In the literature about PDEs it is easy to find books that talk about weak solutions of a partial differential equations. A short reminder of the most usual defintiondefinition is given bellow.
Let's consider a problem:
$$(1) \hspace{1cm} u(x,t)_t + q(u(x,t))_x = f(u(x,t)), \; x \in \mathbb{R}, t \in [0,T], $$ $$(2) \hspace{1cm} u(x,0)=u_{0} (x), \; x \in \mathbb{R}. $$
In order to get the weak solution, we take test function of two variables $\psi(x,t)$ that is smooth and has a compact support on $\mathbb{R} \times [0,T)$, i.e. $\psi(x,t) \in C_{c}^{\infty}(\mathbb{R} \times [0,T))$ and multiply it with the equation (1). Than we integrate everything and perform the integration by parts. At the end, we get that the weak solution $u$ of a problem (1)-(2) is given with:
$$\int_{0}^T \int_{\mathbb{R}} [u \psi_{t} + q(u) \psi_{x}] \; dx dt + \int_{\mathbb{R}} u_{0}(x) \psi (x,0) \; dx = - \int_{0}^T \int_{\mathbb{R}} f \psi \; dx dt,$$
for every test function $\psi(x,t) \in C_{c}^{\infty}(\mathbb{R} \times [0,T))$.
On the other hand, not so usual definition of the weak solution of the above problem (that can be found for example in the papers of E. Feireisl) is given by:
$$\int_{0}^T \phi_t \int_{\mathbb{R}} u \varphi \; dx dt + \int_{0}^T \phi \int_{\mathbb{R}} u \varphi_x \; dx dt + \phi (0) \int_{\mathbb{R}} u_{0} \varphi \; dx = - \int_{0}^T \phi \int_{\mathbb{R}} f \varphi \; dx dt,$$
for every test function $\phi$ that is smooth and has compact support on $[0,T)$ and for every test function $\varphi$ that is smooth and has a compact support on $\mathbb{R}$. So here we used two test functions of single variables $\phi(t)$ and $\varphi(x)$ instead of one test function of two variables $\psi (x,t)$.
I have two questions.
What are the advantages/disadvantages of the second approach where we used two test functions $\phi(t)$ and $\varphi (x)$ comparing to the first approach where we used $\psi (x,t)$?
Could I used just $\phi(t)$ or just $\varphi (x)$ (I am not sure if that $u$ than counts as a weak solution)?
