For $u_t = \Delta u + f(u)$ with $u(0) = u_0 \in C(\mathbb{R}^n)$ bounded, $f$ Lipschitz.

Step 1: Reformulate as integral equation via Duhamel's principle: $u(t) = e^{t\Delta}u_0 + \int_0^t e^{(t-s)\Delta}f(u(s))\,ds$ where $e^{t\Delta}$ is the heat semigroup.

Step 2: Define solution space $X_T = C([0,T]; L^\infty)$ with norm $\|u\|_T = \sup_{t \in [0,T]}\|u(t)\|_{L^\infty}$.

Define operator $\Phi: X_T \to X_T$ by: $\Phi[u](t) = e^{t\Delta}u_0 + \int_0^t e^{(t-s)\Delta}f(u(s))\,ds$

Step 3: Show $\Phi$ maps into $X_T$. Since $\|e^{t\Delta}v\|_{L^\infty} \leq \|v\|_{L^\infty}$ and $|f(u)| \leq C(1 + |u|)$: $\|\Phi[u](t)\|_{L^\infty} \leq \|u_0\|_{L^\infty} + CT\sup_s\|u(s)\|_{L^\infty} \leq M$ for $T$ small enough.

Step 4: Show $\Phi$ is contraction. For $u, v \in X_T$: $\Phi[u] - \Phi[v] = \int_0^t e^{(t-s)\Delta}(f(u(s)) - f(v(s)))\,ds$

Using Lipschitz property $|f(u) - f(v)| \leq L|u - v|$: $\|\Phi[u] - \Phi[v]\|_T \leq LT\|u - v\|_T$

For $T < 1/L$, $\Phi$ is a contraction.

Step 5: Banach fixed point theorem gives unique fixed point $u = \Phi[u]$ in $X_T$, which is the desired solution.

Regularity: Parabolic smoothing ensures $u \in C^\infty((0,T) \times \mathbb{R}^n)$ for $t > 0$.