Damped wave equation with a critical nonlinearity in higher space dimensions

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• 作者：Nakao Hayashiup>aup>up> ; up>up>nhayashi@math.sci.osaka-u.ac.jp" class="auth_mail" title="E-mail the corresponding authorup> ; Pavel I. Naumkinup>bup>up> ; up>up>pavelni@matmor.unam.mx" class="auth_mail" title="E-mail the corresponding authorup>
• 关键词：Damped wave equation ; Critical nonlinearity ; Large time asymptotics ; Higher space dimension
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：801-822
• 全文大小：401 K

文摘

We study the Cauchy problem for nonlinear damped wave equations with a critical defocusing power nonlinearity urce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1630508X&_mathId=si1.gif&_user=111111111&_pii=S0022247X1630508X&_rdoc=1&_issn=0022247X&md5=9394f571bd766bc0e524fbe19a121046">urce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X1630508X-si1.gif">, where n   denotes the space dimension. For ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1630508X&_mathId=si2.gif&_user=111111111&_pii=S0022247X1630508X&_rdoc=1&_issn=0022247X&md5=3f27486c041b98521d9f0644d68962ee" title="Click to view the MathML source">n=1,2,3$n=1,2,3$, global in time existence of small solutions was shown in [4]. In this paper, we generalize the results to any spatial dimension via the method of decomposition of the equation into the high and low frequency components under the assumption that the initial data are small and decay rapidly at infinity. Furthermore we present a sharp time decay estimate of solutions with a logarithmic correction.