without boundary conditions, isn't every positive constant function a positive weak solution of the Trudinger equation, that is:
$\displaystyle\int\int -u^{p-1}\phi_t + |\nabla u|^{p-2}\nabla u\cdot \nabla\phi\,dx\,dt =0$
for all non-negative $\phi\in C^{\infty}_0(\Omega\times(0,T))$? Cause we can use integration by parts on the first term and get $\displaystyle\int\int u^{p-1}_t\phi = 0$.
$p$ would have to be greater than 1 btw. The second term of the integrand vanishes and shouldn't blow up as we can bound it via Cauchy Schwarz by $|\nabla u|^{p-2}\nabla u\cdot \nabla\phi \leq |\nabla u|^{p-1}|\nabla \phi|$ which is ok as $p$ is greater than 1. Is this correct?
