In this article, we consider the nonlinear problem
{array∂u∂t=Au+V(x)um+p−2+λuqarrayinΩ×(0,T),arrayu(x,0)=u0(x)≥0arrayinΩ,array|∇u|p−2∂u∂ν=β(x)up−1arrayon∂Ω×(0,T).$$ \left\{\begin{array}{cc}\frac{\partial u}{\partial t}=\mathbf{A}u+V(x){u}^{m+p-2}+\lambda {u}^q\kern0.30em & \mathrm{in}\kern0.30em \Omega \times \left(0,T\right),\\ {}u\left(x,0\right)={u}_0(x)\ge 0\kern0.30em & \mathrm{in}\kern0.30em \Omega, \\ {}{\left|\nabla u\right|}^{p-2}\frac{\partial u}{\partial \nu }=\beta (x){u}^{p-1}\kern0.30em & \mathrm{on}\kern0.3em \mathrm{\partial \Omega}\times \left(0,T\right).\end{array}\right. $$
where
boldAu=divfalse(mum−1false|∇ufalse|p−2∇ufalse)$$ \mathbf{A}u=\operatorname{div}\left(m{u}^{m-1}{\left|\nabla u\right|}^{p-2}\nabla u\right) $$ is the doubly nonlinear operator. Here,
normalΩ⊂ℝN$$ \Omega \subset {\mathbb{R}}^N $$ is a bounded domain with smooth boundary,
m>0,0.1em10,0.1emλ∈ℝ,0.1emq>0$$ m>0,1<p<N,V\in {L}_{loc}^1\left(\Omega \right),m+p-2>0,\lambda \in \mathbb{R},q>0 $$ and
β∈Lloc1false(∂normalΩfalse)$$ \beta \in {L}_{loc}^1\left(\mathrm{\partial \Omega}\right) $$. We establish some sufficient conditions on the functions
Vfalse(xfalse),0.1emβfalse(xfalse)$$ V(x),\beta (x) $$, and the exponents
m+p$$ m+p $$ and
q$$ q $$, so that the above problem has no positive solutions. Furthermore, various concrete potentials
Vfalse(xfalse)$$ V(x) $$ are taken into account to demonstrate applications of our main result.