A Korteweg-de Vries type of fifth-order equations on a finite domain with point dissipation

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• 作者：Guangyue Gao ; Shu-Ming Sun ; ^{sun@math.vt.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：A KdV type of equations ; Point dissipation ; Closed loop control
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 June 2016
• 年：2016
• 卷：438
• 期：1
• 页码：200-239
• 全文大小：601 K

文摘

The paper discusses initial value problem of a Korteweg–de Vries type of fifth-order equation posed on a periodic domain class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000846&_mathId=si2.gif&_user=111111111&_pii=S0022247X16000846&_rdoc=1&_issn=0022247X&md5=52cfc29dbe5ceea2c9d4d947e7e0d3b1" title="Click to view the MathML source">x∈[0,2π]class="mathContainer hidden">class="mathCode">$x\in [0,2\pi ]$ with periodic boundary conditions class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000846&_mathId=si3.gif&_user=111111111&_pii=S0022247X16000846&_rdoc=1&_issn=0022247X&md5=e2f6c1c4b55b78daeebe24ef0351458a" title="Click to view the MathML source">w_ix(0,t)=w_ix(2π,t)class="mathContainer hidden">class="mathCode">$w_{ix}(0,t)=w_{ix}(2\pi ,t)$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000846&_mathId=si4.gif&_user=111111111&_pii=S0022247X16000846&_rdoc=1&_issn=0022247X&md5=e056e2c5e3fc5cf99b9f63c7248f3899" title="Click to view the MathML source">i=0,2,3,4class="mathContainer hidden">class="mathCode">$i=0,2,3,4$ and an class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000846&_mathId=si5.gif&_user=111111111&_pii=S0022247X16000846&_rdoc=1&_issn=0022247X&md5=0dd75c8840484d506c7513f52f42e4f6" title="Click to view the MathML source">L²class="mathContainer hidden">class="mathCode">$L^2$-stabilizing feedback control law class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000846&_mathId=si6.gif&_user=111111111&_pii=S0022247X16000846&_rdoc=1&_issn=0022247X&md5=018499b609a484c4c65d28e5c1ff7255" title="Click to view the MathML source">w_x(2π,t)=αw_x(0,t)+(1−α)w_xxx(0,t)class="mathContainer hidden">class="mathCode">$w_x(2\pi ,t)=\alpha w_x(0,t)+(1-\alpha )w_{xxx}(0,t)$ where n   is a fixed positive integer, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000846&_mathId=si7.gif&_user=111111111&_pii=S0022247X16000846&_rdoc=1&_issn=0022247X&md5=32ea25b14a00ed756bb8e8e1ae8b46d9" title="Click to view the MathML source">a_jclass="mathContainer hidden">class="mathCode">$a_j$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000846&_mathId=si8.gif&_user=111111111&_pii=S0022247X16000846&_rdoc=1&_issn=0022247X&md5=ecf1f5688eaf26d37428c18b8eba0ba9" title="Click to view the MathML source">j=1,2,⋯,nclass="mathContainer hidden">class="mathCode">$j=1,2,\cdots ,n$, α   are real constants, and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000846&_mathId=si50.gif&_user=111111111&_pii=S0022247X16000846&_rdoc=1&_issn=0022247X&md5=caa1547580558bdf6fd98e504e054207" title="Click to view the MathML source">|α|<1class="mathContainer hidden">class="mathCode">$|\alpha |<1$. It is shown that for class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000846&_mathId=si10.gif&_user=111111111&_pii=S0022247X16000846&_rdoc=1&_issn=0022247X&md5=82c4c69d661528b0239ecef3e852bb73">class="imgLazyJSB inlineImage" height="20" width="128" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16000846-si10.gif">class="mathContainer hidden">class="mathCode">$w_0(x)\in H_{\alpha }^1(0,2\pi )$ with the boundary conditions described above, the problem is locally well-posed for class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000846&_mathId=si11.gif&_user=111111111&_pii=S0022247X16000846&_rdoc=1&_issn=0022247X&md5=e6e800b2d9406f8902c50828fe02c44c">class="imgLazyJSB inlineImage" height="20" width="168" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16000846-si11.gif">class="mathContainer hidden">class="mathCode">$w\in C([0,T];H_{\alpha }^1(0,2\pi ))$ with a conserved volume of w  , class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000846&_mathId=si12.gif&_user=111111111&_pii=S0022247X16000846&_rdoc=1&_issn=0022247X&md5=dabe531451163735eb823af28492caa5">class="imgLazyJSB inlineImage" height="25" width="135" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16000846-si12.gif">class="mathContainer hidden">class="mathCode">$[w]={\int }_0^{2\pi }w(x,t)dx$. Moreover, the solution with small initial condition exists globally and approaches to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000846&_mathId=si13.gif&_user=111111111&_pii=S0022247X16000846&_rdoc=1&_issn=0022247X&md5=7f2d5a90ef7ad6b0069668cbeda60b54" title="Click to view the MathML source">[w₀(x)]/(2π)class="mathContainer hidden">class="mathCode">$[w_0(x)]/(2\pi )$ as class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000846&_mathId=si14.gif&_user=111111111&_pii=S0022247X16000846&_rdoc=1&_issn=0022247X&md5=e357bf694cb481d33d63756aa636b35d" title="Click to view the MathML source">t→+∞class="mathContainer hidden">class="mathCode">$t\to +\infty$.