{"paper_id":"aed05438-7263-4366-8d82-37f9eda53169","body_text":"Non-Autonomous Maximal Regularity Under Besov Regularity in Time | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Non-Autonomous Maximal Regularity Under Besov Regularity in Time Mahdi achache This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-2216560/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We consider the maximal regularity problem for non-autonomous Cauchy problems u′(t) + A(t)u(t) = f (t) t-a.e., u(0) = u0. Each operator A(t) arises from a time depending sesquilinear form a(t) on a Hilbert space H with constant domain V. We prove maximal L p -regularity result for p ≤ 2 under min-imal regularity assumptions on the forms. Our main assumption is that (A(t)) t∈[0,τ ] arepiecewise in the Besov space B p 1/2, 2 p with respect to the variable t. This improves previouslyknown results. We give three examples which illustrate our results. Mathematics Subject Classification (2010): 35K90, 35K45, 47D06. Besov space maximal regularity mon-autonomous evolution equations sesquilinear forms. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {\"props\":{\"pageProps\":{\"initialData\":{\"identity\":\"rs-2216560\",\"acceptedTermsAndConditions\":true,\"allowDirectSubmit\":true,\"archivedVersions\":[],\"articleType\":\"Research Article\",\"associatedPublications\":[],\"authors\":[{\"id\":148089311,\"identity\":\"f6f46da9-fc0c-489e-a7c3-f38a94321934\",\"order_by\":0,\"name\":\"Mahdi achache\",\"email\":\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+klEQVRIiWNgGAWjYFACHgYGxgYgLcFgcOAjiMHA2HiAwYZILQdnNgApIPcAQxqRWph5wVoYGPBq4W8/e/DDzx020fyzmzcett1hU6fbfhhoS8I9nFokzuQlS/aeScudcedYweHcM2kSZmcSQVqKcVtzg8dAmrHtcG7DjRyDw7lthyXMDgC1MP5IwKlD/gaP8W/Gtv+580FaLEFazj8E2YJbi8ENHjOgLQdyN4C0MIK03EjEr8XwTI6ZZW9bcu7GG2kFB3vb0iS33QDakoBHi9zxM8Y3frbZ5c67kbz5w882G36z8+kPH3zAowUHIFnDKBgFo2AUjAIUAAAYpWUWFzk0+gAAAABJRU5ErkJggg==\",\"orcid\":\"\",\"institution\":\"Aix-Marseille University\",\"correspondingAuthor\":true,\"submittingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Mahdi\",\"middleName\":\"\",\"lastName\":\"achache\",\"suffix\":\"\"}],\"badges\":[],\"createdAt\":\"2022-10-29 14:59:14\",\"currentVersionCode\":1,\"declarations\":\"\",\"doi\":\"10.21203/rs.3.rs-2216560/v1\",\"doiUrl\":\"https://doi.org/10.21203/rs.3.rs-2216560/v1\",\"draftVersion\":[],\"editorialEvents\":[],\"editorialNote\":\"\",\"failedWorkflow\":false,\"files\":[{\"id\":38344368,\"identity\":\"a972c1dd-1cca-41ef-9959-203c639b3fec\",\"added_by\":\"auto\",\"created_at\":\"2023-06-11 05:14:34\",\"extension\":\"pdf\",\"order_by\":1,\"title\":\"\",\"display\":\"\",\"copyAsset\":false,\"role\":\"manuscript-pdf\",\"size\":369016,\"visible\":true,\"origin\":\"\",\"legend\":\"\",\"description\":\"\",\"filename\":\"mahdiachacheCopyCopy219.pdf\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-2216560/v1_covered_f9931ec2-9803-4170-869a-773bd1dd85b3.pdf\"}],\"financialInterests\":\"No competing interests reported.\",\"formattedTitle\":\"\\u003cp\\u003eNon-Autonomous Maximal Regularity Under Besov Regularity in Time\\u003c/p\\u003e\",\"fulltext\":[],\"fulltextSource\":\"\",\"fullText\":\"\",\"funders\":[],\"hasAdminPriorityOnWorkflow\":false,\"hasManuscriptDocX\":false,\"hasOptedInToPreprint\":true,\"hasPassedJournalQc\":\"\",\"hasAnyPriority\":false,\"hideJournal\":true,\"highlight\":\"\",\"institution\":\"\",\"isAcceptedByJournal\":false,\"isAuthorSuppliedPdf\":true,\"isDeskRejected\":\"\",\"isHiddenFromSearch\":false,\"isInQc\":false,\"isInWorkflow\":false,\"isPdf\":true,\"isPdfUpToDate\":true,\"isWithdrawnOrRetracted\":false,\"journal\":{\"display\":true,\"email\":\"info@researchsquare.com\",\"identity\":\"researchsquare\",\"isNatureJournal\":false,\"hasQc\":true,\"allowDirectSubmit\":true,\"externalIdentity\":\"\",\"sideBox\":\"\",\"snPcode\":\"\",\"submissionUrl\":\"/submission\",\"title\":\"Research Square\",\"twitterHandle\":\"researchsquare\",\"acdcEnabled\":true,\"dfaEnabled\":false,\"editorialSystem\":\"\",\"reportingPortfolio\":\"\",\"inReviewEnabled\":false,\"inReviewRevisionsEnabled\":true},\"keywords\":\"Besov space, maximal regularity, mon-autonomous evolution equations, sesquilinear forms.\",\"lastPublishedDoi\":\"10.21203/rs.3.rs-2216560/v1\",\"lastPublishedDoiUrl\":\"https://doi.org/10.21203/rs.3.rs-2216560/v1\",\"license\":{\"name\":\"CC BY 4.0\",\"url\":\"https://creativecommons.org/licenses/by/4.0/\"},\"manuscriptAbstract\":\"\\u003cp\\u003eWe consider the maximal regularity problem for non-autonomous Cauchy problems u′(t) + A(t)u(t) = f (t) t-a.e., u(0) = u0. 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