Publication Date:
2018
Description:
〈p〉Publication date: 15 November 2018〈/p〉
〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 10〈/p〉
〈p〉Author(s): Enzo Vitillaro〈/p〉
〈h5〉Abstract〈/h5〉
〈div〉
〈p〉The aim of this paper is to study the problem〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mrow〉〈mo〉{〈/mo〉〈mtable〉〈mtr〉〈mtd columnalign="left"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉P〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈mi〉f〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉u〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mspace width="2em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mrow〉〈mtext〉in 〈/mtext〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mo〉∞〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈mo〉×〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mtext〉,〈/mtext〉〈/mrow〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈/mtd〉〈mtd columnalign="left"〉〈mrow〉〈mtext〉on 〈/mtext〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mo〉∞〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈mo〉×〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Γ〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mtext〉,〈/mtext〉〈/mrow〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo〉+〈/mo〉〈msub〉〈mrow〉〈mo〉∂〈/mo〉〈/mrow〉〈mrow〉〈mi〉ν〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo〉−〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="normal"〉Γ〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉Q〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈mi〉g〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉u〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mspace width="2em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mrow〉〈mtext〉on 〈/mtext〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mo〉∞〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈mo〉×〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Γ〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mtext〉,〈/mtext〉〈/mrow〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mtd〉〈mtd columnalign="left"〉〈mrow〉〈mtext〉in 〈/mtext〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Ω〈/mi〉〈/mrow〉〈mo〉‾〈/mo〉〈/mover〉〈mtext〉,〈/mtext〉〈/mrow〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/math〉〈/span〉 where Ω is a bounded open subset of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/math〉 boundary (〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈mi〉N〈/mi〉〈mo〉≥〈/mo〉〈mn〉2〈/mn〉〈/math〉), 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈mi mathvariant="normal"〉Γ〈/mi〉〈mo〉=〈/mo〉〈mo〉∂〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Γ〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉 is relatively open on Γ, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="normal"〉Γ〈/mi〉〈/mrow〉〈/msub〉〈/math〉 denotes the Laplace–Beltrami operator on Γ, 〈em〉ν〈/em〉 is the outward normal to Ω, and the terms 〈em〉P〈/em〉 and 〈em〉Q〈/em〉 represent nonlinear damping terms, while 〈em〉f〈/em〉 and 〈em〉g〈/em〉 are nonlinear perturbations.〈/p〉
〈p〉In the paper we establish local and global existence, uniqueness and Hadamard well-posedness results when source terms can be supercritical or super-supercritical.〈/p〉
〈/div〉
Print ISSN:
0022-0396
Electronic ISSN:
1090-2732
Topics:
Mathematics
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