I have the following problem:

Let the function $f$ be $C^1$ and real valued. let $u(x,t)$ be a solution to the semi linear heat equation given as

$\frac{du}{dt}-\frac{d^2u}{dx^2}=f(u)$.

Let $u(x,t)$ be defined on $[-1,1]\times [0,+\infty]\rightarrow R$.

Furthermore, $u$ are of $C^3$ class, satisfying the Neumann boundaries:

$\frac{du}{dx}(-1,t)=\frac{du}{dx}(1,t)=0, t\geq 0$.

Let $s>0$ and define the set $A=\{(x,t)|0\leq t\leq s, -1\leq x\leq 1\}$.

I should now proof that the connected set of $\{(x,t)| v(x,t)>0\}$ is intersecting with the set $\{(x,0)|-1\leq x\leq 1\}$.

The function $v(x,t)=\frac{du}{dx}(x,t)$ and satisfies that $v(a,s)=v(b,s)=0$ and $v(y,s)>0$ for every $a<y<b$.

The only thing I could find was that a connected space can not be seperated into two non-empty spaces, but I dont know how that can be used here.

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