[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85814-en":3,"doc-seo-85814-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85814,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Regularity and high-order time stepping for semilinear subdiffusion equations with singular initial data beyond the L∞ framework","This paper develops a numerical analysis for semilinear subdiffusion equations with singular initial data outside the L∞ setting. The main challenge is that the nonlinear term exhibits stronger singular behavior than earlier frameworks, making the usual Lipschitz arguments in the base space insufficient. The study shifts to weaker fractional Sobolev-type spaces, using smoothing properties of subdiffusion solution operators and refined nonlinear assumptions. Well-posedness, regularity for mild solutions, and pointwise-in-time error bounds for exponential convolution quadrature are proved, and numerical tests validate the expected convergence rates.","arXiv :2607 . 10100v1 [math .NA] 11 Jul 2026  \nRegularity and high-order time stepping for semilinear subdiffusion equations with singular initial data beyond the  \nL ∞ framework  \nRunjie Zhang∗1 and Dongling Wang†1  \n1 Hunan Research Center of the Basic Discipline Fundamental Algorithmic Theory  \nand Novel Computational Methods, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, China  \nJuly 14, 2026  \nAbstract  \nThis paper aims to analyze a numerical scheme for semilinear subdiffusion problems with singular initial data beyond the L∞ framework. The main difficulty lies in the stronger singular behavior of the nonlinear term compared with previous analyses. Since the singular initial datum is too rough to guarantee a uniform L∞ bound for the solution, the usual Lipschitz framework in the base space is no longer sufficient. The analysis must instead be carried out in weaker fractional Sobolev-type spaces, where nonlinear composition is more delicate and the term f(u(t)) may exhibit an amplified singularity relative to that of u(t) . To overcome this difficulty, we exploit the smoothing properties of the subdiffusion solution operators and formulate suitable nonlinear assumptions in fractional operator spaces. These smoothing estimates allow part of the singularity to be transferred from the nonlinear term to the solution operators, where it can be controlled. Under these assumptions, we establish well-posedness and regularity results for the mild solution and derive a pointwise-in-time error estimate for the exponential convolution quadrature method. Numerical experiments confirm the predicted convergence rates.  \nKeywords: semilinear subdiffusion equation, singular initial data, convolution quadrature, exponential integrator  \nMathematics Subject Classification (2010): 35R11, 65M15, 65R20  \n1 Introduction  \nThis paper analyzes high-order time-stepping methods for the semilinear subdiffusion problem  \n(∂uαt(u0)() +u0A, u (t) = f(u(t)), 0 \u003C t ≤ T,  \n(1)  \n∗ Email: [rjzhangxtu@163.com](rjzhangxtu@163.com)  \n†Corresponding author: Wang. Email: [wdymath@xtu.edu.cn. The](wdymath@xtu.edu.cn. The) research of Dongling Wang is supported in part by the National Natural Science Foundation of China under Grants 12271463 and the Natural Science Foundation of Hunan Province under Grant 2026JJ50363 .  \non the Hilbert space X = L2 (Ω), where Ω is a convex polygonal domain in Rd , d = 1 , 2 , 3. The operator ∂αtu denotes the Caputo fractional time derivative of order α ∈ (0 , 1) in time, defined by  \n∂αtu := Γ(1~~ ~~~~ ~~α) Z0 t (t − s)−α ddsu (s) ds, (2)  \nwith Γ(z) = R0∞ sz−1e−sds being the gamma function. The spatial operator A : D (A) ⊂ X → X is the sectorial operator induced by a second-order elliptic operator with suitable boundary conditions. The nonlinear term f is assumed to be a smooth function on R. The initial value is allowed to be nonsmooth: u0 ∈ D (Aγ ) for some 0 \u003C γ ≤ d/4 . A function u ∈ C([0, T];L2 (Ω)) is called a mild solution of (1) if it satisfies the integral equation  \nu (t) = F(t)u0 + Z0 t E (t − s)f (u(s))ds, (3)  \nwhere F(t) and E(t) are the solution operators defined in (4) below. The mild and strong formulations are equivalent whenever the solution possesses sufficient regularity.  \nThe subdiffusion equation in the form (1) has attracted considerable attention in the development of stable and accurate numerical methods, along with rigorous numerical analysis, owing to its remarkable ability to model a wide range of anomalously slow transport processes. Two major strategies are available for approximating the fractional derivative (2): convolution quadrature (CQ) [1, 2 , 3 , 4] and L1-type methods [5, 6 , 7 , 8] . In the stability analysis of L1-type methods, deriving explicit bounds for discrete Gronwall inequalities is of great significance, and several such inequalities have been established in the literature [9, 10 , 11] . For a comprehensive treatment of these two st","cbCaidgDT3lVdgpP","https://ap.wps.com/l/cbCaidgDT3lVdgpP","pdf",357011,1,18,"English","en",105,"# Introduction\n## Fractional subdiffusion model and mild solutions\n## Numerical strategies: convolution quadrature and L1-type methods\n## Challenges from initial-time singularity\n## Related work and higher-order schemes","[{\"question\":\"Why does the analysis need to go beyond the L∞ framework for singular initial data?\",\"answer\":\"Because the solution cannot be guaranteed to have a uniform L∞ bound when the initial datum is sufficiently rough, and the usual Lipschitz framework in the base space no longer controls the nonlinear term.\"},{\"question\":\"What spaces are used to handle the nonlinear term’s amplified singularity?\",\"answer\":\"The analysis is carried out in weaker fractional Sobolev-type spaces, where nonlinear composition becomes more delicate and the term f(u(t)) may show amplified singular behavior relative to u(t).\"},{\"question\":\"How is the exponential convolution quadrature method justified in the paper?\",\"answer\":\"By leveraging smoothing estimates of the subdiffusion solution operators to transfer part of the singularity from the nonlinear term to the operators, the authors prove pointwise-in-time error estimates and verify the predicted convergence rates numerically.\"}]",1784206412,45,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"regularity-and-high-order-time-stepping-for-semilinear-subdiffusion-equations-with-singular-initial-data-beyond-the-l-framework","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/regularity-and-high-order-time-stepping-for-semilinear-subdiffusion-equations-with-singular-initial-data-beyond-the-l-framework/85814/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why does the analysis need to go beyond the L∞ framework for singular initial data?","Question",{"text":75,"@type":76},"Because the solution cannot be guaranteed to have a uniform L∞ bound when the initial datum is sufficiently rough, and the usual Lipschitz framework in the base space no longer controls the nonlinear term.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What spaces are used to handle the nonlinear term’s amplified singularity?",{"text":80,"@type":76},"The analysis is carried out in weaker fractional Sobolev-type spaces, where nonlinear composition becomes more delicate and the term f(u(t)) may show amplified singular behavior relative to u(t).",{"name":82,"@type":73,"acceptedAnswer":83},"How is the exponential convolution quadrature method justified in the paper?",{"text":84,"@type":76},"By leveraging smoothing estimates of the subdiffusion solution operators to transfer part of the singularity from the nonlinear term to the operators, the authors prove pointwise-in-time error estimates and verify the predicted convergence rates numerically.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & 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