In this paper, we investigate a reaction-diffusion equation ut−duxx = au+ ∫ t 0 u p(x,τ)dτ+k(x) with double free boundaries. We study blowup phenomena in finite time and asymptotic behavior of time-global solutions. Our results show if ∫ h0 −h0 k(x)ψ1dx is large enough, then the blowup occurs. Meanwhile we also prove when T∗<+∞, the solution must blow up in finite time. On the other hand, we prove that the solution decays at an exponential rate and the two free boundaries converge to a finite limit provided the initial datum is small sufficiently. AMS Subject Classifications: 35K20, 35R35, 92B05 Chinese Library Classifications: O175
{"title":"Blowup and Asymptotic Behavior of a Free Boundary Problem with a Nonlinear Memory","authors":"Jiahui Huang","doi":"10.4208/jpde.v33.n3.5","DOIUrl":"https://doi.org/10.4208/jpde.v33.n3.5","url":null,"abstract":"In this paper, we investigate a reaction-diffusion equation ut−duxx = au+ ∫ t 0 u p(x,τ)dτ+k(x) with double free boundaries. We study blowup phenomena in finite time and asymptotic behavior of time-global solutions. Our results show if ∫ h0 −h0 k(x)ψ1dx is large enough, then the blowup occurs. Meanwhile we also prove when T∗<+∞, the solution must blow up in finite time. On the other hand, we prove that the solution decays at an exponential rate and the two free boundaries converge to a finite limit provided the initial datum is small sufficiently. AMS Subject Classifications: 35K20, 35R35, 92B05 Chinese Library Classifications: O175","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"2 1","pages":"249-260"},"PeriodicalIF":0.3,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89965948","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Nash Type Result for Divergence Parabolic Equation Related to Hörmander's Vector Fields","authors":"Lingling Hou Pengcheng Niu","doi":"10.4208/jpde.v33.n4.3","DOIUrl":"https://doi.org/10.4208/jpde.v33.n4.3","url":null,"abstract":"","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"31 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91183798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Multiple Solutions for a Fractional p-Laplacian Equation with Concave Nonlinearities","authors":"global RuichangPei","doi":"10.4208/jpde.v33.n2.1","DOIUrl":"https://doi.org/10.4208/jpde.v33.n2.1","url":null,"abstract":"","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"43 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87208545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Well-Posedness of Solutions for Sixth-Order Cahn-Hilliard Equation Arising in Oil-Water-Surfactant Mixtures","authors":"Haichao Zhao","doi":"10.4208/jpde.v33.n3.1","DOIUrl":"https://doi.org/10.4208/jpde.v33.n3.1","url":null,"abstract":"","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"29 1","pages":"193-207"},"PeriodicalIF":0.3,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84211751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Higher Order Dissipative-Dispersive System and Application","authors":"Samer Israwi","doi":"10.4208/jpde.v33.n3.6","DOIUrl":"https://doi.org/10.4208/jpde.v33.n3.6","url":null,"abstract":"","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"21 1","pages":"261-274"},"PeriodicalIF":0.3,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88305159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Eigenvalues of Elliptic Systems for the Mixed Problem in Perturbations of Lipschitz Domains with Nonhomogeneous Neumann Boundary Conditions","authors":"global sci","doi":"10.4208/jpde.v33.n1.4","DOIUrl":"https://doi.org/10.4208/jpde.v33.n1.4","url":null,"abstract":"","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"13 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84830503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Reaction-diffusion Equations","authors":"J. Shu","doi":"10.4208/jpde.v33.n4.4","DOIUrl":"https://doi.org/10.4208/jpde.v33.n4.4","url":null,"abstract":"","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"72 1","pages":"377-394"},"PeriodicalIF":0.3,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82872683","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Global Solution and Exponential Stability for a Laminated Beam with Fourier Thermal Law","authors":"global CarlosRaposo","doi":"10.4208/jpde.v33.n2.4","DOIUrl":"https://doi.org/10.4208/jpde.v33.n2.4","url":null,"abstract":"","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"43 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76840685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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