{"paper":{"title":"Existence and regularity of solutions to parabolic-elliptic nonlinear systems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Existence of solutions to the parabolic-elliptic system is established for L1 data, with u gaining summability in L^s and L^q W^{1,q} despite the coupling term being only L2.","cross_cats":[],"primary_cat":"math.AP","authors_text":"Marco Picerni","submitted_at":"2026-04-17T14:34:04Z","abstract_excerpt":"In this paper we study the existence and summability of the solutions to the following parabolic-elliptic system of partial differential equations with discontinuous coefficients: \\begin{equation*} \\begin{cases} u_t - \\operatorname{div}(A(x, t) \\nabla u) = -\\operatorname{div}(u M(x) \\nabla \\psi) + f(x, t) & \\text{in } \\Omega_T, \\\\ -\\operatorname{div}(M(x) \\nabla \\psi) = |u|^\\theta & \\text{in } \\Omega_T, \\\\ \\psi(x, t) = 0 & \\text{on } \\partial \\Omega \\times (0, T), \\\\ u(x, t) = 0 & \\text{on } \\partial \\Omega \\times (0, T), \\\\ u(x, 0) = 0 & \\text{in } \\Omega. \\end{cases} \\end{equation*}\n  Here, "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove existence results for data f∈L¹(Ω_T) and a corresponding increase in summability that obeys the L^p-regularity theorems for parabolic equations proved by Aronson-Serrin and by Boccardo-Dall'Aglio-Gallouët-Orsina. In particular, despite the term u M(x)∇ψ not being regular enough (since it only belongs to L²(Ω_T)), the solution u belongs to L^s(Ω_T)∩L^q(0,T;W^{1,q}_0(Ω)) for suitable s>1 and q>1.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The coefficients A(x,t) and M(x) satisfy the standard measurability, boundedness and uniform ellipticity conditions required to invoke the cited parabolic regularity theorems; these assumptions are implicit in the abstract but not stated explicitly.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Existence and higher summability are shown for solutions of a parabolic-elliptic system with discontinuous coefficients, L^1 data, and |u|^θ nonlinearity where θ < 2/N.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Existence of solutions to the parabolic-elliptic system is established for L1 data, with u gaining summability in L^s and L^q W^{1,q} despite the coupling term being only L2.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"258d98225c0c6f9a04650f11507f71568bf65e9fd87e31730667d23993191f64"},"source":{"id":"2604.16100","kind":"arxiv","version":2},"verdict":{"id":"334f3eaa-74d9-4bab-b8c2-f9359e9c7f0f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T08:12:21.821341Z","strongest_claim":"We prove existence results for data f∈L¹(Ω_T) and a corresponding increase in summability that obeys the L^p-regularity theorems for parabolic equations proved by Aronson-Serrin and by Boccardo-Dall'Aglio-Gallouët-Orsina. In particular, despite the term u M(x)∇ψ not being regular enough (since it only belongs to L²(Ω_T)), the solution u belongs to L^s(Ω_T)∩L^q(0,T;W^{1,q}_0(Ω)) for suitable s>1 and q>1.","one_line_summary":"Existence and higher summability are shown for solutions of a parabolic-elliptic system with discontinuous coefficients, L^1 data, and |u|^θ nonlinearity where θ < 2/N.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The coefficients A(x,t) and M(x) satisfy the standard measurability, boundedness and uniform ellipticity conditions required to invoke the cited parabolic regularity theorems; these assumptions are implicit in the abstract but not stated explicitly.","pith_extraction_headline":"Existence of solutions to the parabolic-elliptic system is established for L1 data, with u gaining summability in L^s and L^q W^{1,q} despite the coupling term being only L2."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.16100/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}