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- W3163792878 abstract "Abstract We study the wave inequality with a Hardy potential <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=block> <m:mtable rowspacing=4pt columnspacing=1em> <m:mtr> <m:mtd> <m:mstyle displaystyle=true> <m:msub> <m:mi mathvariant=normal>∂</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>t</m:mi> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>−</m:mo> <m:mrow class=MJX-TeXAtom-ORD> <m:mi class=MJX-tex-mathit mathvariant=italic>Δ</m:mi> </m:mrow> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mfrac> <m:mi>λ</m:mi> <m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mo stretchy=false>|</m:mo> </m:mrow> <m:mi>x</m:mi> <m:msup> <m:mrow class=MJX-TeXAtom-ORD> <m:mo stretchy=false>|</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mfrac> <m:mi>u</m:mi> <m:mo>≥</m:mo> <m:mrow class=MJX-TeXAtom-ORD> <m:mo stretchy=false>|</m:mo> </m:mrow> <m:mi>u</m:mi> <m:msup> <m:mrow class=MJX-TeXAtom-ORD> <m:mo stretchy=false>|</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msup> <m:mspace width=1em /> <m:mtext>in </m:mtext> <m:mo stretchy=false>(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=normal>∞</m:mi> <m:mo stretchy=false>)</m:mo> <m:mo>×</m:mo> <m:mrow class=MJX-TeXAtom-ORD> <m:mi class=MJX-tex-mathit mathvariant=italic>Ω</m:mi> </m:mrow> <m:mo>,</m:mo> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $$begin{array}{} displaystyle partial_{tt}u-{itDelta} u+frac{lambda}{|x|^2}ugeq |u|^pquad mbox{in } (0,infty)times {itOmega}, end{array}$$ where Ω is the exterior of the unit ball in ℝ N , N ≥ 2, p > 1, and λ ≥ − <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mtable rowspacing=4pt columnspacing=1em> <m:mtr> <m:mtd> <m:mstyle displaystyle=true> <m:msup> <m:mfenced open=( close=)> <m:mfrac> <m:mrow> <m:mi>N</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mn>2</m:mn> </m:mfrac> </m:mfenced> <m:mn>2</m:mn> </m:msup> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $begin{array}{} displaystyle left(frac{N-2}{2}right)^2 end{array}$ , under the inhomogeneous boundary condition <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=block> <m:mtable rowspacing=4pt columnspacing=1em> <m:mtr> <m:mtd> <m:mstyle displaystyle=true> <m:mi>α</m:mi> <m:mfrac> <m:mrow> <m:mi mathvariant=normal>∂</m:mi> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant=normal>∂</m:mi> <m:mi>ν</m:mi> </m:mrow> </m:mfrac> <m:mo stretchy=false>(</m:mo> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>x</m:mi> <m:mo stretchy=false>)</m:mo> <m:mo>+</m:mo> <m:mi>β</m:mi> <m:mi>u</m:mi> <m:mo stretchy=false>(</m:mo> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>x</m:mi> <m:mo stretchy=false>)</m:mo> <m:mo>≥</m:mo> <m:mi>w</m:mi> <m:mo stretchy=false>(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=false>)</m:mo> <m:mspace width=1em /> <m:mtext>on </m:mtext> <m:mo stretchy=false>(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=normal>∞</m:mi> <m:mo stretchy=false>)</m:mo> <m:mo>×</m:mo> <m:mi mathvariant=normal>∂</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi class=MJX-tex-mathit mathvariant=italic>Ω</m:mi> </m:mrow> <m:mo>,</m:mo> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $$begin{array}{} displaystyle alpha frac{partial u}{partial nu}(t,x)+beta u(t,x)geq w(x)quadmbox{on } (0,infty)times partial{itOmega}, end{array}$$ where α , β ≥ 0 and ( α , β ) ≠ (0, 0). Namely, we show that there exists a critical exponent p c ( N , λ ) ∈ (1, ∞] for which, if 1 < p < p c ( N , λ ), the above problem admits no global weak solution for any w ∈ L 1 ( ∂ Ω ) with ∫ ∂Ω w ( x ) dσ > 0, while if p > p c ( N , λ ), the problem admits global solutions for some w > 0. To the best of our knowledge, the study of the critical behavior for wave inequalities with a Hardy potential in an exterior domain was not considered in previous works. Some open questions are also mentioned in this paper." @default.
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- W3163792878 date "2021-01-01" @default.
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- W3163792878 title "On the critical behavior for inhomogeneous wave inequalities with Hardy potential in an exterior domain" @default.
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- W3163792878 doi "https://doi.org/10.1515/anona-2020-0181" @default.
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