\(L^ p\)-estimates of solutions of some nonlinear degenerate diffusion equations (Q1061310)

The author investigates the problem \(\partial u/\partial t-\Delta u^{m+1}+f(x,t,u)=0\) in \(\Omega\times (0,\infty)\) \(u(x,0)=u_ 0\geq 0\), \(u=0\) on \(\partial \Omega\). Here \(\Omega\) is a bounded domain in \(R^ N\) with smooth boundary, m is a positive constant and f satisfies appropriate regularity properties. Two related problems are discussed as well. The author then proves existence and uniqueness of (positive) weak solutions along with \(L^ p\)-estimates.