Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 105 / 4 / b

The assumed linear estimate and the first bound giveChoose $b>2\alpha \Vert \psi {\Vert }_{H^1}$ and then choose $\tau >0$ so small that $\alpha \beta {\tau }^{1/2}{b}^3\le b/2$. This proves that $A$ maps $X_{b,\tau }$ into itself.

For $w,\tilde{w}\in X_{b,\tau }$, linearity of the heat equation and the second cubic estimate giveShrinking $\tau$ again makes $2\alpha \gamma {\tau }^{1/2}{b}^2<1$. Thus $A$ is a contraction of the closed ball $X_{b,\tau }$ in the Banach space $L^{\infty }((0,\tau );H_0^1(U))$.