A weighted estimate of the gradient for generalized solutions of quasilinear degenerate parabolic equations (Q810747)

The authors prove a theorem on the existence of generalized solutions of quasilinear degenerate parabolic equations, a model example of which is \[ \partial u/\partial t=\sum^{n}_{i=1}(\partial /\partial x_ i)(| u|^{\sigma}\partial u/\partial x_ i)=f(x,t),\quad x\in \Omega,\quad \sigma >0;\quad u(x,0)=0,\quad u|_ S=0, \] \(\Omega\) is a bounded domain. The main point of the proof is establishing under some assumptions the following estimate for any solution \[ | u|^{\sigma - 2+1/\gamma} | u_ x|^ 2\leq M,\quad \gamma \in (0;0.5). \]