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 $$\begin{aligned} \partial _t u^q - {{\,\textrm{div}\,}}{\big (|\nabla u|^{p-2}\nabla u \big )}=0 \qquad \text{ in } \Omega _T, \end{aligned}$$ ∂ t u q - div ( | ∇ u | p - 2 ∇ u ) = 0 in Ω T , with $$q>0$$ q > 0 and $$p>1$$ p > 1 and $$\Omega _T:=\Omega \times (0,T)\subset \mathbb {R}^{n+1}$$ Ω T : = Ω × ( 0 , T ) ⊂ R n + 1 . Instead of requiring a lower bound for the sub- or super-solutions in the whole domain $$\Omega _T$$ 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.