A comparison principle for doubly nonlinear parabolic partial differential equations

Abstract

In this paper, we derive a comparison principle for non-negative weak sub- and super-solutions to doubly nonlinear parabolic partial differential equations whose prototype is (Formula presented.) with q>0 and p>1 and ΩT:=Ω×(0,T)⊂Rn+1. Instead of requiring a lower bound for the sub- or super-solutions in the whole domain ΩT, we only assume the lateral boundary data to be strictly positive. The main results yield some applications. Firstly, we obtain uniqueness of non-negative weak solutions to the associated Cauchy–Dirichlet problem. Secondly, we prove that any weak solution is also a viscosity solution.