Let us consider the following pde on the domain $(0,T)\times(0,1)$

$ \dot{p}(t,x)+v(t)p_{x}(t,x)+v'(t)\int_{0}^{1} \rho(t,s)p_{s}(t,s)\ ds=0 $

with initial data $p(0,x)=p_{0}(x)$ and boundary data $p(t,0)=a(t)$ both in a certain function space. Let furthermore $\rho\in L^{2}((0,T)\times (0,1))$ and $v\in C^{1}([0,T])$.

For initial and boundary data in $H^{1}(0,1), H^{1}(0,T)$ respectively, and the regularity assumption, that $p_{0}(0)=a(0)$ we can show that the solution is in the space

$ C([0,T]; H^{1}(0,1))\cap C([0,1]; H^{1}(0,T)). $

Now we are interested in a "weaker" solution something like $C([0,1],L^{2}(0,T))$ and vice versa. Usually you define such a solution by multiplying the pde with a more regular test function $w(x,t)$ and perform an Integration over $(0,T)\times (0,1)$. Then you shift the derivatives to the test function.

In the case under consideration we can't do this, since we can't shift the space derivative of $p$ in the integral.

So the question is, how we can define a weaker solution for this problem. Are there any references, where we can find more information to our problem...

Using Google and the standard text books was of no success.

Thank you very much

Alex