Doubly nonlinear parabolic equations with Robin boundary conditions

Abstract

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).$$ \left\{\begin{array}{cc}\frac{\partial u}{\partial t}&amp;amp;amp;#x0003D;\mathbf{A}u&amp;amp;amp;#x0002B;V(x){u}&amp;amp;amp;#x0005E;{m&amp;amp;amp;#x0002B;p-2}&amp;amp;amp;#x0002B;\lambda {u}&amp;amp;amp;#x0005E;q\kern0.30em &amp;amp;amp; \mathrm{in}\kern0.30em \Omega \times \left(0,T\right),\\ {}u\left(x,0\right)&amp;amp;amp;#x0003D;{u}_0(x)\ge 0\kern0.30em &amp;amp;amp; \mathrm{in}\kern0.30em \Omega, \\ {}{\left&amp;amp;amp;#x0007C;\nabla u\right&amp;amp;amp;#x0007C;}&amp;amp;amp;#x0005E;{p-2}\frac{\partial u}{\partial \nu }&amp;amp;amp;#x0003D;\beta (x){u}&amp;amp;amp;#x0005E;{p-1}\kern0.30em &amp;amp;amp; \mathrm{on}\kern0.3em \mathrm{\partial \Omega}\times \left(0,T\right).\end{array}\right. $$ where boldAu=divfalse(mum−1false|∇ufalse|p−2∇ufalse)$$ \mathbf{A}u&amp;amp;amp;#x0003D;\operatorname{div}\left(m{u}&amp;amp;amp;#x0005E;{m-1}{\left&amp;amp;amp;#x0007C;\nabla u\right&amp;amp;amp;#x0007C;}&amp;amp;amp;#x0005E;{p-2}\nabla u\right) $$ is the doubly nonlinear operator. Here, normalΩ⊂ℝN$$ \Omega \subset {\mathbb{R}}&amp;amp;amp;#x0005E;N $$ is a bounded domain with smooth boundary, m>0,0.1em1<p<N,0.1emV∈Lloc1false(normalΩfalse),0.1emm+p−2>0,0.1emλ∈ℝ,0.1emq>0$$ m&amp;amp;gt;0,1&amp;amp;lt;p&amp;amp;lt;N,V\in {L}_{loc}&amp;amp;amp;#x0005E;1\left(\Omega \right),m&amp;amp;amp;#x0002B;p-2&amp;amp;gt;0,\lambda \in \mathbb{R},q&amp;amp;gt;0 $$ and β∈Lloc1false(∂normalΩfalse)$$ \beta \in {L}_{loc}&amp;amp;amp;#x0005E;1\left(\mathrm{\partial \Omega}\right) $$. We establish some sufficient conditions on the functions Vfalse(xfalse),0.1emβfalse(xfalse)$$ V(x),\beta (x) $$, and the exponents m+p$$ m&amp;amp;amp;#x0002B;p $$ 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.