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/ID (518) /P 1150 0 R /A 1175 0 R >> endobj 1175 0 obj << /O /Layout /BBox [ 342 542.35 590.02 567.65 ] >> endobj 1176 0 obj << /K 43 /Alt ($\\|\(x_{1},y_{1}\)-\(x_{2},y_{2}\)\\|\\leq \\delta $) /S /MATH /Pg 95 0 R /ID (519) /P 1150 0 R /A 1177 0 R >> endobj 1177 0 obj << /O /Layout /BBox [ 211 539.11 341.32 546.89 ] >> endobj 1178 0 obj << /K [ 46 47 48 49 50 51 52 ] /Alt (\\begin{equation*} \\left \\| \\overline{T}\(t\)[\\phi ]\(x_{1},y_{1}\)-\\overline{T}\(t\)[\\phi ]\(x_{2},y_{2}\) \\right \\| <\\frac{\\varepsilon}{3},\\quad \\left \\| \\overline{T}\(t\)[\\psi ]\(x_{1},y_{1}\)- \\overline{T}\(t\)[\\psi ]\(x_{2},y_{2}\)\\right \\| <\\frac{\\varepsilon}{3t}. \\end{equation*}) /S /DISPLAYMATH /Pg 95 0 R /ID (534) /P 1063 0 R /A 1179 0 R >> endobj 1179 0 obj << /O /Layout /BBox [ 65 509.78 420.09 528.37 ] /Placement /Block >> endobj 1180 0 obj << /K 54 /S /P /Pg 95 0 R /P 1063 0 R /ID (565) >> endobj 1181 0 obj << /K [ 56 57 58 59 60 61 62 63 64 65 66 67 68 ] /Alt (\\begin{equation*} \\begin{aligned} &\\left \\| Q[t,\\phi ]\(x_{1},y_{1}\)-Q[t,\\phi ]\(x_{2},y_{2}\)\\right \\| \\\\ &\\leq \\left \\|\\overline{T}\(t\)[\\phi ]\(x_{1},y_{1}\)-\\overline{T}\(t\)[ \\phi ]\(x_{2},y_{2}\)\\right \\| \\\\ &+\\left \\| \\int _{0}^{t-\\frac{\\varepsilon}{6d}}\\overline{T}\(t-s\)[{ \\mathbf{f}}\(\\cdot ,\\cdot ,Q[s,\\phi ]\(\\cdot ,\\cdot \)\)]\(x_{1},y_{1}\)- \\overline{T}\(t-s\)[{\\mathbf{f}}\(\\cdot ,\\cdot ,Q[s,\\phi ]\(\\cdot ,\\cdot \)\)]\(x_{2},y_{2}\) \\,{\\mathrm{d}}s\\right \\| \\\\ &+\\left \\| \\int _{t-\\frac{\\varepsilon}{6d}}^{t}\\overline{T}\(t-s\)[{ \\mathbf{f}}\(\\cdot ,\\cdot ,Q[s,\\phi ]\(\\cdot ,\\cdot \)\)]\(x_{1},y_{1}\)- \\overline{T}\(t-s\)[{\\mathbf{f}}\(\\cdot ,\\cdot ,Q[s,\\phi ]\(\\cdot ,\\cdot \)\)]\(x_{2},y_{2}\) \\,{\\mathrm{d}}s\\right \\| \\\\ &\\leq \\frac{\\varepsilon}{3}+\\int _{0}^{t-\\frac{\\varepsilon}{6d}} \\frac{\\varepsilon}{3t}\\,{\\mathrm{d}}s+\\frac{\\varepsilon}{3}<\\varepsilon . \\end{aligned} \\end{equation*}) /S /DISPLAYMATH 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0 R 49 ] /S /P /Pg 159 0 R /P 1741 0 R /ID (1163) >> endobj 1980 0 obj << /K [ << /Obj 164 0 R /Type /OBJR >> 44 ] /S /Link /Pg 159 0 R /P 1979 0 R /ID (1164) >> endobj 1981 0 obj << /K 46 /Alt ($\\lim \\limits _{\\|x\\|\\rightarrow \\infty}\\left [ \\sup \\left \\lbrace \\| \\psi \(x,\\cdot \)\\|:\\psi \\in \\omega \(\\phi \)\\right \\rbrace \\right ] =0 $) /S /MATH /Pg 159 0 R /ID (1136) /P 1979 0 R /A 1982 0 R >> endobj 1982 0 obj << /O /Layout /BBox [ 182 382.63 343.14 400.47 ] >> endobj 1983 0 obj << /K 48 /Alt ($\\alpha _{0}>s_{0}$) /S /MATH /Pg 159 0 R /ID (1137) /P 1979 0 R /A 1984 0 R >> endobj 1984 0 obj << /O /Layout /BBox [ 51 373.4 81.15 379.53 ] >> endobj 1985 0 obj << /K [ 51 52 ] /Alt (\\begin{equation*} \\psi _{k}\(x,\\cdot \)-b^{*}\(\\eta _{\\infty}^{*}\)_{k}\(x,\\cdot \)< \\frac{\\gamma}{2},\\quad \\psi \\in \\omega \(\\phi \),\\,\\|x\\|>\\alpha _{0}. \\end{equation*}) /S /DISPLAYMATH /Pg 159 0 R /ID (1138) /P 1741 0 R /A 1986 0 R >> endobj 1986 0 obj << /O /Layout /BBox [ 133 342.93 352.72 361.37 ] /Placement /Block >> endobj 1987 0 obj << /K [ 54 1988 0 R 56 ] /S /P /Pg 159 0 R /P 1741 0 R /ID (1165) >> endobj 1988 0 obj << /K 55 /Alt ($\\gamma $) /S /MATH /Pg 159 0 R /ID (1139) /P 1987 0 R /A 1989 0 R >> endobj 1989 0 obj << /O /Layout /BBox [ 155 321.95 161.69 328.38 ] >> endobj 1990 0 obj << /K [ 58 59 ] /Alt (\\begin{equation*} \\gamma =\\sup \\left \\lbrace \\psi _{k}\(x,y\)-b^{*}\(\\eta _{\\infty}^{*}\)_{k}\(x,y\): \(k,x,y,\\psi \)\\in K\\times \(B_{\\alpha _{0}}\\backslash B_{s_{0}}\)\\times\\overline{\\Omega}\\times \\omega \(\\phi \)\\right \\rbrace . \\end{equation*}) /S /DISPLAYMATH /Pg 159 0 R /ID (1140) /P 1741 0 R /A 1991 0 R >> endobj 1991 0 obj << /O /Layout /BBox [ 78 296.51 405.92 308.99 ] /Placement /Block >> endobj 1992 0 obj << /K [ 61 1993 0 R 63 1995 0 R 66 ] /S /P /Pg 159 0 R /P 1741 0 R /ID (1166) >> endobj 1993 0 obj << /K 62 /Alt ($\\omega \(\\phi \)$) /S /MATH /Pg 159 0 R /ID (1141) /P 1992 0 R /A 1994 0 R >> endobj 1994 0 obj << /O /Layout /BBox [ 212 275.08 233.06 283.84 ] >> endobj 1995 0 obj << /K [ 64 65 ] /Alt ($\(k_{0},x^{*},y^{*},\\psi ^{*},\\psi ^{**}\)\\in K\\times \(B_{\\alpha _{0}} \\backslash B_{s_{0}}\)\\times \\overline{\\Omega}\\times \\omega \(\\phi \) \\times \\omega \(\\phi \)$) /S /MATH /Pg 159 0 R /ID (1142) /P 1992 0 R /A 1996 0 R >> endobj 1996 0 obj << /O /Layout /BBox [ 341 274.98 609.99 289 ] >> endobj 1997 0 obj << /K 68 /Alt (\\begin{equation*} \\psi ^{*}_{k_{0}}\(x^{*},y^{*}\)-b^{*}\(\\eta _{\\infty}^{*}\)_{k_{0}}\(x^{*},y^{*}\)= \\gamma >0~~\\mbox{and}~~\\psi ^{*}=Q[n_{0},\\psi ^{**}]. \\end{equation*}) /S /DISPLAYMATH /Pg 159 0 R /ID (1143) /P 1741 0 R /A 1998 0 R >> endobj 1998 0 obj << /O /Layout /BBox [ 114 235.89 370.02 247.65 ] /Placement /Block >> endobj 1999 0 obj << /K [ 70 2000 0 R 72 2002 0 R 74 2003 0 R 76 2005 0 R 79 ] /S /P /Pg 159 0 R /P 1741 0 R /ID (1167) >> endobj 2000 0 obj << /K 71 /Alt (${\\mathbf{f}}$) /S /MATH /Pg 159 0 R /ID (1144) /P 1999 0 R /A 2001 0 R >> endobj 2001 0 obj << /O /Layout /BBox [ 169 217 176.02 223.92 ] >> endobj 2002 0 obj << /K [ << /Obj 165 0 R /Type /OBJR >> 73 ] /S /Link /Pg 159 0 R /P 1999 0 R /ID (1168) >> endobj 2003 0 obj << /K 75 /Alt (${\\mathbf{f}}\(x,y,{\\mathbf{u}}\)\\leq {\\mathrm{D}}_{\\mathbf{u}}{\\mathbf{f}}\(x,y,{\\mathbf{0}}\){\\mathbf{u}}$) /S /MATH /Pg 159 0 R /ID (1145) /P 1999 0 R /A 2004 0 R >> endobj 2004 0 obj << /O /Layout /BBox [ 305 214.99 430.95 223.92 ] >> endobj 2005 0 obj << /K [ 77 78 ] /Alt ($\(x,y,{\\mathbf{u}}\)\\in \\mathbb{R}^{m}\\times \\overline{\\Omega}\\times\\mathbb{R}^{N}_{+}$) /S /MATH /Pg 159 0 R /ID (1146) /P 1999 0 R /A 2006 0 R >> endobj 2006 0 obj << /O /Layout /BBox [ 51 203.43 161.64 216.03 ] >> endobj 2007 0 obj << /K 81 /Alt (\\begin{equation*} Q[n_{0},\\psi ^{**}]\\leq \\mathbb{L}[n_{0},\\psi ^{**};{\\mathrm{D}}_{\\mathbf{u}}{ \\mathbf{f}}\(\\cdot ,\\cdot ,{\\mathbf{0}}\)]. \\end{equation*}) /S /DISPLAYMATH /Pg 159 0 R /ID (1147) /P 1741 0 R /A 2008 0 R >> endobj 2008 0 obj << /O /Layout /BBox [ 166 179.02 318.14 188.65 ] /Placement /Block >> endobj 2009 0 obj << /K [ 83 2010 0 R 85 2012 0 R 87 ] /S /P /Pg 159 0 R /P 1741 0 R /ID (1169) >> endobj 2010 0 obj << /K 84 /Alt ($b^{*},\\, x^{*},\\,y^{*},\\,\\rho ^{*},\\,\\psi ^{*},\\,\\psi ^{**},\\,s_{0}$) /S /MATH /Pg 159 0 R /ID (1148) /P 2009 0 R /A 2011 0 R >> endobj 2011 0 obj << /O /Layout /BBox [ 160 155.99 273.44 165.15 ] >> endobj 2012 0 obj << /K [ << /Obj 166 0 R /Type /OBJR >> 86 ] /S /Link /Pg 159 0 R /P 2009 0 R /ID (1170) >> endobj 2013 0 obj << /K [ 89 90 91 << /MCID 0 /Pg 167 0 R /Type /MCR >> ] /Alt (\\begin{align*} &\\psi ^{*}\(x^{*},y^{*}\)-b^{*}\\eta _{\\infty}^{*}\(x^{*},y^{*}\) \\\\ &= Q[n_{0},\\psi ^{**}]\(x^{*},y^{*}\)-b^{*}\\eta _{\\infty}^{*}\(x^{*},y^{*}\) \\\\ &\\leq \\mathbb{L}[n_{0},\\psi ^{**};{\\mathrm{D}}_{\\mathbf{u}}{\\mathbf{f}}\(\\cdot , \\cdot ,{\\mathbf{0}}\)]\(x^{*},y^{*}\)-\\frac{b^{*}}{\(\\rho ^{*}\)^{n_{0}}}\( \\widehat{\\mathbb{L}[1,\\cdot ; {\\mathrm{D}}_{\\mathbf{u}}{\\mathbf{f}}\(\\cdot ,\\cdot ,{\\mathbf{0}}\)]}_{\\sigma ^{*},1+d_{0}}\)^{n_{0}}[ \\eta _{\\infty}^{*}]\(x^{*},y^{*}\) \\\\ &\\leq \\mathbb{L}[n_{0},\\psi ^{**};{\\mathrm{D}}_{\\mathbf{u}}{\\mathbf{f}}\(\\cdot , \\cdot ,{\\mathbf{0}}\)]\(x^{*},y^{*}\)-\\frac{b^{*}}{\(\\rho ^{*}\)^{n_{0}}} \\mathbb{L}[n_{0},\\eta _{\\infty}^{*}; {\\mathrm{D}}_{\\mathbf{u}}{\\mathbf{f}}\(\\cdot , \\cdot ,{\\mathbf{0}}\)]\(x^{*},y^{*}\) \\\\ &\\leq \\mathbb{L}[n_{0},\\psi ^{**};{\\mathrm{D}}_{\\mathbf{u}}{\\mathbf{f}}\(\\cdot , \\cdot ,{\\mathbf{0}}\)]\(x^{*},y^{*}\)-b^{*}\\mathbb{L}[n_{0},\\eta _{\\infty}^{*}; {\\mathrm{D}}_{\\mathbf{u}}{\\mathbf{f}}\(\\cdot ,\\cdot ,{\\mathbf{0}}\)]\(x^{*},y^{*}\) \\\\ &\\leq \\mathbb{L}[n_{0},\\psi ^{**}-b^{*}\\eta _{\\infty}^{*};{\\mathrm{D}}_{ \\mathbf{u}}{\\mathbf{f}}\(\\cdot ,\\cdot ,{\\mathbf{0}}\)]\(x^{*},y^{*}\) \\\\ &\\leq [\(\\psi ^{*}\)_{k_{0}}\(x^{*},y^{*}\)-b^{*}\(\\eta _{\\infty}^{*}\)_{k_{0}}\(x^{*},y^{*}\)] \\times \\mathbb{L}[n_{0},{\\mathbf{1}};{\\mathrm{D}}_{\\mathbf{u}}{\\mathbf{f}}\(\\cdot ,\\cdot ,{ \\mathbf{0}}\)]\(x^{*},y^{*}\) \\\\ &\\leq \\delta _{0}\(\(\\psi ^{*}\)_{k_{0}}\(x^{*},y^{*}\)-b^{*}\(\\eta _{ \\infty}^{*}\)_{k_{0}}\(x^{*},y^{*}\)\) {\\mathbf{1}}, \\end{align*}) /S /DISPLAYALIGN /Pg 159 0 R /ID (1149) /P 1741 0 R /A 2014 0 R >> endobj 2014 0 obj << /O /Layout /BBox [ 51 124.04 2642.41 147.14 ] /Placement /Block >> endobj 2015 0 obj << /K [ 2 2016 0 R 4 2018 0 R 8 2020 0 R 10 2022 0 R ] /S /P /Pg 167 0 R /P 1741 0 R /ID (1224) >> endobj 2016 0 obj << /K 3 /Alt ($\(\\psi ^{*}\)_{k_{0}}\(x^{*},y^{*}\)-b^{*}\(\\eta _{\\infty}^{*}\)_{k_{0}}\(x^{*},y^{*}\)> \\gamma >0$) /S /MATH /Pg 167 0 R /ID (1172) /P 2015 0 R /A 2017 0 R >> endobj 2017 0 obj << /O /Layout /BBox [ 171 528.19 347.73 538.15 ] >> endobj 2018 0 obj << /K [ 5 6 7 ] /Alt ($0<\\delta _{0}<\\frac{1}{\\sqrt{n}}\\leq 1$) /S /MATH /Pg 167 0 R /ID (1173) /P 2015 0 R /A 2019 0 R >> endobj 2019 0 obj << /O /Layout /BBox [ 364 524.88 431.06 540.11 ] >> endobj 2020 0 obj << /K 9 /Alt ($\\omega \(\\phi \)\\leq b^{*} \\eta _{\\infty}^{*}$) /S /MATH /Pg 167 0 R /ID (1174) /P 2015 0 R /A 2021 0 R >> endobj 2021 0 obj << /O /Layout /BBox [ 152 513.32 207.76 523.15 ] >> endobj 2022 0 obj << /K 11 /Alt (MATH) /S /MATH /Pg 167 0 R /ID (1175) /P 2015 0 R /A 2023 0 R >> endobj 2023 0 obj << /O /Layout /BBox [ 352 515.9 359.54 521.04 ] >> endobj 2024 0 obj << /K [ 15 2025 0 R 17 2026 0 R 19 ] /S /P /Pg 167 0 R /P 489 0 R /ID (1225) >> endobj 2025 0 obj << /K [ << /Obj 169 0 R /Type /OBJR >> 16 ] /S /Link /Pg 167 0 R /P 2024 0 R /ID (1226) >> endobj 2026 0 obj << /K 18 /Alt ($N = m = 1$) /S /MATH /Pg 167 0 R /ID (1176) /P 2024 0 R /A 2027 0 R >> endobj 2027 0 obj << /O /Layout /BBox [ 415 479.85 460.18 486.69 ] >> endobj 2028 0 obj << /K [ 20 21 2029 0 R ] /Alt (\\begin{equation} \\label{equ4.1} \\begin{cases} \\partial _{t} u=d\\Delta u+c\\partial _{x} u+\\partial _{u}f\(x, y, 0\)u , & \(t, x, y\) \\in \(0, \\infty \) \\times \\mathbb{R} \\times \\Omega , \\\\ \\partial _{\\nu} u\(t, x, y\) = 0, & \(t, x, y\) \\in \(0, \\infty \) \\times\\mathbb{R} \\times \\partial \\Omega , \\\\ u\(0, x, y\) = \\phi \(x, y\), & \(x, y\) \\in \\mathbb{R} \\times\\overline{\\Omega}. \\end{cases} \\end{equation}) /S /DISPLAYMATH /Pg 167 0 R /ID (1178) /P 489 0 R /A 2031 0 R >> endobj 2029 0 obj << /K 2030 0 R /S /EQNUMBER /Pg 167 0 R /P 2028 0 R /ID (1179) >> endobj 2030 0 obj << /K 23 /S /EQNUM /Pg 167 0 R /P 2029 0 R /ID (1177) >> endobj 2031 0 obj << /O /Layout /BBox [ 51 409.97 433.57 453.01 ] /Placement /Block >> endobj 2032 0 obj << /K [ 26 2033 0 R 28 2035 0 R 30 2037 0 R 32 2038 0 R 34 2040 0 R 36 ] /S /P /Pg 167 0 R /P 489 0 R /ID (1227) >> endobj 2033 0 obj << /K 27 /Alt ($h\(x,y\) = \\partial _{u}f\(x, y, 0\)$) /S /MATH /Pg 167 0 R /ID (1180) /P 2032 0 R /A 2034 0 R >> endobj 2034 0 obj << /O /Layout /BBox [ 80 388.99 170.06 397.84 ] >> endobj 2035 0 obj << /K 29 /Alt ($L = d\\Delta + c\\partial _{x} + h\(x, y\)$) /S /MATH /Pg 167 0 R /ID (1181) /P 2032 0 R /A 2036 0 R >> endobj 2036 0 obj << /O /Layout /BBox [ 264 388.99 362.67 397.84 ] >> endobj 2037 0 obj << /K [ << /Obj 170 0 R /Type /OBJR >> 31 ] /S /Link /Pg 167 0 R /P 2032 0 R /ID (1228) >> endobj 2038 0 obj << /K 33 /Alt ($-L$) /S /MATH /Pg 167 0 R /ID (1182) /P 2032 0 R /A 2039 0 R >> endobj 2039 0 obj << /O /Layout /BBox [ 249 378.17 263.74 385.68 ] >> endobj 2040 0 obj << /K 35 /Alt ($\\mathbb{R}\\times \\Omega $) /S /MATH /Pg 167 0 R /ID (1183) /P 2032 0 R /A 2041 0 R >> endobj 2041 0 obj << /O /Layout /BBox [ 277 378.17 304.15 385.84 ] >> endobj 2042 0 obj << /K 37 /Alt (\\begin{equation*} \\lambda _{1}\(-L,\\mathbb{R}\\times \\Omega \)=\\sup \\left \\{\\lambda \\in\\mathbb{R}: \\exists \\psi >0,\\,\(L+\\lambda \) \\psi \\leq 0 \\, \\text { a.e. in }\\,\\mathbb{R}\\times \\Omega ,\\, \\partial _{\\nu} \\psi\\geq 0 \\,\\text { on }\\, \\mathbb{R}\\times \\partial \\Omega \\right \\}, \\end{equation*}) /S /DISPLAYMATH /Pg 167 0 R /ID (1184) /P 489 0 R /A 2043 0 R >> endobj 2043 0 obj << /O /Layout /BBox [ 55 350.84 429.09 359.84 ] /Placement /Block >> endobj 2044 0 obj << /K [ 38 2045 0 R 40 2047 0 R 42 2049 0 R 44 2050 0 R 46 2052 0 R 48 ] /S /P /Pg 167 0 R /P 489 0 R /ID (1229) >> endobj 2045 0 obj << /K 39 /Alt ($\\psi \\in W^{2,p}\(\(-r,r\)\\times \\Omega \),\\,\\forall r>0,\\,p>n+1$) /S /MATH /Pg 167 0 R /ID (1185) /P 2044 0 R /A 2046 0 R >> endobj 2046 0 obj << /O /Layout /BBox [ 78 327.99 252.59 338.7 ] >> endobj 2047 0 obj << /K 41 /Alt ($\\lambda _{1}:=\\lambda _{1}\(-L,\\mathbb{R}\\times \\Omega \) $) /S /MATH /Pg 167 0 R /ID (1186) /P 2044 0 R /A 2048 0 R >> endobj 2048 0 obj << /O /Layout /BBox [ 275 328.27 363.31 336.84 ] >> endobj 2049 0 obj << /K [ << /Obj 171 0 R /Type /OBJR >> 43 ] /S /Link /Pg 167 0 R /P 2044 0 R /ID (1230) >> endobj 2050 0 obj << /K 45 /Alt ($r>0$) /S /MATH /Pg 167 0 R /ID (1187) /P 2044 0 R /A 2051 0 R >> endobj 2051 0 obj << /O /Layout /BBox [ 108 317.86 129.68 324.69 ] >> endobj 2052 0 obj << /K 47 /Alt ($\(\\lambda \(r\),\\varphi _{r}\)\\in \\mathbb{R}_{+}\\times W^{2,p}\(\(-r,r\) \\times \\Omega ,\\mathbb{R}_{+}\) $) /S /MATH /Pg 167 0 R /ID (1188) /P 2044 0 R /A 2053 0 R >> endobj 2053 0 obj << /O /Layout /BBox [ 265 315.88 435.74 326.7 ] >> endobj 2054 0 obj << /K [ 49 50 2055 0 R ] /Alt (\\begin{equation} \\label{equ4.2} \\begin{cases} -L\\varphi _{r}=\\lambda \(r\)\\varphi _{r}, \\quad &\\text {a.e. in }\\,\(-r,r\) \\times \\Omega , \\\\ \\partial _{\\nu} \\varphi _{r}=0, & \\text{on}~~ \(-r,r\)\\times \\partial\\Omega , \\\\ \\varphi _{r}=0, & \\text{on}~~ \\{\\pm r\\}\\times \\overline{\\Omega}, \\end{cases} \\end{equation}) /S /DISPLAYMATH /Pg 167 0 R /ID (1190) /P 489 0 R /A 2057 0 R >> endobj 2055 0 obj << /K 2056 0 R /S /EQNUMBER /Pg 167 0 R /P 2054 0 R /ID (1191) >> endobj 2056 0 obj << /K 52 /S /EQNUM /Pg 167 0 R /P 2055 0 R /ID (1189) >> endobj 2057 0 obj << /O /Layout /BBox [ 51 247.97 433.57 291.01 ] /Placement /Block >> endobj 2058 0 obj << /K [ 55 2059 0 R 57 2061 0 R 59 2063 0 R 61 2065 0 R 63 2066 0 R 65 2068 0 R 67 2070 0 R 70 2072 0 R 72 2074 0 R 74 2076 0 R 76 2078 0 R 78 2080 0 R 80 ] /S /P /Pg 167 0 R /P 489 0 R /ID (1231) >> endobj 2059 0 obj << /K 56 /Alt ($\\varphi _{r}\(0,y_{0}\)=1$) /S /MATH /Pg 167 0 R /ID (1192) /P 2058 0 R /A 2060 0 R >> endobj 2060 0 obj << /O /Layout /BBox [ 189 226.99 242.01 235.84 ] >> endobj 2061 0 obj << /K 58 /Alt ($y_{0} \\in \\Omega $) /S /MATH /Pg 167 0 R /ID (1193) /P 2058 0 R /A 2062 0 R >> endobj 2062 0 obj << /O /Layout /BBox [ 277 226.99 304.53 235.69 ] >> endobj 2063 0 obj << /K 60 /Alt ($r$) /S /MATH /Pg 167 0 R /ID (1194) /P 2058 0 R /A 2064 0 R >> endobj 2064 0 obj << /O /Layout /BBox [ 426 229 430.77 233.53 ] >> endobj 2065 0 obj << /K [ << /Obj 172 0 R /Type /OBJR >> 62 ] /S /Link /Pg 167 0 R /P 2058 0 R /ID (1232) >> endobj 2066 0 obj << /K 64 /Alt ($\\lambda \(r\)\\searrow \\lambda _{1}$) /S /MATH /Pg 167 0 R /ID (1195) /P 2058 0 R /A 2067 0 R >> endobj 2067 0 obj << /O /Layout /BBox [ 189 213.11 230.35 221.84 ] >> endobj 2068 0 obj << /K 66 /Alt ($\\varphi _{r}\\to \\varphi $) /S /MATH /Pg 167 0 R /ID (1196) /P 2058 0 R /A 2069 0 R >> endobj 2069 0 obj << /O /Layout /BBox [ 235 213.12 265.95 219.38 ] >> endobj 2070 0 obj << /K [ 68 69 ] /Alt ($C_{{\\mathrm{loc}}}^{1}\(\\mathbb{R}\\times \\overline{\\Omega}\)$) /S /MATH /Pg 167 0 R /ID (1197) /P 2058 0 R /A 2071 0 R >> endobj 2071 0 obj << /O /Layout /BBox [ 279 211.57 330.2 223.99 ] >> endobj 2072 0 obj << /K 71 /Alt ($W_{{\\mathrm{loc}}}^{2,p}\(\\mathbb{R}\\times \\Omega \)$) /S /MATH /Pg 167 0 R /ID (1198) /P 2058 0 R /A 2073 0 R >> endobj 2073 0 obj << /O /Layout /BBox [ 391 211.57 447.08 225.28 ] >> endobj 2074 0 obj << /K 73 /Alt ($p>1$) /S /MATH /Pg 167 0 R /ID (1199) /P 2058 0 R /A 2075 0 R >> endobj 2075 0 obj << /O /Layout /BBox [ 96 200.99 119.44 209.69 ] >> endobj 2076 0 obj << /K 75 /Alt ($r\\to \\infty $) /S /MATH /Pg 167 0 R /ID (1200) /P 2058 0 R /A 2077 0 R >> endobj 2077 0 obj << /O /Layout /BBox [ 135 203 165.04 207.65 ] >> endobj 2078 0 obj << /K 77 /Alt ($\\varphi $) /S /MATH /Pg 167 0 R /ID (1201) /P 2058 0 R /A 2079 0 R >> endobj 2079 0 obj << /O /Layout /BBox [ 214 201.12 220.2 207.38 ] >> endobj 2080 0 obj << /K 79 /Alt ($\\varphi \(0,y_{0}\)=1$) /S /MATH /Pg 167 0 R /ID (1202) /P 2058 0 R /A 2081 0 R >> endobj 2081 0 obj << /O /Layout /BBox [ 275 200.99 324.46 209.84 ] >> endobj 2082 0 obj << /K [ 81 2083 0 R ] /Alt (\\begin{equation} \\label{equ4.3} \\begin{cases} -L\\varphi =\\lambda _{1}\\varphi ,\\quad &\\text{a.e. in}~~ \\mathbb{R} \\times \\Omega , \\\\ \\partial _{\\nu }\\varphi =0,&\\text{on}~~\\mathbb{R}\\times \\partial\\Omega . \\end{cases} \\end{equation}) /S /DISPLAYMATH /Pg 167 0 R /ID (1204) /P 489 0 R /A 2085 0 R >> endobj 2083 0 obj << /K 2084 0 R /S /EQNUMBER /Pg 167 0 R /P 2082 0 R /ID (1205) >> endobj 2084 0 obj << /K 83 /S /EQNUM /Pg 167 0 R /P 2083 0 R /ID (1203) >> endobj 2085 0 obj << /O /Layout /BBox [ 51 157.55 433.57 187.43 ] /Placement /Block >> endobj 2086 0 obj << /K [ 86 2087 0 R 88 2089 0 R 90 2091 0 R 92 2092 0 R 94 ] /S /P /Pg 167 0 R /P 489 0 R /ID (1233) >> endobj 2087 0 obj << /K 87 /Alt ($\\Omega $) /S /MATH /Pg 167 0 R /ID (1206) /P 2086 0 R /A 2088 0 R >> endobj 2088 0 obj << /O /Layout /BBox [ 88 139 94.79 146.75 ] >> endobj 2089 0 obj << /K 89 /Alt ($\\mathbb{R}^{n}$) /S /MATH /Pg 167 0 R /ID (1207) /P 2086 0 R /A 2090 0 R >> endobj 2090 0 obj << /O /Layout /BBox [ 231 139 241.5 147.24 ] >> endobj 2091 0 obj << /K [ << /Obj 173 0 R /Type /OBJR >> 91 ] /S /Link /Pg 167 0 R /P 2086 0 R /ID (1234) >> endobj 2092 0 obj << /K 93 /Alt ($\\varepsilon > 0$) /S /MATH /Pg 167 0 R /ID (1208) /P 2086 0 R /A 2093 0 R >> endobj 2093 0 obj << /O /Layout /BBox [ 99 127 119.43 134.77 ] >> endobj 2094 0 obj << /K 95 /Alt (\\begin{equation*} \\widetilde{\\Omega} := \\bigl\\{ y \\in \\mathbb{R}^{n} : \\mathrm{dist}\(y, \\Omega \) < \\varepsilon \\bigr\\} \\end{equation*}) /S /DISPLAYMATH /Pg 167 0 R /ID (1209) /P 489 0 R /A 2095 0 R >> endobj 2095 0 obj << /O /Layout /BBox [ 181 97.51 304.55 109.67 ] /Placement /Block >> endobj 2096 0 obj << /K [ 96 2097 0 R 99 2099 0 R 101 2101 0 R 103 2103 0 R 108 2105 0 R 110 ] /S /P /Pg 167 0 R /P 489 0 R /ID (1235) >> endobj 2097 0 obj << /K [ 97 98 ] /Alt ($y \\in \\overline{\\widetilde{\\Omega}} \\backslash \\Omega $) /S /MATH /Pg 167 0 R /ID (1211) /P 2096 0 R /A 2098 0 R >> endobj 2098 0 obj << /O /Layout /BBox [ 138 72.98 174.43 85.96 ] >> endobj 2099 0 obj << /K 100 /Alt ($\\bar{\\Omega}$) /S /MATH /Pg 167 0 R /ID (1213) /P 2096 0 R /A 2100 0 R >> endobj 2100 0 obj << /O /Layout /BBox [ 293 75 300.85 83.76 ] >> endobj 2101 0 obj << /K 102 /Alt ($\\pi \(y\)$) /S /MATH /Pg 167 0 R /ID (1214) /P 2096 0 R /A 2102 0 R >> endobj 2102 0 obj << /O /Layout /BBox [ 356 72.99 375.81 81.84 ] >> endobj 2103 0 obj << /K [ 104 105 106 107 ] /Alt ($\\mathcal{R}: \\mathbb{R} \\times \\left \( \\overline{\\widetilde{\\Omega}} \\backslash 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373.99 174.56 382.84 ] >> endobj 2497 0 obj << /K 73 /Alt (\\begin{equation*} \\begin{aligned} &\\|T\(t\)[\\phi ]\(x_{1},y_{1}\)-T\(t\)[\\phi ]\(x_{2},y_{2}\)\\| \\\\ &=\\|S_{t}^{2}[\\psi \(t,\\cdot ,y_{1}\)]\(x_{1}\)-S_{t}^{2}[\\psi \(t,\\cdot ,y_{2}\)]\(x_{2}\) \\| \\\\ &\\leq \\|S_{t}^{2}[\\psi \(t,\\cdot ,y_{1}\)]\(x_{1}\)-S_{t}^{2}[\\psi \(t, \\cdot ,y_{1}\)]\(x_{2}\)\\|+\\|S_{t}^{2}[\\psi \(t,\\cdot ,y_{1}\)]\(x_{2}\)-S_{t}^{2}[ \\psi \(t,\\cdot ,y_{2}\)]\(x_{2}\)\\|. \\end{aligned} \\end{equation*}) /S /DISPLAYMATH /Pg 208 0 R /ID (1538) /P 2436 0 R /A 2498 0 R >> endobj 2498 0 obj << /O /Layout /BBox [ 60 311.58 424.63 359.4 ] /Placement /Block >> endobj 2499 0 obj << /K [ 75 2500 0 R 77 ] /S /P /Pg 208 0 R /P 2436 0 R /ID (1560) >> endobj 2500 0 obj << /K [ << /Obj 212 0 R /Type /OBJR >> 76 ] /S /Link /Pg 208 0 R /P 2499 0 R /ID (1561) >> endobj 2501 0 obj << /K [ 79 80 ] /Alt (\\begin{equation*} \\|S_{t}^{2}[\\psi \(t,\\cdot ,y_{1}\)]\(x_{1}\)-S_{t}^{2}[\\psi \(t,\\cdot ,y_{1}\)]\(x_{2}\) \\|<\\frac{\\varepsilon}{2}. \\end{equation*}) /S /DISPLAYMATH /Pg 208 0 R /ID (1539) /P 2436 0 R /A 2502 0 R >> endobj 2502 0 obj << /O /Layout /BBox [ 147 257.93 338.33 276.37 ] /Placement /Block >> endobj 2503 0 obj << /K [ 82 2504 0 R 84 2506 0 R 86 2508 0 R 88 ] /S /P /Pg 208 0 R /P 2436 0 R /ID (1562) >> endobj 2504 0 obj << /K 83 /Alt ($R^{*}=\\max \\{R,\\tilde{R}\\}$) /S /MATH /Pg 208 0 R /ID (1541) /P 2503 0 R /A 2505 0 R >> endobj 2505 0 obj << /O /Layout /BBox [ 66 233.11 126.52 252.16 ] >> endobj 2506 0 obj << /K 85 /Alt ($G\(t, y, \\bar{y}\)$) /S /MATH /Pg 208 0 R /ID (1543) /P 2503 0 R /A 2507 0 R >> endobj 2507 0 obj << /O /Layout /BBox [ 243 232.99 296.92 241.84 ] >> endobj 2508 0 obj << /K 87 /Alt ($x\\in B_{R^{*}}\(0\)$) /S /MATH /Pg 208 0 R /ID (1544) /P 2503 0 R /A 2509 0 R >> endobj 2509 0 obj << /O /Layout /BBox [ 318 233.27 368.45 243.23 ] >> endobj 2510 0 obj << /K [ 90 91 92 ] /Alt (\\begin{equation*} \\|S_{t}^{1}[\\phi \(x,\\cdot 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2409 0 R (1515) 2412 0 R (1517) 2452 0 R (1518) 2454 0 R (1519) 2456 0 R (152) 652 0 R (1520) 2459 0 R (1521) 2462 0 R (1522) 2464 0 R (1523) 2467 0 R (1524) 2469 0 R (1525) 2471 0 R (1526) 2474 0 R (1527) 2476 0 R (1528) 2478 0 R (1529) 2480 0 R (153) 654 0 R (1530) 2482 0 R (1532) 2484 0 R (1533) 2486 0 R (1534) 2489 0 R (1535) 2491 0 R (1536) 2493 0 R (1537) 2495 0 R (1538) 2497 0 R (1539) 2501 0 R (154) 656 0 R (1541) 2504 0 R (1543) 2506 0 R (1544) 2508 0 R (1545) 2510 0 R (1546) 2513 0 R (155) 658 0 R (1553) 2444 0 R (1554) 2451 0 R (1555) 2458 0 R (1556) 2461 0 R (1557) 2466 0 R (1558) 2473 0 R (1559) 2488 0 R (156) 660 0 R (1560) 2499 0 R (1561) 2500 0 R (1562) 2503 0 R (1563) 2512 0 R (1565) 2516 0 R ] /Limits [ (1487) (1565) ] >> endobj 2952 0 obj << /Names [ (1566) 2519 0 R (1567) 2521 0 R (1568) 2523 0 R (1569) 2525 0 R (157) 662 0 R (1570) 2527 0 R (1571) 2530 0 R (1572) 2532 0 R (1573) 2534 0 R (1575) 2536 0 R (1576) 2538 0 R (1577) 2542 0 R (1578) 2544 0 R (1579) 2546 0 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677 0 R (1640) 2616 0 R (1641) 2618 0 R (1642) 2617 0 R (1643) 2620 0 R (1644) 2619 0 R (1645) 2621 0 R (1646) 2622 0 R (1647) 2623 0 R (1648) 2625 0 R (1649) 2624 0 R (165) 679 0 R (1650) 2627 0 R (1651) 2626 0 R (1652) 2628 0 R (1653) 2629 0 R (1654) 2630 0 R (1655) 2632 0 R (1656) 2631 0 R (1657) 2634 0 R (1658) 2633 0 R (1659) 2635 0 R (166) 682 0 R (1660) 2636 0 R (1661) 2637 0 R (1662) 2639 0 R (1663) 2638 0 R (1664) 2641 0 R (1665) 2640 0 R (1666) 2642 0 R (1667) 2643 0 R (1668) 2644 0 R (1669) 2646 0 R (167) 684 0 R (1670) 2645 0 R (1671) 2648 0 R (1672) 2647 0 R (1673) 2649 0 R (1674) 2650 0 R (1675) 2651 0 R (1676) 2653 0 R (1677) 2652 0 R (1678) 2655 0 R (1679) 2654 0 R (168) 686 0 R (1680) 2656 0 R (1681) 2657 0 R (1682) 2658 0 R (1683) 2660 0 R (1684) 2659 0 R (1685) 2662 0 R (1686) 2661 0 R (1687) 2663 0 R (1688) 2664 0 R (1689) 2665 0 R ] /Limits [ (1631) (1689) ] >> endobj 2954 0 obj << /Names [ (169) 688 0 R (1690) 2667 0 R (1691) 2666 0 R (1692) 2669 0 R (1693) 2668 0 R (1694) 2670 0 R (1695) 2671 0 R (1696) 2672 0 R (1697) 2674 0 R (1698) 2673 0 R (1699) 2676 0 R (17) 520 0 R (170) 690 0 R (1700) 2675 0 R (1701) 2677 0 R (1702) 2678 0 R (1703) 2679 0 R (1704) 2681 0 R (1705) 2680 0 R (1706) 2683 0 R (1707) 2682 0 R (1708) 2684 0 R (1709) 2685 0 R (171) 692 0 R (1710) 2686 0 R (1711) 2688 0 R (1712) 2687 0 R (1713) 2690 0 R (1714) 2689 0 R (1715) 2691 0 R (1716) 2692 0 R (1717) 2693 0 R (1718) 2695 0 R (1719) 2696 0 R (172) 694 0 R (1720) 2697 0 R (1721) 2699 0 R (1722) 2698 0 R (1723) 2701 0 R (1724) 2700 0 R (1725) 2702 0 R (1726) 2703 0 R (1727) 2704 0 R (1728) 2706 0 R (1729) 2705 0 R (173) 696 0 R (1730) 2707 0 R (1731) 2708 0 R (1732) 2709 0 R (1733) 2711 0 R (1734) 2710 0 R (1735) 2713 0 R (1736) 2712 0 R (1737) 2714 0 R (1738) 2715 0 R (1739) 2716 0 R (1740) 2718 0 R (1741) 2717 0 R (1742) 2720 0 R (1743) 2719 0 R (1744) 2721 0 R (1745) 2722 0 R (1746) 2723 0 R (1747) 2725 0 R ] /Limits [ (169) (1747) ] >> endobj 2955 0 obj << /Names [ (1748) 2724 0 R (1749) 2727 0 R (1750) 2726 0 R (1751) 2728 0 R (1752) 2729 0 R (1753) 2730 0 R (1754) 2732 0 R (1755) 2731 0 R (1756) 2734 0 R (1757) 2733 0 R (1758) 2735 0 R (1759) 2736 0 R (1760) 2737 0 R (1761) 2739 0 R (1762) 2738 0 R (1763) 2741 0 R (1764) 2740 0 R (1770) 2570 0 R (1771) 2575 0 R (1772) 2582 0 R (1773) 2583 0 R (1774) 2584 0 R (1775) 2694 0 R (1777) 2742 0 R (1778) 2743 0 R (1779) 2744 0 R (1780) 2746 0 R (1781) 2745 0 R (1782) 2748 0 R (1783) 2747 0 R (1784) 2749 0 R (1785) 2750 0 R (1786) 2751 0 R (1787) 2753 0 R (1788) 2752 0 R (1789) 2755 0 R (179) 635 0 R (1790) 2754 0 R (1791) 2756 0 R (1792) 2757 0 R (1793) 2758 0 R (1794) 2760 0 R (1795) 2759 0 R (1796) 2761 0 R (1797) 2762 0 R (1798) 2763 0 R (1799) 2765 0 R (180) 636 0 R (1800) 2764 0 R (1801) 2767 0 R (1802) 2766 0 R (1803) 2768 0 R (1804) 2769 0 R (1805) 2770 0 R (1806) 2772 0 R (1807) 2771 0 R (1808) 2774 0 R (1809) 2773 0 R (181) 637 0 R (1810) 2775 0 R (1811) 2776 0 R (1812) 2777 0 R (1813) 2779 0 R (1814) 2778 0 R ] /Limits [ (1748) (1814) ] >> endobj 2956 0 obj << /Names [ (1815) 2780 0 R (1816) 2781 0 R (1817) 2782 0 R (1818) 2784 0 R (1819) 2783 0 R (182) 638 0 R (1820) 2786 0 R (1821) 2785 0 R (1822) 2787 0 R (1823) 2788 0 R (1824) 2789 0 R (1825) 2791 0 R (1826) 2790 0 R (1827) 2793 0 R (1828) 2792 0 R (1829) 2794 0 R (183) 639 0 R (1830) 2795 0 R (1831) 2796 0 R (1832) 2798 0 R (1833) 2797 0 R (1834) 2800 0 R (1835) 2799 0 R (1836) 2801 0 R (1837) 2802 0 R (1838) 2803 0 R (1839) 2805 0 R (184) 640 0 R (1840) 2804 0 R (1841) 2807 0 R (1842) 2806 0 R (1843) 2808 0 R (1844) 2809 0 R (1845) 2810 0 R (1846) 2812 0 R (1847) 2811 0 R (1848) 2814 0 R (1849) 2813 0 R (185) 641 0 R (1850) 2815 0 R (1851) 2816 0 R (1852) 2817 0 R (1853) 2819 0 R (1854) 2818 0 R (1855) 2821 0 R (1856) 2820 0 R (1857) 2822 0 R (1858) 2823 0 R (1859) 2824 0 R (186) 642 0 R (1860) 2826 0 R (1861) 2825 0 R (1862) 2828 0 R (1863) 2827 0 R (1864) 2829 0 R (1865) 2830 0 R (1866) 2831 0 R (1867) 2833 0 R (1868) 2832 0 R (1869) 2835 0 R (187) 643 0 R (1870) 2834 0 R (1871) 2836 0 R (1872) 2837 0 R ] /Limits [ (1815) (1872) ] >> endobj 2957 0 obj << /Names [ (1873) 2838 0 R (1874) 2840 0 R (1875) 2839 0 R (1876) 2842 0 R (1877) 2841 0 R (1878) 2843 0 R (1879) 2844 0 R (188) 644 0 R (1880) 2845 0 R (1881) 2847 0 R (1882) 2846 0 R (1883) 2849 0 R (1884) 2848 0 R (1885) 2850 0 R (1886) 2851 0 R (1887) 2852 0 R (1888) 2854 0 R (1889) 2853 0 R (189) 645 0 R (1890) 2856 0 R (1891) 2855 0 R (1892) 2857 0 R (1893) 2858 0 R (1894) 2859 0 R (1895) 2861 0 R (1896) 2860 0 R (1897) 2863 0 R (1898) 2862 0 R (1899) 2864 0 R (19) 522 0 R (190) 646 0 R (1900) 2865 0 R (1901) 2866 0 R (1902) 2868 0 R (1903) 2867 0 R (1904) 2870 0 R (1905) 2869 0 R (1906) 2871 0 R (1907) 2872 0 R (1908) 2873 0 R (1909) 2875 0 R (191) 651 0 R (1910) 2874 0 R (1911) 2877 0 R (1912) 2876 0 R (1913) 2878 0 R (1914) 2879 0 R (1915) 2880 0 R (1916) 2882 0 R (1917) 2881 0 R (1918) 2884 0 R (1919) 2883 0 R (192) 668 0 R (1920) 2885 0 R (1921) 2886 0 R (1922) 2887 0 R (1923) 2889 0 R (1924) 2888 0 R (1925) 2891 0 R (1926) 2890 0 R (1927) 2892 0 R (1928) 2893 0 R (1929) 2894 0 R (193) 681 0 R ] /Limits [ (1873) (193) ] >> endobj 2958 0 obj << /Names [ (1930) 2896 0 R (1931) 2895 0 R (1932) 2898 0 R (1933) 2897 0 R (1934) 2899 0 R (1935) 2900 0 R (1936) 2901 0 R (1937) 2903 0 R (1938) 2902 0 R (1939) 2905 0 R (194) 683 0 R (1940) 2904 0 R (1941) 2906 0 R (1942) 2907 0 R (1943) 2908 0 R (1944) 2910 0 R (1945) 2909 0 R (195) 685 0 R (196) 687 0 R (198) 700 0 R (199) 703 0 R (200) 706 0 R (201) 709 0 R (202) 714 0 R (203) 716 0 R (204) 718 0 R (205) 720 0 R (206) 722 0 R (207) 724 0 R (208) 727 0 R (209) 729 0 R (21) 530 0 R (210) 731 0 R (211) 733 0 R (212) 735 0 R (213) 737 0 R (214) 739 0 R (215) 741 0 R (216) 743 0 R (217) 745 0 R (218) 747 0 R (219) 749 0 R (22) 531 0 R (220) 752 0 R (221) 754 0 R (223) 756 0 R (225) 758 0 R (226) 760 0 R (227) 762 0 R (228) 764 0 R (229) 767 0 R (23) 533 0 R (230) 769 0 R (231) 771 0 R (232) 773 0 R (233) 776 0 R (234) 778 0 R (235) 780 0 R (236) 783 0 R (237) 785 0 R (238) 787 0 R (239) 789 0 R (240) 791 0 R (241) 793 0 R ] /Limits [ (1930) (241) ] >> endobj 2959 0 obj << /Names [ (242) 795 0 R (243) 797 0 R (244) 799 0 R (245) 801 0 R (246) 803 0 R (247) 805 0 R (248) 807 0 R (249) 809 0 R (25) 532 0 R (250) 811 0 R (251) 813 0 R (252) 815 0 R (253) 817 0 R (254) 819 0 R (255) 821 0 R (256) 823 0 R (257) 825 0 R (258) 827 0 R (259) 829 0 R (26) 534 0 R (260) 832 0 R (261) 834 0 R (262) 835 0 R (263) 836 0 R (264) 837 0 R (265) 838 0 R (266) 840 0 R (267) 842 0 R (268) 844 0 R (269) 846 0 R (27) 536 0 R (270) 848 0 R (271) 850 0 R (277) 698 0 R (278) 699 0 R (279) 701 0 R (280) 702 0 R (281) 704 0 R (282) 705 0 R (283) 707 0 R (284) 708 0 R (285) 711 0 R (286) 712 0 R (287) 713 0 R (288) 726 0 R (289) 751 0 R (29) 535 0 R (290) 766 0 R (291) 775 0 R (292) 782 0 R (293) 831 0 R (295) 852 0 R (296) 853 0 R (297) 854 0 R (298) 855 0 R (299) 857 0 R (3) 503 0 R (30) 537 0 R (300) 859 0 R (301) 861 0 R (302) 863 0 R (303) 865 0 R (304) 867 0 R (305) 868 0 R ] /Limits [ (242) (305) ] >> endobj 2960 0 obj << /Names [ (306) 869 0 R (307) 870 0 R (308) 872 0 R (309) 874 0 R (310) 876 0 R (311) 878 0 R (312) 879 0 R (313) 880 0 R (314) 881 0 R (315) 884 0 R (316) 886 0 R (317) 887 0 R (318) 888 0 R (327) 889 0 R (328) 891 0 R (329) 892 0 R (33) 538 0 R (330) 893 0 R (331) 894 0 R (332) 896 0 R (333) 898 0 R (334) 901 0 R (335) 903 0 R (336) 905 0 R (337) 907 0 R (338) 909 0 R (339) 911 0 R (34) 539 0 R (340) 913 0 R (341) 916 0 R (342) 920 0 R (343) 918 0 R (344) 919 0 R (345) 923 0 R (346) 927 0 R (347) 925 0 R (348) 926 0 R (35) 525 0 R (354) 883 0 R (355) 900 0 R (356) 906 0 R (357) 908 0 R (358) 910 0 R (359) 912 0 R (36) 527 0 R (360) 914 0 R (361) 915 0 R (362) 917 0 R (363) 922 0 R (364) 924 0 R (366) 931 0 R (367) 933 0 R (368) 935 0 R (369) 937 0 R (370) 939 0 R (371) 941 0 R (372) 943 0 R (373) 945 0 R (374) 947 0 R (375) 949 0 R (376) 951 0 R (377) 953 0 R (378) 955 0 R (379) 957 0 R ] /Limits [ (306) (379) ] >> endobj 2961 0 obj << /Names [ (380) 959 0 R (381) 961 0 R (382) 963 0 R (383) 965 0 R (384) 967 0 R (385) 970 0 R (386) 972 0 R (387) 974 0 R (388) 976 0 R (389) 980 0 R (39) 490 0 R (390) 982 0 R (391) 985 0 R (392) 986 0 R (393) 987 0 R (394) 988 0 R (395) 989 0 R (396) 991 0 R (397) 993 0 R (398) 994 0 R (399) 995 0 R (4) 504 0 R (40) 491 0 R (400) 996 0 R (401) 998 0 R (402) 1000 0 R (403) 1001 0 R (404) 1002 0 R (405) 1003 0 R (406) 1005 0 R (407) 1007 0 R (408) 1009 0 R (409) 1011 0 R (41) 494 0 R (410) 1012 0 R (411) 1013 0 R (412) 1014 0 R (413) 1016 0 R (414) 1018 0 R (415) 1020 0 R (416) 1022 0 R (417) 1024 0 R (418) 1026 0 R (419) 1028 0 R (42) 496 0 R (420) 1030 0 R (421) 1033 0 R (422) 1035 0 R (423) 1038 0 R (424) 1039 0 R (425) 1040 0 R (426) 1041 0 R (427) 1042 0 R (428) 1044 0 R (429) 1046 0 R (43) 498 0 R (430) 1047 0 R (431) 1048 0 R (432) 1049 0 R (433) 1051 0 R (434) 1053 0 R (435) 1054 0 R (436) 1055 0 R (437) 1056 0 R ] /Limits [ (380) (437) ] >> endobj 2962 0 obj << /Names [ (438) 1058 0 R (439) 1060 0 R (44) 500 0 R (440) 1062 0 R (441) 1063 0 R (442) 1064 0 R (443) 1065 0 R (444) 1067 0 R (445) 1069 0 R (446) 1071 0 R (447) 1073 0 R (448) 1075 0 R (449) 1077 0 R (45) 493 0 R (450) 1079 0 R (451) 1081 0 R (452) 1083 0 R (458) 929 0 R (459) 930 0 R (46) 497 0 R (460) 934 0 R (461) 954 0 R (462) 956 0 R (463) 969 0 R (464) 978 0 R (465) 979 0 R (466) 984 0 R (467) 1032 0 R (468) 1037 0 R (47) 499 0 R (470) 1085 0 R (471) 1088 0 R (472) 1090 0 R (473) 1092 0 R (474) 1094 0 R (475) 1096 0 R (476) 1098 0 R (477) 1100 0 R (48) 502 0 R (486) 1102 0 R (487) 1104 0 R (488) 1106 0 R (489) 1108 0 R (490) 1110 0 R (491) 1112 0 R (492) 1114 0 R (493) 1116 0 R (494) 1118 0 R (495) 1121 0 R (496) 1123 0 R (497) 1125 0 R (498) 1128 0 R (499) 1130 0 R (5) 505 0 R (50) 506 0 R (500) 1133 0 R (501) 1135 0 R (502) 1137 0 R (503) 1139 0 R (504) 1142 0 R (505) 1146 0 R (506) 1148 0 R (507) 1151 0 R (508) 1153 0 R ] /Limits [ (438) (508) ] >> endobj 2963 0 obj << /Names [ (509) 1155 0 R (51) 507 0 R (510) 1157 0 R (511) 1159 0 R (512) 1161 0 R (513) 1163 0 R (514) 1165 0 R (515) 1168 0 R (516) 1170 0 R (517) 1172 0 R (518) 1174 0 R (519) 1176 0 R (52) 509 0 R (525) 1084 0 R (526) 1087 0 R (527) 1120 0 R (528) 1127 0 R (529) 1132 0 R (53) 511 0 R (530) 1141 0 R (531) 1144 0 R (532) 1145 0 R (534) 1178 0 R (535) 1181 0 R (536) 1185 0 R (537) 1187 0 R (538) 1189 0 R (539) 1193 0 R (54) 513 0 R (540) 1197 0 R (541) 1195 0 R (542) 1196 0 R (543) 1200 0 R (544) 1202 0 R (545) 1204 0 R (546) 1206 0 R (547) 1208 0 R (548) 1210 0 R (549) 1212 0 R (55) 515 0 R (550) 1215 0 R (551) 1217 0 R (554) 1220 0 R (555) 1222 0 R (557) 1224 0 R (56) 516 0 R (563) 1150 0 R (564) 1167 0 R (565) 1180 0 R (566) 1183 0 R (567) 1184 0 R (568) 1191 0 R (569) 1192 0 R (57) 517 0 R (570) 1194 0 R (571) 1199 0 R (572) 1207 0 R (573) 1214 0 R (575) 1229 0 R (576) 1231 0 R (577) 1234 0 R (578) 1236 0 R (579) 1238 0 R (58) 519 0 R ] /Limits [ (509) (58) ] >> endobj 2964 0 obj << /Names [ (580) 1240 0 R (581) 1241 0 R (582) 1242 0 R (583) 1243 0 R (584) 1244 0 R (585) 1246 0 R (586) 1248 0 R (587) 1250 0 R (588) 1252 0 R (589) 1254 0 R (59) 521 0 R (590) 1255 0 R (591) 1256 0 R (592) 1257 0 R (593) 1259 0 R (594) 1261 0 R (595) 1263 0 R (596) 1265 0 R (597) 1267 0 R (598) 1269 0 R (599) 1271 0 R (60) 523 0 R (600) 1273 0 R (601) 1275 0 R (602) 1277 0 R (603) 1279 0 R (604) 1280 0 R (605) 1281 0 R (606) 1282 0 R (607) 1284 0 R (608) 1286 0 R (609) 1288 0 R (61) 524 0 R (610) 1291 0 R (611) 1293 0 R (612) 1296 0 R (613) 1298 0 R (614) 1300 0 R (615) 1303 0 R (616) 1305 0 R (617) 1306 0 R (618) 1307 0 R (619) 1308 0 R (62) 526 0 R (620) 1309 0 R (621) 1311 0 R (622) 1312 0 R (623) 1313 0 R (624) 1314 0 R (625) 1316 0 R (626) 1318 0 R (627) 1319 0 R (628) 1320 0 R (629) 1321 0 R (63) 529 0 R (630) 1323 0 R (631) 1326 0 R (632) 1328 0 R (633) 1331 0 R (634) 1334 0 R (635) 1337 0 R 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domain,Reaction-diffusion systems,Shifting effect,Threshold dynamics) /Subject (Journal of Differential Equations, 458 \(2026\) 114084. 10.1016/j.jde.2025.114084) /CrossMarkDomains#5B2#5D (sciencedirect.com) /Author (Qian Guo) >> endobj 2975 0 obj << /Dests 351 0 R >> endobj 2976 0 obj << /Length 5075 /Subtype /XML /Type /Metadata >> stream 1 https://www.elsevier.com/tdm/tdmrep-policy.json application/pdf doi:10.1016/j.jde.2025.114084 Elsevier Inc. Journal of Differential Equations, 458 (2026) 114084. 10.1016/j.jde.2025.114084 Asymptotic annihilation Cylindrical domain Reaction-diffusion systems Shifting effect Threshold dynamics Threshold dynamics of a reaction-diffusion system in a cylinder with shifting effect Qian Guo Taishan Yi Yurong Zhang Xingfu Zou journal Journal of Differential Equations © 2025 Elsevier Inc. 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