Uniform stabilization of the fourth order Schrödinger equation

详细信息    查看全文

• 作者：Belkacem Aksas ; Salah-Eddine Rebiai ; ^{rebiai@hotmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Fourth order Schrö ; dinger equation ; Boundary stabilization ; Internal stabilization ; Exponential stability
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：446
• 期：2
• 页码：1794-1813
• 全文大小：350 K

文摘

We study both boundary and internal stabilization problems for the fourth order Schrödinger equation in a smooth bounded domain Ω of mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305777&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305777&_rdoc=1&_issn=0022247X&md5=cc3c55eae7737e8b8d13e4af5cd76f77" title="Click to view the MathML source">RⁿmathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll">mathvariant="double-struck">Rnmath>. We first consider the boundary stabilization problem. By introducing suitable dissipative boundary conditions, we prove that the solution decays exponentially in an appropriate energy space. In the internal stabilization problem, by assuming that the damping term is effective on a neighborhood of a part of the boundary, we prove the exponential decay of the mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305777&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305777&_rdoc=1&_issn=0022247X&md5=8659a0c5ce66f47416cd479b74f0870b" title="Click to view the MathML source">L²(Ω)mathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll">L2(mathvariant="normal">Ω)math>-energy of the solution. Both results are established by using multiplier techniques and compactness/uniqueness arguments.