# Question on proper choice of test function

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.