Let $\Omega$ be a $\mathcal{C}^2-$compact manifold in $\mathbb R^2$ and consider the non-homogeneous heat equation: $$ \partial_t v - \Delta v = f \quad \text{in } \ \Omega \times (0,T), $$ where $f \in L^1(\Omega \times (0,T))$. We can say that $v \in L^1(0,T;W^{1,s}_0(\Omega))$ for $s<2$ is a weak solution if: \begin{equation*} \tag{$\star$} -\int_0^T \int_{\Omega} v \partial_t \phi +\int_0^T \int_{\Omega} \nabla v\nabla \phi = \int_0^T \int_{\Omega} f \phi \end{equation*}

for every $\phi \in C^1((0,T);W^{1,s'}(\Omega))$ where $\frac{1}{s}+\frac{1}{s'}=1$


If $$ \psi(s) := \begin{cases} n &\text{if } s>n\\ s &\text{if } -n\le s \le n\\ -n &\text{if } s<-n \end{cases} $$ can we choose the test function for $(\star)$ to be: $\psi(v)1_{[0,\tau]}$, where $1_{[0,\tau]}$ is the indicator function for any $\tau \le T$ ? If the answer is negative, then can we assume sufficiently better regularity for $\phi$ in order to allow this specific test function?

I'm having a hard time getting my head around the regularity of $\phi$ and now with the indicator function also involved, I'm getting more confused.


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.