Boundary layer analysis of nonlinear reaction-diffusion equations in a polygonal domain

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• 作者：Chang-Yeol Jung^a ; ^{cjung@unist.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Eunhee Park^b ; ^{parkeh@indiana.edu" class="auth_mail" title="E-mail the corresponding author} ; Roger Temam^b ; ^{temam@indiana.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Singular perturbations ; Nonlinear reaction&ndash ; diffusion ; Corner layers
• 刊名：Nonlinear Analysis
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：148
• 期：Complete
• 页码：161-202
• 全文大小：970 K

文摘

We propose a boundary layer analysis which fits a domain with corners. In particular, we consider nonlinear reaction–diffusion problems posed in a polygonal domain having a small diffusive coefficient mmlsi4" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302279&_mathId=si4.gif&_user=111111111&_pii=S0362546X16302279&_rdoc=1&_issn=0362546X&md5=037d990fbd6470ff0d100a7399c989d4" title="Click to view the MathML source">ε>0mathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll"><mi>εmi><mo>>mo><mn>0mn>math>. We present the full analysis of the singular behaviours at any orders with respect to the parameter mmlsi5" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302279&_mathId=si5.gif&_user=111111111&_pii=S0362546X16302279&_rdoc=1&_issn=0362546X&md5=78916b131d5de0516321d19825319992" title="Click to view the MathML source">εmathContainer hidden">mathCode"><math altimg="si5.gif" overflow="scroll"><mi>εmi>math> where we use a systematic nonlinear treatment initiated in Jung et al. (2016). The boundary layers are formed near the polygonal boundaries and two adjacent ones overlap at a corner mmlsi6" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302279&_mathId=si6.gif&_user=111111111&_pii=S0362546X16302279&_rdoc=1&_issn=0362546X&md5=d9d1e1fe9519062e030b4e6e16defb3d" title="Click to view the MathML source">PmathContainer hidden">mathCode"><math altimg="si6.gif" overflow="scroll"><mi>Pmi>math> and the overlapping produces additional layers, the so-called corner layers. It is noteworthy that the boundary layers are also degenerate due to the singularities of the solutions involving a negative power of the radial distance to the corner mmlsi6" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302279&_mathId=si6.gif&_user=111111111&_pii=S0362546X16302279&_rdoc=1&_issn=0362546X&md5=d9d1e1fe9519062e030b4e6e16defb3d" title="Click to view the MathML source">PmathContainer hidden">mathCode"><math altimg="si6.gif" overflow="scroll"><mi>Pmi>math> which are present in the Laplace operator on a sector (sector corresponding to the part of the polygon near the corner). The corner layers are then designed to absorb both the singularities and the interaction of the two boundary layers at mmlsi6" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302279&_mathId=si6.gif&_user=111111111&_pii=S0362546X16302279&_rdoc=1&_issn=0362546X&md5=d9d1e1fe9519062e030b4e6e16defb3d" title="Click to view the MathML source">PmathContainer hidden">mathCode"><math altimg="si6.gif" overflow="scroll"><mi>Pmi>math>.