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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 $\frac{1}{s}+\frac{1}{s'}=1$

QUESTION:

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.

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