How can I show that \int_{0}^{t}\int_{M}|\nabla u^{q}|^{p}\leq t \int_{M}|\nabla u^{q}|^{p}(0) for weak solutions of \partial_{t}u=\Delta_{p}u^{q} on a Riemannian manifold? Please explain also the notion of weak solutions and do not use a Sobolev inequality on the manifold. Can I use as a test function f_{n}(t)u^{q} where f is a function such that f_{t}(t)=1/n*(t-t_{1}) whenever t_{1}\leq t\leq t_{1}+n
MathBot Answer:
MathBot is working on a solution to your problem.
