{"title":"Harnack Type Inequalities and Multiple Solutions in Cones of Nonlinear Problems","authors":"Diana-Raluca Herlea, D. O’Regan, R. Precup","doi":"10.4171/zaa/1655","DOIUrl":"https://doi.org/10.4171/zaa/1655","url":null,"abstract":"","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":"22 1","pages":"151-170"},"PeriodicalIF":1.2,"publicationDate":"2020-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87019628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Existence and New General Decay Results for a Viscoelastic Timoshenko System","authors":"J. Hassan, S. Messaoudi, M. Zahri","doi":"10.4171/zaa/1657","DOIUrl":"https://doi.org/10.4171/zaa/1657","url":null,"abstract":"","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":"35 1","pages":"185-222"},"PeriodicalIF":1.2,"publicationDate":"2020-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85468683","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the exact controllability and stabilization of a system of two wave equations coupled by velocities with an internal, local control acting on only one equation. We distinguish two cases. In the first one, when the waves propagate at the same speed: using a frequency domain approach combined with multiplier technique, we prove that the system is exponentially stable when the coupling region satisfies the geometric control condition GCC. Following a result of Haraux ([11]), we establish the main indirect observability inequality. This results leads, by the HUM method, to prove that the total system is exactly controllable by means of locally distributed control. In the second case, when the waves propagate at different speed, we establish an exponential decay rate in the weak energy space. Consequently, the system is exactly controllable using a result of [11]. Finally, numerically, we provide results that insure the theoretical results of [13].
{"title":"Exact Controllability and Stabilization of Locally Coupled Wave Equations: Theoretical Results","authors":"S. Gerbi, Chiraz Kassem, Amina Mortada, A. Wehbe","doi":"10.4171/ZAA/1673","DOIUrl":"https://doi.org/10.4171/ZAA/1673","url":null,"abstract":"In this paper, we study the exact controllability and stabilization of a system of two wave equations coupled by velocities with an internal, local control acting on only one equation. We distinguish two cases. In the first one, when the waves propagate at the same speed: using a frequency domain approach combined with multiplier technique, we prove that the system is exponentially stable when the coupling region satisfies the geometric control condition GCC. Following a result of Haraux ([11]), we establish the main indirect observability inequality. This results leads, by the HUM method, to prove that the total system is exactly controllable by means of locally distributed control. In the second case, when the waves propagate at different speed, we establish an exponential decay rate in the weak energy space. Consequently, the system is exactly controllable using a result of [11]. Finally, numerically, we provide results that insure the theoretical results of [13].","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2020-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44455065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we obtain new estimates of the sequential Caputo fractional derivatives of a function at its extremum points. We derive comparison principles for the linear fractional differential equations, and apply these principles to obtain lower and upper bounds of solutions of linear and nonlinear fractional differential equations. The extremum principle is then applied to show that the initial-boundary-value problem for nonlinear anomalous diffusion possesses at most one classical solution and this solution depends continuously on the initial and boundary data. This answers positively to the open problem about maximum principle for the space and time-space fractional PDEs posed by Luchko in 2011. The extremum principle for an elliptic equation with a fractional derivative and for the fractional Laplace equation are also proved.
{"title":"Maximum Principle for Space and Time-Space Fractional Partial Differential Equations","authors":"M. Kirane, B. Torebek","doi":"10.4171/ZAA/1685","DOIUrl":"https://doi.org/10.4171/ZAA/1685","url":null,"abstract":"In this paper we obtain new estimates of the sequential Caputo fractional derivatives of a function at its extremum points. We derive comparison principles for the linear fractional differential equations, and apply these principles to obtain lower and upper bounds of solutions of linear and nonlinear fractional differential equations. The extremum principle is then applied to show that the initial-boundary-value problem for nonlinear anomalous diffusion possesses at most one classical solution and this solution depends continuously on the initial and boundary data. This answers positively to the open problem about maximum principle for the space and time-space fractional PDEs posed by Luchko in 2011. The extremum principle for an elliptic equation with a fractional derivative and for the fractional Laplace equation are also proved.","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2020-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42671671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Measures of Non-Compactness and Sobolev–Lorentz Spaces","authors":"Ondřej Bouchala","doi":"10.4171/zaa/1649","DOIUrl":"https://doi.org/10.4171/zaa/1649","url":null,"abstract":"","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":"64 1","pages":"27-40"},"PeriodicalIF":1.2,"publicationDate":"2020-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/zaa/1649","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72502313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the decay rates of solutions to thermoelastic systems in materials where, in contrast to classical thermoelastic models for Kirchhoff type plates, two temperatures are involved, related by an elliptic equation. The arising initial value problems deal with systems of partial differential equations involving Schrödinger like equations, hyperbolic and elliptic equations. Depending on the model – with Fourier or with Cattaneo type heat conduction – we obtain polynomial decay rates without or with regularity loss. This way we obtain another example where the loss of regularity in the Cauchy problem corresponds to the loss of exponential stability in bounded domains. The well-posedness is done using semigroup theory in appropriate space reflecting the different regularity compared to the classical single temperature case, and the (optimal) decay estimates are obtained with sophisticated pointwise estimates in Fourier space.
{"title":"The Cauchy Problem for Thermoelastic Plates with Two Temperatures","authors":"R. Racke, Yoshihiro Ueda","doi":"10.4171/ZAA/1653","DOIUrl":"https://doi.org/10.4171/ZAA/1653","url":null,"abstract":"We consider the decay rates of solutions to thermoelastic systems in materials where, in contrast to classical thermoelastic models for Kirchhoff type plates, two temperatures are involved, related by an elliptic equation. The arising initial value problems deal with systems of partial differential equations involving Schrödinger like equations, hyperbolic and elliptic equations. Depending on the model – with Fourier or with Cattaneo type heat conduction – we obtain polynomial decay rates without or with regularity loss. This way we obtain another example where the loss of regularity in the Cauchy problem corresponds to the loss of exponential stability in bounded domains. The well-posedness is done using semigroup theory in appropriate space reflecting the different regularity compared to the classical single temperature case, and the (optimal) decay estimates are obtained with sophisticated pointwise estimates in Fourier space.","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":"21 1","pages":"103-129"},"PeriodicalIF":1.2,"publicationDate":"2020-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78877597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Blow-Up Criterion of Strong Solution with Vacuum for the 2D Nonhomogeneous Density-Temperature-Dependent Boussinesq Equations","authors":"Z. Ye","doi":"10.4171/zaa/1652","DOIUrl":"https://doi.org/10.4171/zaa/1652","url":null,"abstract":"","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":"54 1","pages":"83-101"},"PeriodicalIF":1.2,"publicationDate":"2020-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82714329","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Global Bifurcation from Intervals for Problems with Pucci's Operator","authors":"Hua Luo, Guowei Dai","doi":"10.4171/zaa/1651","DOIUrl":"https://doi.org/10.4171/zaa/1651","url":null,"abstract":"","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":"268 1","pages":"67-81"},"PeriodicalIF":1.2,"publicationDate":"2020-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73253417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Fractional $p$-Laplacian Problems with Hardy Terms and Critical Exponents","authors":"A. Fiscella, Hadi Mirzaee","doi":"10.4171/zaa/1647","DOIUrl":"https://doi.org/10.4171/zaa/1647","url":null,"abstract":"","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2019-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/zaa/1647","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43865936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Blow-Up Phenomena for the Periodic Two-Component Degasperis–Procesi System","authors":"Xingxing Liu","doi":"10.4171/zaa/1646","DOIUrl":"https://doi.org/10.4171/zaa/1646","url":null,"abstract":"","PeriodicalId":54402,"journal":{"name":"Zeitschrift fur Analysis und ihre Anwendungen","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2019-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45764702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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