Motivation for weak solution of a PDE (initial condition)

Question

The following question came to me when reading the famous paper of ALT and LUCKHAUS: "Quasilinear elliptic-parabolic differential equations"

When looking at a (nonlinear degenerate) PDE like

$$ \partial_t u - div (A(x,t,u,\nabla u)) = f(x,t) \text{ in } \Omega_T$$ $$ u(x,0) = u_0 \text{ in } \Omega$$ $$ u(x,t) = 0 \text{ in } \partial \Omega \times (0,T) $$ $$ A(x,t,u,\nabla u) \cdot n = 0 \text{ in } \partial \Omega \times (0,T) $$

one defines as weak solution as function u which satiesfies the following 2 equations:

$$ \int_{\Omega_T} \partial_t u \, \phi dxdt + \int_{\Omega_T}(u - u_0) \partial_t \phi dxdt = 0$$ $$ \int_{\Omega_T} \partial_t u \, \phi dxdt + \int_{\Omega_T} A(x,t,u,\nabla u) \nabla \phi dxdt = \int_{\Omega_T} f \phi dxdt $$

Now I wonder about the first one. The deduction of the second is ok to me, as I can use the standard way (applying Gauß to the original PDE) - But I can't seem to find the deduction / motivation of equation 1.

Any help or reference is welcome.

Answer 1

The motivation is the initial data. Depending on the regularity of the solution it allows to give a weak sense to it.

Answer 2

It is just a mistake, as there are too many in math literature. the first one should read $$\int_{\Omega_T}(\phi\partial_tu+u\partial_t\phi)\,dxdt+\int_{\Omega}u_0\phi(0,x)\,dx=0.$$ In practice, one writes only one identity, not two: $$\int_{\Omega_T}(u\partial_t\phi-A(x,t,u,\nabla u)\cdot\nabla\phi+f\phi)\,dxdt+\int_{\Omega}u_0\phi(0,x)\,dx=0.$$