{"paper":{"title":"Refined estimates of the propagation speed in porous medium equation of combustion type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For a family of combustion nonlinearities the asymptotic spreading speed in the porous medium equation receives an explicit characterization of the o(1) correction.","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fan Wu, Suying Liu","submitted_at":"2026-05-13T11:21:58Z","abstract_excerpt":"We are concerned with the Cauchy problem $u_{t}=(u^{m})_{xx}+f(u)$, where the nonliearity $f(u)$ is of combustion type and the initial data is compactly supported. In \\cite{lou2024convergence}, among other things, the authors prove that by considering a multiple of a given initial data, there is a critical value such that the corresponding transition solution spreads at the asymptotic speed $2y_{0}\\sqrt{t}[1+o(1)]\\ \\text{as} \\ t\\rightarrow\\infty$, while the lower order term $o(1)$ remains unknown. In this paper, for a family of functions of combustion type, we refine the estimates of the asymp"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"for a family of functions of combustion type, we refine the estimates of the asymptotic speed of the transition solution and provide a precise characterization of the lower order term o(1). Our result also reveals that there is no unified characterization of the lower order term for general combustion type functions f.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The initial data is a multiple of a fixed compactly supported function and f belongs to a specific family of combustion-type nonlinearities that permits the refined expansion; the precise conditions on this family are not stated in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For a family of combustion-type nonlinearities, the transition solution spreads at speed 2 y0 sqrt(t) with a precisely characterized lower-order term that depends on the specific form of f and is not universal across all such f.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For a family of combustion nonlinearities the asymptotic spreading speed in the porous medium equation receives an explicit characterization of the o(1) correction.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7c3be9b986bf2f10e5a78b0b18e6b1308d1cac8a6ed12ce1ad1c7f80520e3dad"},"source":{"id":"2605.13361","kind":"arxiv","version":1},"verdict":{"id":"9489f9e7-6726-47c7-ab5f-eb643b7d97f1","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T20:17:39.641287Z","strongest_claim":"for a family of functions of combustion type, we refine the estimates of the asymptotic speed of the transition solution and provide a precise characterization of the lower order term o(1). Our result also reveals that there is no unified characterization of the lower order term for general combustion type functions f.","one_line_summary":"For a family of combustion-type nonlinearities, the transition solution spreads at speed 2 y0 sqrt(t) with a precisely characterized lower-order term that depends on the specific form of f and is not universal across all such f.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The initial data is a multiple of a fixed compactly supported function and f belongs to a specific family of combustion-type nonlinearities that permits the refined expansion; the precise conditions on this family are not stated in the abstract.","pith_extraction_headline":"For a family of combustion nonlinearities the asymptotic spreading speed in the porous medium equation receives an explicit characterization of the o(1) correction."},"references":{"count":15,"sample":[{"doi":"","year":1988,"title":"S. Angenent. The zero set of a solution of a parabolic equation.J. Reine Angew. Math, 390(1988), 79–96. 23","work_id":"dcfd367b-263f-43a2-b98f-d8355fd4b0f2","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1982,"title":"D. Aronson, M. G. Crandall, and L. A. Peletier. Stabilization of solutions of a degenerate nonlinear diffusion problem.Nonlinear Analysis: Theory, Methods and Applications,6 (1982), 1001–1022","work_id":"4b33f269-5732-4568-824b-f7674e111078","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1980,"title":"D. G. Aronson. Density-dependent interaction–diffusion systems. InDynamics and modelling of reactive systems, (1980), 161–176, Elsevier,","work_id":"3555d820-4a92-4db6-8f37-19e82bf3ce9c","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1991,"title":"A. de Pablo. and J. L. V´ azquez. 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