 Solution

The Banach fixed-point theorem gives a unique $u\in X_{b,\tau }$ with $A(u)=u$. By the definition of $A$, this fixed point is a weak solution of the linear heat problem with forcing $f=-u^3$. It therefore satisfieswith the prescribed initial and homogeneous Dirichlet data. Hence the nonlinear heat equation has a local weak solution in $L^{\infty }((0,\tau );H_0^1(U))$.