Consider $u_t = \Delta u + f(u)$ in $\mathbb{R}^n$ with $u(0) = u_0 \in L^\infty$ and $f$ locally Lipschitz.

There exists $T > 0$ (depending on $\|u_0\|_{L^\infty}$ and Lipschitz constant of $f$) such that a unique classical solution exists on $[0, T)$.

If $f(u)/u \to -\infty$ as $|u| \to \infty$ (negative feedback), then $T = \infty$ (global existence).

For $u_t = \Delta u + u^p$ on $\mathbb{R}^n$ with non-negative, non-trivial $u_0$:

• If $1 < p \leq 1 + 2/n$: All solutions blow up in finite time
• If $p > 1 + 2/n$: Small data gives global existence, large data blows up

The critical exponent $p_c = 1 + 2/n$ (Fujita exponent) is sharp.

For $u_t + F(u, Du, D^2u) = 0$ with $F$ smooth and satisfying structural conditions (parabolicity, growth bounds), local-in-time solutions exist in Hölder or Sobolev spaces via contraction mapping on solution operator written as integral equation.

For $u_t = \Delta\phi(u)$ with monotone $\phi$, weak solutions exist globally using compactness methods. Uniqueness may fail, but entropy solutions (satisfying additional inequalities) are unique.