Doubly nonlinear parabolic equations with Robin boundary conditions

摘要

In this article, we consider the nonlinear problem ∂u∂t=Au+V(x)um+p−2+λuqinΩ×(0,T),u(x,0)=u0(x)≥0inΩ,∇up−2∂u∂ν=β(x)up−1on∂Ω×(0,T).$$ left{begin{array}{cc}frac{partial u}{partial t}amp;amp;amp;#x0003D;mathbf{A}uamp;amp;amp;#x0002B;V(x){u}amp;amp;amp;#x0005E;{mamp;amp;amp;#x0002B;p-2}amp;amp;amp;#x0002B;lambda {u}amp;amp;amp;#x0005E;qkern0.30em amp;amp;amp; mathrm{in}kern0.30em Omega times left(0,Tright), {}uleft(x,0right)amp;amp;amp;#x0003D;{u}_0(x)ge 0kern0.30em amp;amp;amp; mathrm{in}kern0.30em Omega, {}{leftamp;amp;amp;#x0007C;nabla urightamp;amp;amp;#x0007C;}amp;amp;amp;#x0005E;{p-2}frac{partial u}{partial nu }amp;amp;amp;#x0003D;beta (x){u}amp;amp;amp;#x0005E;{p-1}kern0.30em amp;amp;amp; mathrm{on}kern0.3em mathrm{partial Omega}times left(0,Tright).end{array}right. $$ where Au=div(mum−1∇up−2∇u)$$ mathbf{A}uamp;amp;amp;#x0003D;operatorname{div}left(m{u}amp;amp;amp;#x0005E;{m-1}{leftamp;amp;amp;#x0007C;nabla urightamp;amp;amp;#x0007C;}amp;amp;amp;#x0005E;{p-2}nabla uright) $$ is the doubly nonlinear operator. Here, Ω⊂ℝN$$ Omega subset {mathbb{R}}amp;amp;amp;#x0005E;N $$ is a bounded domain with smooth boundary, m>0,10,λ∈ℝ,q>0$$ mamp;amp;0,1amp;amp;pamp;amp;N,Vin {L}_{loc}amp;amp;amp;#x0005E;1left(Omega right),mamp;amp;amp;#x0002B;p-2amp;amp;0,lambda in mathbb{R},qamp;amp;0 $$ and β∈Lloc1(∂Ω)$$ beta in {L}_{loc}amp;amp;amp;#x0005E;1left(mathrm{partial Omega}right) $$. We establish some sufficient conditions on the functions V(x),β(x)$$ V(x),beta (x) $$, and the exponents m+p$$ mamp;amp;amp;#x0002B;p $$ and q$$ q $$, so that the above problem has no positive solutions. Furthermore, various concrete potentials V(x)$$ V(x) $$ are taken into account to demonstrate applications of our main result.