Large time behavior for the fast diffusion equation with critical absorption

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• 作者：Said Benachour^a ; ^{said.benachour@univ-lorraine.fr" class="auth_mail" title="E-mail the corresponding author} ; Razvan Gabriel Iagar^b ; ^c ; ^{razvan.iagar@icmat.es" class="auth_mail" title="E-mail the corresponding author} ; Philippe Laurenç ; ot^d ; ^{laurenco@math.univ-toulouse.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35B33 ; 35B40 ; 35B45 ; 35K67
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 June 2016
• 年：2016
• 卷：260
• 期：11
• 页码：8000-8024
• 全文大小：379 K

文摘

We study the large time behavior of nonnegative solutions to the Cauchy problem for a fast diffusion equation with critical zero order absorption with 16000747&_mathId=si2.gif&_user=111111111&_pii=S0022039616000747&_rdoc=1&_issn=00220396&md5=7710bdef4dc8eb3a0d40568a26dfd765" title="Click to view the MathML source">m_c:=(N−2)₊/N<m<1$m_c:={(N-2)}_{+}/N<m<1$ and 16000747&_mathId=si3.gif&_user=111111111&_pii=S0022039616000747&_rdoc=1&_issn=00220396&md5=15b341ab8c2b1d45f4b5f56b1825c864" title="Click to view the MathML source">q=m+2/N$q=m+2/N$. Given an initial condition 16000747&_mathId=si4.gif&_user=111111111&_pii=S0022039616000747&_rdoc=1&_issn=00220396&md5=a1bcccbe1cf8669ace34a40cd23f54a4" title="Click to view the MathML source">u₀$u_0$ decaying arbitrarily fast at infinity, we show that the asymptotic behavior of the corresponding solution u   is given by a Barenblatt profile with a logarithmic scaling, thereby extending a previous result requiring a specific algebraic lower bound on 16000747&_mathId=si4.gif&_user=111111111&_pii=S0022039616000747&_rdoc=1&_issn=00220396&md5=a1bcccbe1cf8669ace34a40cd23f54a4" title="Click to view the MathML source">u₀$u_0$. A by-product of our analysis is the derivation of sharp gradient estimates and a universal lower bound, which have their own interest and hold true for general exponents 16000747&_mathId=si42.gif&_user=111111111&_pii=S0022039616000747&_rdoc=1&_issn=00220396&md5=366d3a7e9d40cbfd63d3fd12a57806c1" title="Click to view the MathML source">q>1$q>1$.