Nonlinear parabolic equations with Robin boundary conditions and Hardy-Leray type inequalities

We are primarily concerned with the absence of positive solutions following problem, \[ { ∂ u t = mathvariant="normal">Δ 0 , T ≥<!-- ≥ <mml:mi>ν<!-- ν <mml:mi>β<!-- β <mml:mtext>on fence="true" stretchy="true" symmetric="true" \begin {cases} \frac {\partial u}{\partial t}=\Delta ( u^m)+V(x)u^{m}+ \lambda u^q &amp; \text {in}\quad \Omega \times (0, ) ,\\ u(x,0)=u_{0}(x)\geq {in} \quad \nu }=\beta (x) {on }\partial (0,T), \end {cases} \] where &gt; 1 encoding="application/x-tex">0&gt;m&gt;1 , alttext="upper element-of L Subscript l o c 1 right-parenthesis"> ∈<!-- ∈ <mml:msubsup> L l o c encoding="application/x-tex">V\in L_{loc}^1(\Omega ) alttext="beta encoding="application/x-tex">\beta \in L_{loc}^1(\partial alttext="lamda double-struck R"> mathvariant="double-struck">R encoding="application/x-tex">\lambda \mathbb {R} alttext="q 0"> encoding="application/x-tex">q&gt;0 alttext="normal subset-of R N"> ⊂<!-- ⊂ <mml:mi>N encoding="application/x-tex">\Omega \subset {R}^N is bounded open subset alttext="double-struck encoding="application/x-tex">\mathbb smooth boundary alttext="partial-differential Omega"> encoding="application/x-tex">\partial \Omega and alttext="StartFraction EndFraction"> encoding="application/x-tex">\frac } outer derivative alttext="u"> encoding="application/x-tex">u . Moreover, we also present some new sharp Hardy Leray type inequalities remainder terms that provide us concrete potentials to use partial differential equation our interest.