A comparison principle for doubly nonlinear parabolic partial differential equations

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 ∂ _t u^q - div (|∇ u|^p-2∇ u )=0 in Ω _T, with q>0 and p>1 and Ω _T:=Ω× (0,T)⊂ℝ^n+1 . Instead of requiring a lower bound for the sub- or super-solutions in the whole domain Ω _T , 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.