{"title":"Initial-boundary value problems to semilinear multi-term fractional differential equations","authors":"S. Siryk, Nataliya Vasylyeva","doi":"10.3934/cpaa.2023068","DOIUrl":null,"url":null,"abstract":"For $\\nu,\\nu_i,\\mu_j\\in(0,1)$, we analyze the semilinear integro-differential equation on the one-dimensional domain $\\Omega=(a,b)$ in the unknown $u=u(x,t)$ \\[ \\mathbf{D}_{t}^{\\nu}(\\varrho_{0}u)+\\sum_{i=1}^{M}\\mathbf{D}_{t}^{\\nu_{i}}(\\varrho_{i}u) -\\sum_{j=1}^{N}\\mathbf{D}_{t}^{\\mu_{j}}(\\gamma_{j}u) -\\mathcal{L}_{1}u-\\mathcal{K}*\\mathcal{L}_{2}u+f(u)=g(x,t), \\] where $\\mathbf{D}_{t}^{\\nu},\\mathbf{D}_{t}^{\\nu_{i}}, \\mathbf{D}_{t}^{\\mu_{j}}$ are Caputo fractional derivatives, $\\varrho_0=\\varrho_0(t)>0,$ $\\varrho_{i}=\\varrho_{i}(t)$, $\\gamma_{j}=\\gamma_{j}(t)$, $\\mathcal{L}_{k}$ are uniform elliptic operators with time-dependent smooth coefficients, $\\mathcal{K}$ is a summable convolution kernel. Particular cases of this equation are the recently proposed advanced models of oxygen transport through capillaries. Under certain structural conditions on the nonlinearity $f$ and orders $\\nu,\\nu_i,\\mu_j$, the global existence and uniqueness of classical and strong solutions to the related initial-boundary value problems are established via the so-called continuation arguments method. The crucial point is searching suitable a priori estimates of the solution in the fractional H\\\"{o}lder and Sobolev spaces. The problems are also studied from the numerical point of view.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications on Pure and Applied Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/cpaa.2023068","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
For $\nu,\nu_i,\mu_j\in(0,1)$, we analyze the semilinear integro-differential equation on the one-dimensional domain $\Omega=(a,b)$ in the unknown $u=u(x,t)$ \[ \mathbf{D}_{t}^{\nu}(\varrho_{0}u)+\sum_{i=1}^{M}\mathbf{D}_{t}^{\nu_{i}}(\varrho_{i}u) -\sum_{j=1}^{N}\mathbf{D}_{t}^{\mu_{j}}(\gamma_{j}u) -\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u+f(u)=g(x,t), \] where $\mathbf{D}_{t}^{\nu},\mathbf{D}_{t}^{\nu_{i}}, \mathbf{D}_{t}^{\mu_{j}}$ are Caputo fractional derivatives, $\varrho_0=\varrho_0(t)>0,$ $\varrho_{i}=\varrho_{i}(t)$, $\gamma_{j}=\gamma_{j}(t)$, $\mathcal{L}_{k}$ are uniform elliptic operators with time-dependent smooth coefficients, $\mathcal{K}$ is a summable convolution kernel. Particular cases of this equation are the recently proposed advanced models of oxygen transport through capillaries. Under certain structural conditions on the nonlinearity $f$ and orders $\nu,\nu_i,\mu_j$, the global existence and uniqueness of classical and strong solutions to the related initial-boundary value problems are established via the so-called continuation arguments method. The crucial point is searching suitable a priori estimates of the solution in the fractional H\"{o}lder and Sobolev spaces. The problems are also studied from the numerical point of view.
期刊介绍:
CPAA publishes original research papers of the highest quality in all the major areas of analysis and its applications, with a central theme on theoretical and numeric differential equations. Invited expository articles are also published from time to time. It is edited by a group of energetic leaders to guarantee the journal''s highest standard and closest link to the scientific communities.