Uniqueness and stability of traveling waves for cellular neural networks with multiple delays

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• 作者：Zhi-Xian Yu^a ; ^{zxyu0902@163.com" class="auth_mail" title="E-mail the corresponding author} ; Ming Mei^b ; ^c ; ^{mmei@champlaincollege.qc.ca" class="auth_mail" title="E-mail the corresponding author} ; ^{ming.mei@mcgill.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35C07 ; 92D25 ; 35B35
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 January 2016
• 年：2016
• 卷：260
• 期：1
• 页码：241-267
• 全文大小：366 K

文摘

In this paper, we investigate the properties of traveling waves to a class of lattice differential equations for cellular neural networks with multiple delays. Following the previous study [38] on the existence of the traveling waves, here we focus on the uniqueness and the stability of these traveling waves. First of all, by establishing the a priori   asymptotic behavior of traveling waves and applying Ikehara's theorem, we prove the uniqueness (up to translation) of traveling waves class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615004404&_mathId=si1.gif&_user=111111111&_pii=S0022039615004404&_rdoc=1&_issn=00220396&md5=cdce590a09f5bc3099ab08dea854e35f" title="Click to view the MathML source">蠒(n−ct)class="mathContainer hidden">class="mathCode">$蠒(n-ct)$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615004404&_mathId=si2.gif&_user=111111111&_pii=S0022039615004404&_rdoc=1&_issn=00220396&md5=8d953b80620a9e1b627df0daf89f89eb" title="Click to view the MathML source">c≤c_鈦?/sub>class="mathContainer hidden">class="mathCode">$c\le c_{鈦?/mo>}$ for the cellular neural networks with multiple delays, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615004404&_mathId=si3.gif&_user=111111111&_pii=S0022039615004404&_rdoc=1&_issn=00220396&md5=fc9d8840a84220a6349fde72aea9ea11" title="Click to view the MathML source">c_鈦?/sub><0class="mathContainer hidden">class="mathCode">$c_{鈦?/mo>}<0$ is the critical wave speed. Then, by the weighted energy method together with the squeezing technique, we further show the global stability of all non-critical traveling waves for this model, that is, for all monotone waves with the speed class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615004404&_mathId=si4.gif&_user=111111111&_pii=S0022039615004404&_rdoc=1&_issn=00220396&md5=207d080a79b6dec1b971efaec873e8e5" title="Click to view the MathML source">c<c_鈦?/sub>class="mathContainer hidden">class="mathCode">$c<c_{鈦?/mo>}$, the original lattice solutions converge time-exponentially to the corresponding traveling waves, when the initial perturbations around the monotone traveling waves decay exponentially at far fields, but can be arbitrarily large in other locations.