The heat equation $u_t - \Delta u = 0$ models diffusion of eg. a chemical in some domain. This equation comes from considering $$\frac{d}{dt}\int_\Omega u = \int_{\Omega} \nabla \cdot F$$ where the flux $F$ is chosen $F=\nabla u$.
If we choose $F= |\nabla u|^{p-2}\nabla u$, $p \geq 2$, we get the $p$-Laplacian equation $$u_t - \Delta_p u = 0.$$ What does this equation supposed to model?
I have heard "it models diffusion at a slow rate if $p \geq 2$". Is it right? I don't see what $p$ has to do with the rate of dispersion.
The diffusion is proportional to the gradient of the concentration: $$ -k\,\nabla u. $$ The minus sign comes from the fact that diffusion goes from higher to lower concentration. $k$ depends on the chemical being diffused and on the medium in which the dispersion takes place. If $k$ is constant, we get the equation $u_t-k\,\Delta u=0$. But $k$ may deppend for instance on the direction of the diffusion (anisotropic medium). Then we get the equation $u_t=\nabla\cdot(k(x)\,\nabla u)$. $k$ may depend also on the concentration $u$. For instance, if $k=u^p$, we get the porous medium equation $u_t=\nabla\cdot(u^p\nabla u)$. In the $p$-Laplacian equation, $k$ depends on the gradient of the concentration. If $p>2$, then $|\nabla u|^{p-2}$ is small if the gradient is small. This implies that the diffusion is slow.