Degenerate parabolic equations with singular lower order terms. (Q2343874)

Existence and regularity are studied for suitably defined weak solutions to second-order degenerate parabolic equations with singular right-hand side of the form \[ \partial_t u - \operatorname {div}( a(x,t,\nabla u)) = \frac {f(x,t)}{u^\gamma}, \quad x\in \Omega, \quad t\in (0,T), \] supplemented with homogeneous Dirichlet boundary conditions and a non-negative initial condition \(u_0\in L^\infty (\Omega)\) which is bounded from below by a positive constant on each compact subset of \(\Omega \). Here, \(\Omega \) is a bounded open set of \(\mathbb {R}^N\), \(N\geq 2\), \(a\) is a Carathéodory function satisfying the Leray-Lions structure conditions for some \(p\geq 2\), \(f\in L^m(\Omega \times (0,T))\) for some \(m\geq 1\), and \(\gamma \) is a positive parameter. Existence is shown for any \(m\geq 1\) but the regularity of the weak solution is better if \(m\geq p(N+2)/[p(N+2)-N(1-\gamma)_+]\). Owing to the singularity of the right-hand side, an important step of the proof is to show positive lower bounds on compact subsets of \(\Omega \) which is achieved by a repeated use of Harnack's inequality.