By the method of up-sub solutions, we consider a class of parabolic equations with nonlocal source. In the paper, we discuss the relation of the coefficients and the importance of the initial value. We get the sufficient conditions for the global existence and finite blow-up of the solutions.
We study a nonlinear parabolic system governing the biological dynamic in the soil. We prove global existence (in time) and uniqueness of weak and positive solution for this reaction-diffusion semilinear system in a bounded domain, completed with homogeneous Neumann boundary conditions and positive initial conditions.
Weak Solution to a Parabolic Nonlinear System Arising in Biological Dynamic in the Soil