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


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.


Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.