# Existence of a Solution for a Nonlinear Heat Equation

Let $$\Omega \subset \mathbb{R}^n$$ be a bounded set with smooth boundary and let $$f:\mathbb{R}^+\rightarrow\mathbb{R}^+$$ be Lipschitz continuous, nondecreasing and $$f(0)=0$$. Consider the following boundary value problem \begin{aligned}u_t-\Delta u &= f(u), & &\text{in }\Omega\times(0,\infty) \\ u&= 0, &&\text{on } \partial\Omega\times(0,\infty) \\ u &= u_0 \geq 0, & &\text{on }\Omega\times\{t=0\}.\end{aligned}

In a paper that I'm reading the author uses the unique global classical solution to this problem, however, it is not clear to me why such a solution exists. Does someone know how to prove this or does someone know where I can find a proof for this?

So far I tried to construct a fixed point argument. For $$0 and $$<\alpha<1$$ let $$D_M$$ be the space of all functions $$v$$ on $$\Omega\times(0,\infty)$$ that are Hölder continuous with Hölder exponent $$\alpha$$, Hölder norm smaller than $$M$$, and $$v(x,0)=u_0(x)$$ on $$\Omega$$. We define a operator $$T$$ on $$D_M$$ such that $$w=Tv$$ is the solution to the linear problem

\begin{aligned}w_t-\Delta w &= f(v), & &\text{in }\Omega\times(0,\infty) \\ w&= 0, &&\text{on } \partial\Omega\times(0,\infty) \\ w &= u_0 \geq 0, & &\text{on }\Omega\times\{t=0\}.\end{aligned}

Parabolic Schauder results give us the existence of such a $$w$$ that is Hölder continuous with exponent $$\alpha$$. Now I'm stuck with proving that the Hölder norm of $$w$$ is also bounded by $$M$$.

## 1 Answer

Maybe you should have a look at Chapter 6: Some Nonlinear Evolution Equations, in the book:

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag New York, Inc, 1983.