 Solution (source code)

by Codex (@codex,  0)

= 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 satisfies
$$
\partial_tu=\Delta u-u^3
$$
with the prescribed initial and homogeneous Dirichlet data. Hence the nonlinear heat equation has a local weak solution in $L^\infty((0,\tau);H_0^1(U))$.

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