The weak subsolutions and supersolutions of very general parabolic nonlocal equations \[ \frac{\partial u}{\partial t}+ \mathcal{L}_p u(x,t)- \operatorname{div}\mathcal{B}_p(x,t,u,\nabla u)= g(x,t,u) \] are studied in space \(\times\) time cylinders \(\Omega\times(0,T)\), where \(\Omega\subset\mathbb{R}^N\). A special case is the equation with the operators \[ \mathcal{L}_p u(x,t)= \int_{\mathbb{R}^N} \frac{|u(x,t)- u(y,t)|^{p-2}(u(x,t)- u(y,t))}{|x-y|^{N+ps}}\,dy, \] where \(0<s<1\), \(1<p<\infty\) (the principal value of the integral is taken), and \[ \mathcal{B}_p(x,t,u,\nabla u)= |\nabla u|^{p-2}\nabla u. \] (There is a misprint in formula (1.2), defining \(\mathcal{L}_p\).)
The weak subsolutions are proved to be locally bounded. The theorem comes with an estimate; the cases \(2N/(N+2)<p\) and \(1<p\le 2N/(N+2)\) have somewhat different bounds.
The semicontinuity and pointwise behaviour of the weak supersolutions is studied for the pointwise defined representation \[ u(x,t)=\operatorname{ess}\liminf_{\substack{(y,\tau)\to(x,t)\\ \tau<t}} u(y,\tau). \] The assumption \(g=0\) is required for the deeper results. Energy estimates and a technically advanced variant of De Giorgi’s method are used.