Strong maximum principles for parabolic nonlinear problems with nonlocal inequalities together with integrals

The author has previously studied strong maximum principles for nonlinear parabolic problems with initial and nonlocal inequalities. The purpose here is to extend those results to strong maximum principles for nonlinear parabolic problems with nonlocal inequalities together with integrals. The results obtained in this paper can be applied in the theories of diffusion and heat conduction, since considered here integrals in nonlocal inequalities can be interpreted as mean amounts of the diffused substance or mean temperatures of the investigated medium.