Degenerate integrodifferential equations of parabolic type with Robin boundary conditions: L^p-theory

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• 作者：Angelo Favini ^{angelo.favini@unibo.it" class="auth_mail" title="E-mail the corresponding author} ; Alfredo Lorenzi ; Hiroki Tanabe
• 关键词：Direct problems of parabolic type for time depending operator coefficients in Banach spaces ; Linear parabolic integro-differential equations ; Existence and uniqueness of solutions
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 March 2017
• 年：2017
• 卷：447
• 期：1
• 页码：579-665
• 全文大小：1096 K

文摘

The aim of this paper consists in solving integrodifferential problem of type (1.1)–(1.2) that may degenerate both in space and time. More precisely, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306254&_mathId=si1.gif&_user=111111111&_pii=S0022247X16306254&_rdoc=1&_issn=0022247X&md5=0c06ca5c374334610c0b7bc49ae2a788" title="Click to view the MathML source">{M_p(t)}_t∈[0,T]class="mathContainer hidden">class="mathCode">${\{M_p(t)\}}_{t\in [0,T]}$ is a family of multiplication operators related to a scalar function class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306254&_mathId=si2.gif&_user=111111111&_pii=S0022247X16306254&_rdoc=1&_issn=0022247X&md5=808295934f80fddcc716d3b0fad9d295" title="Click to view the MathML source">m(t,x)class="mathContainer hidden">class="mathCode">$m(t,x)$ that may vanish, while class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306254&_mathId=si3.gif&_user=111111111&_pii=S0022247X16306254&_rdoc=1&_issn=0022247X&md5=95273915cbc238f4e24c9d154cb50624" title="Click to view the MathML source">{L_p(t)}_t∈[0,T]class="mathContainer hidden">class="mathCode">${\{L_p(t)\}}_{t\in [0,T]}$ is the realization of a family of linear second-order differential operators, with smooth coefficients, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306254&_mathId=si4.gif&_user=111111111&_pii=S0022247X16306254&_rdoc=1&_issn=0022247X&md5=7f06420a679909f2698211882a1ee3be" title="Click to view the MathML source">{L(t)}_t∈[0,T]class="mathContainer hidden">class="mathCode">${\{\mathcal{L}(t)\}}_{t\in [0,T]}$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306254&_mathId=si5.gif&_user=111111111&_pii=S0022247X16306254&_rdoc=1&_issn=0022247X&md5=b9931c8acc3bd2bd6dba858b12a2d1bf" title="Click to view the MathML source">{L_p(t)}class="mathContainer hidden">class="mathCode">$\{L_p(t)\}$ being invertible for all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306254&_mathId=si6.gif&_user=111111111&_pii=S0022247X16306254&_rdoc=1&_issn=0022247X&md5=d1bc6ba1e411dc4d76834851b140ddb3" title="Click to view the MathML source">t∈[0,T]class="mathContainer hidden">class="mathCode">$t\in [0,T]$. Moreover, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306254&_mathId=si7.gif&_user=111111111&_pii=S0022247X16306254&_rdoc=1&_issn=0022247X&md5=a7cb9bbb6f2d4ceaac94aa297035411d" title="Click to view the MathML source">{B_p(t,s)}_{t,s∈[0,T],s≤t}class="mathContainer hidden">class="mathCode">${\{B_p(t,s)\}}_{t,s\in [0,T],s\le t}$ is the realization of a family class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306254&_mathId=si8.gif&_user=111111111&_pii=S0022247X16306254&_rdoc=1&_issn=0022247X&md5=fd261c209f3e56b4ae3b8b9756d3ceea" title="Click to view the MathML source">{B(t,s)}_{t∈[0,T],s≤t}class="mathContainer hidden">class="mathCode">${\{\mathcal{B}(t,s)\}}_{t\in [0,T],s\le t}$ of linear second-order differential operators with smooth coefficients. Finally, the scalar functions a and b   are such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306254&_mathId=si9.gif&_user=111111111&_pii=S0022247X16306254&_rdoc=1&_issn=0022247X&md5=d2159625ea410f6af7018049afe8fa97" title="Click to view the MathML source">1/aclass="mathContainer hidden">class="mathCode">$1/a$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306254&_mathId=si10.gif&_user=111111111&_pii=S0022247X16306254&_rdoc=1&_issn=0022247X&md5=3487379ee4921f02ca430917b1618934" title="Click to view the MathML source">b/aclass="mathContainer hidden">class="mathCode">$b/a$ are Hölder-continuous with suitable Hölder exponents.