Error Bounds in the Final Value Problem for the Heat Equation

Consider the following problem. Given the positive constants $\delta $, M, T and $f(x)$ in $L^2 (\Omega )$, find all solutions of $u_t = \Delta u$ in $\Omega \times (0,T]$, $u = 0$ on $\partial \Omega \times (0,T]$, such that $\| {u( \cdot ,T) - f} \|_{L^2 } \leqq \delta $, $\| {u( \cdot ,0) - f} \|_{L^2 } \leqq M$. It is known that if $u_1 (x,t)$, $u_2 (x,t)$ are any two solutions, then \[ \left\| {u_1 ( \cdot ,t) - u_2 ( \cdot ,t)} \right\|_{L^2 } \leqq 2M^{{{(T - t)} / T}} \delta ^{{t / T}} .\] Let N be the dimension of $\Omega $, q an integer $ \geqq 0$, and let $\sigma > {N / 2} + q$. We show that there is a constant K such that for $0 < t < T$, \[ \mathop {\max }\limits_{| \beta | \leqq q} \left\| {D^\beta u_1 ( \cdot ,t) - D^\beta u_2 ( \cdot ,t)} \right\|_\infty \leqq K\left\{ {t^{{{ - \sigma } / 2}} + (T - t)^{{{ - \sigma } / 2}} + \left( {\frac{{\log ({M / \delta })}}{T}} \right)^{{\sigma / 2}} } \right\}M^{{{(T - t)} / T}} \delta ^{{t / T}} .\]