Integrability and boundedness of local solutions to singular and degenerate quasilinear parabolic equations (Q1854082)

A general quasilinear parabolic equation of the form \[ u_t - \text{div} a(x,t,u, \nabla u) = b(x,t,u, \nabla u) \tag{1} \] is considered. This equation, under suitable structure conditions, is modeled after the equation \[ u_t - \text{div} (|\nabla u |^{p-2} \nabla u) = f(x,t) + \text{div} g(x,t), \] which is degenerate if \(p > 2\) and singular if \(p < 2\). The result gives the description of the quality of local weak solutions of (1) with respect to the parameters in the structure conditions. The conditions under which the weak solution of (1) is (i) locally bounded, (ii) belongs to \(L_{q,\text{loc}}\) for all \(q \geq 1\) or (iii) belongs to \(L_{q,\text{loc}}\) for \(q \in [1,q^*[\) with some \(q^* > 1\) are given. To prove the results, the author uses a local energy inequality and the estimates in \(L_q^{\text{weak}}\) spaces. The results are almost optimal in the sense that they agree with the well-known results in the linear case.