

Steps 1.1 and 1.2 assign the two different values $1$ and $0$ to $u (0, 0)$ , a contradiction; hence no function continuous on $[0, π] \times [0, T]$ attains both data sets.

3.1step 2.1F2given∎

The obstruction is exactly the incompatibility of the limiting initial and lateral values at the parabolic corner, and it is independent of any interior solution theory: even a mild solution of the heat equation with these data cannot be made continuous up to the corner.

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Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)
Per Kristen Jakobsen, An Introduction to Partial Differential Equations (2019) (standard reference, not scraped)