A note on comparison principle for p -laplacian evolution type equation

In this paper we provide a comparison principle for the weak solutions u(· ,t),v(· ,t) of two similar evolution p-Laplacian equations, both with source terms in a divergent and non-divergent form. Once we treat with signal solutions defined in all space ℝ^n , for all t in a maximal existence interval [0,T_*) , the arguments presented here differ from the ones used to prove the comparison principle in bounded domains. We suppose p≥ n , p>2 and also consider some additional natural assumptions. The initial conditions u(· ,0) and v(· ,0) are supposed to belong to the space L^1(ℝ^n) ∩ L^∞(ℝ^n) . An useful proposition to prove the comparison principle will be presented and the contraction of the L^1 norm of u-v for a particular case will be shown.