{"id":"f8bd32cb-5ed1-4a4b-a713-537df6f7d9ac","arxiv_id":"2506.08159","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a weighted porous medium equation with a rough density, every non-negative solution has a unique initial measure trace in a sharp Morrey-type class, and no two such solutions share the same trace.","lead":"This paper completes the Widder theory (initial trace, existence, uniqueness) for a weighted porous medium equation with rough, possibly singular densities. It identifies the sharp class of initial measures, proves uniqueness from the trace, and gives a smoothing estimate that is new even for the classical equation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Prop. 3.8 is stated for all γ<2 but proven only for γ<N/2; for N=3 and γ∈[3/2,2), the local comparison underlying Theorems 2.6, 2.9, 2.12, and 2.14 is imported from [36, Prop. 4.1] without reproducing the transfer to bounded Cauchy–Dirichlet problems.","rationale":"The reader's weakest_assumption and my stress-test identify the same load-bearing point: Proposition 3.8 is the hinge for the trace theorem, uniqueness, smoothing, and local boundedness, and its full range is not proved in this manuscript. I go slightly further by noting that the cited [36, Proposition 4.1] is described as an R^N result, so its transfer to bounded domains with boundary data is a nontrivial step that the text does not supply. I do not see a demonstrated mathematical error elsewhere, and the rest of the argument is extensive and largely self-consistent. The appropriate verdict is therefore CONDITIONAL rather than REJECT: the paper should be accepted only after the full-range local comparison is either proved in the manuscript or checked, cited with a precise statement, and shown to apply to bounded Cauchy–Dirichlet problems. If the check passes, the verdict would return to ACCEPT with no further substantive concern.","tokens_in":67088,"tokens_out":6602,"duration_ms":82819,"concrete_test":"Check whether [36, Proposition 4.1] and its ball-removal argument apply verbatim to the bounded Cauchy–Dirichlet comparison needed in Proposition 3.8 for all γ∈[0,2). Concretely, take N=3, γ=7/4, Ω=B_1, and write out the duality estimate after deleting a small ball B_δ around the singularity, tracking boundary terms on ∂B_δ and ∂Ω. If the estimate closes without any γ<3/2 assumption, the citation covers the gap; if boundary terms reintroduce γ<N/2, then Proposition 3.8 lacks proof in the stated range. A minimal bibliographic check is to verify whether [36, Prop. 4.1] states a bounded-domain version, since the text describes it only for the whole R^N.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central Widder-theory claim rests on Proposition 3.8, a local comparison principle for very weak sub/supersolutions of the Cauchy–Dirichlet problem (3.47). The statement allows every γ∈[0,2), but the proof in Section 3.2 is explicitly carried out under the additional restriction γ<N/2. The paragraph immediately before Proposition 3.8 says the remaining range is handled by [36, Proposition 4.1], but that proof is not reproduced here. This is not a cosmetic gap: when N=3, the missing interval [3/2,2) is nonempty, and ρ may be singular like |x|^{-γ}. Moreover, [36, Prop. 4.1] is described in the text as a result for the whole R^N, whereas Proposition 3.8 requires a bounded smooth domain with boundary data; passing from R^N to bounded domains with boundary integrals is not a formality. The comparison is used in Corollary 3.9, in the approximation/identification step inside the proof of Theorem 2.12, in the preliminary uniqueness-class claim of Theorem 2.9, and in the local boundedness argument, so the gap propagates to the main results. If [36, Prop. 4.1] does not cover the bounded-domain, all-γ setting with the exact hypotheses used here, then the proof of the uniqueness theorem is incomplete for N=3 and γ≥3/2. I am not claiming the results are false, only that the argument as printed has a load-bearing verifiability gap at this point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a complete Widder theory for the weighted porous medium equation ρ u_t = Δ(u^m) on R^N × (0,T), N ≥ 3, m > 1, under the rough weight condition (1.6): C(1+|x|)^{-γ} ≤ ρ(x) ≤ C|x|^{-γ} with γ ∈ [0,2). The main results are: (i) Theorem 2.6, an Aronson–Caffarelli-type initial trace theorem showing that every non-negative very weak solution has a unique Radon measure trace μ satisfying the sharp growth estimate (2.10); (ii) Theorem 2.8, existence of very weak solutions for (possibly sign-changing) measure data in the Morrey-type space X, with precise lifetime and L^∞ smoothing bounds; (iii) Theorem 2.9, a uniqueness result for non-negative solutions with the same initial trace and no global growth assumptions; (iv) Theorem 2.12, sharp local smoothing estimates for possibly sign-changing, merely local, bounded solutions; and (v) Theorem 2.14, local boundedness of non-negative very weak solutions, which in turn implies they are strong energy solutions. The proofs combine potential-theoretic estimates, Green's function techniques, parabolic Moser iteration, and a non-linear approximation scheme. The authors explicitly state that several classical tools of the unweighted theory (scale invariance, Aronson–Bénilan inequality, Aleksandrov's reflection principle, continuity of solutions) are unavailable, and they replace them by new arguments.","tokens_in":67306,"tokens_out":2422,"duration_ms":31818,"significance":"If the results are correct, this is a substantial contribution: it completes the Widder program for (1.1) under a very mild, merely measurable weight condition that covers both singular and degenerate densities. The sharp dimensionally consistent exponents in (1.7)–(1.8), the explicit trade-off between the two terms in the trace estimate (2.10), and the fact that the smoothing estimate (2.21) is new even in the classical case ρ=1 give the paper high visibility. The paper is also careful in discussing optimality against explicit or semi-explicit solutions (Subsection 2.5), and in stating precisely which hypotheses are used. The main theorems are stated with full hypotheses and the proof is long and detailed. The main concern is that a load-bearing comparison principle, Proposition 3.8, is proved only under the extra restriction γ < N/2 and the remaining range is imported from the authors' previous paper [36] without a full transfer to the bounded-domain setting. The reader's report and the stress-test note agree that this is the principal verifiability gap.","major_comments":[{"comment":"Proposition 3.8 is stated for every γ ∈ [0,2), but the printed proof is explicitly carried out under the additional restriction γ < N/2, and the paragraph immediately before the statement says the remaining range is handled by [36, Proposition 4.1]. This is not a cosmetic remark: for N = 3 the missing interval [3/2, 2) is nonempty, and ρ may be singular like |x|^{-γ}. The point is load-bearing because Proposition 3.8 is used in Corollary 3.9, in the local approximation step in the end of the proof of Theorem 2.12, in the identification step inside the proof of Theorem 2.9, and in the local boundedness argument of Section 6. I am not claiming the results are false, but as printed the proof of the main theorems has a verifiability gap in the range γ ∈ [3/2, 2) when N = 3. The authors should either reproduce the transfer from [36, Proposition 4.1] to the bounded Cauchy–Dirichlet setting with the exact hypotheses of Definition 3.5, or prove Proposition 3.8 directly for all γ ∈ [0,2). Without that, the uniqueness theorem is not fully supported in this range.","section":"§3.2, Proposition 3.8"},{"comment":"The proof of the preliminary claim (5.34) uses Corollary 2.15, which in turn requires Theorem 2.14 (local boundedness) and the local approximation based on Proposition 3.8. Moreover, the step ‘thanks to [36, Proposition 4.1 and Remark 4.2]’ asserts that u|_{t>τ} coincides with the constructed solution of Proposition 5.3 for almost every τ. This importation is used to obtain u ∈ C((0,T);L^1(Φ_α)); if the comparison principle of Proposition 3.8 is not available in the full range, this coincidence statement is not justified either. The authors should explicitly state which of the conclusions of [36] are being invoked here and how the local-to-global comparison is obtained under the same hypotheses as the present paper.","section":"§5.3, proof of Theorem 2.9, Preliminary claim"},{"comment":"The proof of Theorem 2.14 relies on the time-regularized approximation u_ε and identifies the limit with u via the inequality (6.32) and the convergence (6.34). The identification step uses the very weak formulation of (3.47), which is only available for almost every r and τ_1 by Lemma 3.6; the text fixes representative r and τ_1 but does not explain how the final local boundedness statement is obtained on a full cylinder rather than on the a.e. selected ones. This is likely a minor gap in exposition, but since Theorem 2.14 is used in the proofs of Theorems 2.6 and 2.9, the authors should clarify the a.e. selection and the covering argument (Remark 6.6 addresses the local case but not the a.e. selection issue).","section":"§6, proof of Theorem 2.14"},{"comment":"Theorem 5.2 is stated for a non-negative finite Radon measure and is imported from [26] with an explanation that the additional structural assumption (5.1) used there is not needed. The explanation in the proof of Theorem 5.2 is brief; in particular, the density argument in Ẇ^1(R^N) ∩ L^2_ρ(R^N) and the integration-by-parts identity (5.2) are only sketched. Since this theorem is used in Step 3 of the proof of Theorem 2.9 to identify the limit w_k, the authors should either provide a complete proof or state precisely which result from [26, 35] they are relying on and why it applies to the present weight condition (1.6).","section":"§5.1, Theorem 5.2"}],"minor_comments":[{"comment":"The text says the definition of ∥μ∥_{1,r} is independent of r ≥ 1, but the equivalence constants depend on the fixed r; this is standard but should be stated explicitly to avoid confusion when comparing growth estimates with different r.","section":"§2.1, Definition 2.4"},{"comment":"The sentence introducing Proposition 3.8 refers to [36, Proposition 4.1] for the full range, but the proposition itself is stated without any reference; the reader would benefit from a clear statement that the present proof covers only γ < N/2 and that the remaining range is a direct transfer of [36, Proposition 4.1] with the boundary data as in Definition 3.5.","section":"§3.2, Proposition 3.8"},{"comment":"In the passage from (4.33) to (4.34), the product in (4.32) is said to be bounded using (4.27), but the displayed inequality (4.35) uses p_{k+1} ≥ (1+θ)^{k+1} p_0, which is true but should be explicitly verified from (4.27).","section":"§4, proof of Theorem 2.12"},{"comment":"Lemma 6.4 is quoted from [16, Lemma 4.7] without proof; while this is acceptable for a lemma from a published paper, the statement uses the notation w for a smooth function while w is also used for the Dirichlet potential in (6.3); a change of notation would improve readability.","section":"§6, Lemma 6.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically rich and the main results are likely correct, but the proof of Proposition 3.8 covers only part of the stated parameter range and the rest is imported from [36] without a detailed transfer. Given that this proposition is used repeatedly in the proofs of the central theorems, I think the appropriate editorial decision is major revision, requiring the authors to close the γ ∈ [3/2, 2) gap (which is only relevant in N = 3) or to give a precise and complete statement of the imported result and its applicability to the present setting."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper is a major step: it gives the first sharp Widder theory for the weighted porous medium equation with rough densities, and the main theorems are genuinely new. For non-negative very weak solutions it proves a weighted Aronson-Caffarelli initial trace theorem with the correct growth rate, uniqueness without any global growth assumptions, local boundedness, and an epsilon-optimal local smoothing estimate that is new even for rho=1. The proofs are long but well organized, and the potential-theoretic techniques are a real departure from the unweighted theory, which relied on continuity and scale invariance that are unavailable here.\n\nThe one place I would push back is the local comparison principle, Proposition 3.8. It is stated for all gamma in [0,2), but the printed proof is explicitly carried out only under the stricter condition gamma < N/2. For N=3 and gamma >= 3/2, the result is imported from the authors' earlier paper [36, Prop. 4.1], which is a comparison principle on R^N. The present paper needs it on bounded smooth domains with boundary integrals, and the transfer is not a one-line exercise. The authors mention that the details in [36] involve removing a small ball around the singularity, which suggests the bounded-domain version should be within reach, but it is not actually written down here. This matters: Proposition 3.8 feeds into the global comparison corollary, the approximation step in the smoothing theorem, and both uniqueness and local boundedness. So it is load-bearing.\n\nI do not think this is a fatal flaw. It looks patchable, and the rest of the paper is consistent and detailed. But it is exactly the kind of thing a referee should ask the authors to spell out, either by giving the bounded-domain proof or by stating the reduction to [36] with the required hypotheses. The circularity burden is low: the main theorems do not reduce to prior results, and the reliance on [36] for existence and comparison is legitimate prior work.\n\nWho should read it: anyone working on nonlinear diffusion, weighted PDEs, or initial trace theory. It deserves a serious referee. My recommendation: send to peer review, and in the report ask for a self-contained proof or a precise citation covering the full range of Proposition 3.8.","headline":"A genuinely sharp Widder theory for the weighted PME, with a real but possibly patchable gap in the comparison lemma for N=3, gamma in [3/2,2).","tokens_in":67989,"tokens_out":4207,"would_cite":true,"duration_ms":44575,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R05","35R06","28A33","35A01","35A02","35B45","35J08","35K65"],"pacs":[],"model":"deepseek-v4-flash","headline":"For weighted porous-medium diffusion with a density that can blow up at a point, every non-negative solution has a unique initial measure trace, and two solutions sharing a trace must coincide.","keywords":["weighted porous medium equation","Widder theory","initial traces","Aronson–Caffarelli estimates","very weak solutions","local smoothing estimates","local boundedness","rough weights"],"falsifier":"Construct two bounded very weak sub- and supersolutions of the Cauchy–Dirichlet problem (3.47) in a ball in $\\mathbb{R}^3$ with a weight satisfying (1.6) for some $\\gamma\\in(3/2,2)$ whose ordering reverses at an interior point of the cylinder; such a crossing falsifies Proposition 3.8 in exactly the range the paper imports rather than proves, and Theorem 2.9 falls with it. A computational backstop: simulate the radial Cauchy problem with $\\rho(x)=|x|^{-\\gamma}$ in $\\mathbb{R}^3$ and initial datum a large multiple of the Dirac delta by two independent approximation schemes (mollified data versus monotone truncations of the constructed solution); agreement at positive times is what the paper predicts, and any divergence of the two limits, or a sup-norm blow-up rate other than $(T-t)^{-1/(m-1)}$, would force a revision of the uniqueness or sharpness claims.","tokens_in":3317,"feed_emoji":"🌊","tokens_out":4705,"duration_ms":324465,"temperature":0.7,"pith_summary":"The paper studies the weighted porous medium equation $\\rho(x)u_t=\\Delta(u^m)$ in $\\mathbb{R}^N\\times(0,T)$ for $m>1$, $N\\ge3$, with a density $\\rho$ that is merely measurable, may blow up at the origin, and decays like $(1+|x|)^{-\\gamma}$ at infinity for some $\\gamma\\in[0,2)$. Its aim is to complete the Widder theory for this equation, meaning the sharp three-part bijection between admissible initial data and solutions: every non-negative very weak solution has a unique initial Radon measure trace obeying $|\\mu|(B_R)=O(R^{N-\\gamma+(2-\\gamma)/(m-1)})$ (Theorem 2.6); conversely, every measure with that growth bound is the initial trace of a constructed very weak solution (Theorem 2.8, building on the authors' prior existence theory); and any two non-negative very weak solutions with the same initial trace are equal, with no growth assumptions imposed on them (Theorem 2.9). Because the classical tools behind the unweighted theory — continuity of solutions, exact scale invariance, Aleksandrov's reflection principle, and the Aronson–Bénilan inequality — are unavailable under rough weights, the arguments are rebuilt from Green's-function inequalities, a parabolic Moser iteration driven by weighted Sobolev–Poincaré estimates, and local potential theory. The paper also establishes two results it claims are of independent interest: a sharp local smoothing estimate for unsigned local solutions that appears to be new even in the unweighted case, and local boundedness of non-negative very weak solutions, which in particular makes them locally finite energy solutions.","feed_headline":"One initial measure pins down each rough-density diffusion solution","feed_subtitle":"Even rough, singular densities leave the equation well-posed: one initial measure per solution, and vice versa.","key_machinery":"The central object is the Euclidean Green's function $G(x,y)=c_N|x-y|^{2-N}$ of $-\\Delta$, used through smooth approximations $G_n$ increasing to $G$ with $-\\Delta_xG_n\\to\\delta_y$. For constructed solutions with bounded integrable data the paper derives the monotonicity $t^{m/(m-1)}u^m(t)$ essentially non-decreasing (a consequence of $\\rho u_t\\ge-\\rho u/((m-1)t)$) and the basic inequality $\\int_{\\mathbb{R}^N}[u(t_0)-u(t_1)]G(x,x_0)\\rho\\,dx\\le(m-1)t_1^{m/(m-1)}t_0^{-1/(m-1)}u^m(x_0,t_1)$; combined with a dichotomy on the time scale $t\\sim R^{(N-\\gamma)(m-1)+2-\\gamma}/M^{m-1}$, this yields the Aronson–Caffarelli estimate for constructed solutions. A second mechanism is the 'fake scaling' pair $\\lambda=(N-\\gamma)/((N-\\gamma)(m-1)+2-\\gamma)$ and $\\theta=(2-\\gamma)/(N-\\gamma)$, satisfying $\\lambda(m-1)+\\theta\\lambda=1$; every quantitative rate in the paper is expressed through these exponents, which replace the exact scale invariance the weighted equation lacks. A third mechanism is the local comparison principle (Proposition 3.8) for very weak sub- and supersolutions of the Cauchy–Dirichlet problem, obtained by a duality argument with a backward linear equation; it is the hinge for passing from constructed solutions to arbitrary ones, and the paper's own proof covers only $\\gamma<N/2$, with the remaining range inherited from the authors' previous work. The smoothing estimates run on a parabolic Moser iteration powered by the weighted Sobolev–Poincaré inequality $\\|f\\|_{L^{2^*}_\\rho(B_R)}\\le C(R^{-2+\\gamma}\\|f\\|^2_{L^2_\\rho(B_R)}+\\|\\nabla f\\|^2_{L^2(B_R)})^{1/2}$ with $2^*=2(N-\\gamma)/(N-2)$, while local boundedness runs on the dual local potential $W=w-h$ solving $-\\Delta W=u\\rho$ and $-W_t=u^m$, refined through a Calderón–Zygmund bootstrap.","core_discovery":"On the paper's own terms, the discovery is that the weighted equation (1.1) has a complete and sharp Widder theory under the sole hypothesis (1.6). The admissible initial data are the Radon measures in the Morrey-type space $X$ defined by $|\\mu|(B_R)=O(R^{N-\\gamma+(2-\\gamma)/(m-1)})$, and the matching solution class is the set of non-negative very weak solutions growing at most like $\\|u(t)\\|_{L^\\infty(B_R)}=O(R^{(2-\\gamma)/(m-1)})$; within this pair, existence (Theorem 2.8, with lifetime $T(\\mu)=C/[\\ell(\\mu)]^{m-1}$ for data outside $X_0$), necessity of the trace (Theorem 2.6), and uniqueness (Theorem 2.9) all hold, so the initial-value problem with measure datum $\\mu$ is solvable if and only if $\\mu\\in X$, and for non-negative $\\mu$ the non-negative solution is unique. The authors stress that the rates are optimal: the spatial exponents are saturated by explicit finite-time blow-up profiles such as $U(x,t)=|x|^{(2-\\gamma)/(m-1)}(T-t)^{-1/(m-1)}$ in the model case $\\rho(x)=|x|^{-\\gamma}$, and the time dependence in the trace estimate (2.10) is matched both by those blow-up solutions and by the weighted Barenblatt solutions at large times. The same sharpness analysis is applied to the local smoothing estimates (2.21): optimal in $R\\to\\infty$, optimal in time for $m\\in(1,2)$, and optimal up to an arbitrarily small $\\varepsilon$ for $m>1$, with the necessity of that $\\varepsilon$ left as an open question.","pith_inferences":["The same Green's-function-plus-trace-class template plausibly transfers to neighboring equations that also lack scale invariance and solution continuity — weighted fast diffusion ($m<1$), nonlinear diffusion with merely measurable coefficients, or diffusion on negatively curved manifolds with decaying curvature — with the 'fake scaling' exponents replaced by spectral data of the underlying space.","A practical corollary the paper leaves implicit: any numerical scheme whose outputs are non-negative very weak solutions with the correct initial trace automatically converges to the unique solution, making the constructed solution a canonical reference profile for benchmarking solvers of weighted degenerate parabolic equations.","The open $\\varepsilon$ in the $m>1$ smoothing branch is a testable feature: compute the near-$t=0$ profile of the friendly-giant solution $V(x,t)=W(x)t^{-1/(m-1)}$ for $\\rho(x)=|x|^{-\\gamma}$; if the sharp asymptotics show no $\\varepsilon$ correction, the exponent is an artifact, whereas its presence would reveal a genuine logarithmic gap between $L^1_\\rho$-data control and sup-norm smoothing."],"forward_implications":["The initial-value problem (1.2) is solvable in the very weak sense exactly for data measures in $X$, with the explicit lifetime formula (2.11); for non-negative $\\mu\\in X$ the non-negative solution is unique, so the weighted equation possesses the full Widder bijection between data class and solution class that the unweighted porous medium equation was known to have.","Every non-negative very weak solution obeys the quantitative laws $\\int_{B_R}u(t)\\rho=O(R^{N-\\gamma+(2-\\gamma)/(m-1)})$ and $\\|u(t)\\|_{L^\\infty(B_R)}=O(R^{(2-\\gamma)/(m-1)})$ for every $t>0$, and both exponents are optimal because explicit blow-up solutions and weighted Barenblatt solutions saturate them.","The weak and strong notions of solution coincide a posteriori: non-negative very weak solutions are locally bounded, have locally finite energy, and belong to $C((\\tau_1,\\tau_2);L^p_{\\rho,loc})$ for every $p<\\infty$ (Corollary 2.16).","Constructed solutions have curve regularity that the authors note is new even for $\\rho=1$: $u\\in\\mathrm{Lip}_{loc}((0,T(\\mu));X)\\cap W^{1,\\infty}_{loc}((0,T(\\mu));L^1(\\Phi_\\alpha))$, together with an ordering principle ($\\mu\\le\\nu$ implies $u(t)\\le v(t)$) and, for $\\mu\\in X_0$, the spatial decay $|x|^{-(2-\\gamma)/(m-1)}u(x,t)\\to0$.","The local smoothing estimates (2.21) hold for unsigned purely local solutions and are sharp as $R\\to\\infty$; they are sharp in time for $m\\in(1,2)$ and sharp up to an arbitrarily small exponent $\\varepsilon$ for $m>1$, whose necessity remains open."],"supporting_citations":[{"why":"Proves the classical unweighted initial-trace theorem with growth $R^{N+2/(m-1)}$; Theorem 2.6 is the weighted analogue and borrows its statement template and sharpness notion.","marker":"[4]"},{"why":"Proves unweighted uniqueness for non-negative solutions without growth assumptions; Theorem 2.9 is its weighted extension, and the Moser-iteration strategy of this reference is the backbone of Theorem 2.12.","marker":"[15]"},{"why":"Establishes the unweighted counterpart of local boundedness for very weak solutions; the potential-based bootstrap of Theorem 2.14 adapts this strategy to the rough weight.","marker":"[16]"},{"why":"Prior paper for the same equation: supplies existence for data in $X$, the uniqueness criterion in the class $Y$, and the local comparison principle that Proposition 3.8 relies on for $\\gamma\\ge N/2$.","marker":"[36]"},{"why":"Source of the Green's-function method used in Proposition 3.4: the monotonicity of $t^{m/(m-1)}u^m$ and the pointwise inequality (3.17) that yield the Aronson–Caffarelli estimate for constructed solutions.","marker":"[12]"},{"why":"Provides the operator-theoretic existence and uniqueness for finite Radon measure data (Theorem 5.2) that identifies the approximate solutions used in the proof of Theorem 2.9.","marker":"[26]"},{"why":"Proved the smoothing estimate (3.12) for smooth positive weights; Proposition 3.3 adapts it to weights satisfying only (1.6), and the estimate drives the mass and Green's-function arguments.","marker":"[41]"},{"why":"Standard monograph supplying the unweighted comparison principle and the Cauchy–Dirichlet construction used in Corollary 3.9 and in the local approximation of very weak solutions.","marker":"[44]"}],"fun_headline_variants":["Rough weights still yield sharp Widder theory for porous mediums","Sharp well-posedness for weighted porous medium with measure data","Measure data meets rough densities: optimal Widder theory","One measure, one solution: rough-density PME solved","No scale invariance needed: optimal trace for weighted PME"],"cache_read_input_tokens":69888,"weakest_assumption_plain":"Everything rests on a local comparison principle that lets every non-negative very weak solution be approximated from below by smooth solutions of bounded-data problems and thereby compared with them; the paper's own proof of that principle covers only the roughness range $\\gamma<N/2$ (restrictive only in dimension $N=3$), and for the rest it relies on a proposition imported from the authors' previous paper without reproducing its proof, so if that inherited result fails at the stated level of generality, the uniqueness theorem and the whole Widder theory collapse.","fun_headline_variants_meta":{"raw":{"variants":["Rough weights still yield sharp Widder theory for porous mediums","Sharp well-posedness for weighted porous medium with measure data","Measure data meets rough densities: optimal Widder theory","One measure, one solution: rough-density PME solved","No scale invariance needed: optimal trace for weighted PME"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000262,"raw_usage":{"total_tokens":1698,"prompt_tokens":1148,"completion_tokens":550,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":764,"completion_tokens_details":{"reasoning_tokens":468}},"tokens_in":764,"tokens_out":550,"duration_ms":6545,"temperature":1.0,"reasoning_tokens":468,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:18:46.013390+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct two bounded very weak sub- and supersolutions of the Cauchy–Dirichlet problem (3.47) in a ball in $\\mathbb{R}^3$ with a weight satisfying (1.6) for some $\\gamma\\in(3/2,2)$ whose ordering reverses at an interior point of the cylinder; such a crossing falsifies Proposition 3.8 in exactly the range the paper imports rather than proves, and Theorem 2.9 falls with it. A computational backstop: simulate the radial Cauchy problem with $\\rho(x)=|x|^{-\\gamma}$ in $\\mathbb{R}^3$ and initial datum a large multiple of the Dirac delta by two independent approximation schemes (mollified data versus monotone truncations of the constructed solution); agreement at positive times is what the paper predicts, and any divergence of the two limits, or a sup-norm blow-up rate other than $(T-t)^{-1/(m-1)}$, would force a revision of the uniqueness or sharpness claims.","supporting_citations":[{"cited_title":"Aronson, L.A","cited_arxiv_id":null,"evidence_quote":"Proves the classical unweighted initial-trace theorem with growth $R^{N+2/(m-1)}$; Theorem 2.6 is the weighted analogue and borrows its statement template and sharpness notion."},{"cited_title":"Dahlberg, C.E","cited_arxiv_id":null,"evidence_quote":"Proves unweighted uniqueness for non-negative solutions without growth assumptions; Theorem 2.9 is its weighted extension, and the Moser-iteration strategy of this reference is the backbone of Theorem 2.12."},{"cited_title":"Dahlberg, C.E","cited_arxiv_id":null,"evidence_quote":"Establishes the unweighted counterpart of local boundedness for very weak solutions; the potential-based bootstrap of Theorem 2.14 adapts this strategy to the rough weight."},{"cited_title":"Muratori, T","cited_arxiv_id":null,"evidence_quote":"Prior paper for the same equation: supplies existence for data in $X$, the uniqueness criterion in the class $Y$, and the local comparison principle that Proposition 3.8 relies on for $\\gamma\\ge N/2$."},{"cited_title":"Bonforte, J.L","cited_arxiv_id":null,"evidence_quote":"Source of the Green's-function method used in Proposition 3.4: the monotonicity of $t^{m/(m-1)}u^m$ and the pointwise inequality (3.17) that yield the Aronson–Caffarelli estimate for constructed solutions."},{"cited_title":"Grillo, M","cited_arxiv_id":null,"evidence_quote":"Provides the operator-theoretic existence and uniqueness for finite Radon measure data (Theorem 5.2) that identifies the approximate solutions used in the proof of Theorem 2.9."},{"cited_title":"Reyes, J.L","cited_arxiv_id":null,"evidence_quote":"Proved the smoothing estimate (3.12) for smooth positive weights; Proposition 3.3 adapts it to weights satisfying only (1.6), and the estimate drives the mass and Green's-function arguments."},{"cited_title":"The Porous Medium Equation. Mathematical Theory","cited_arxiv_id":null,"evidence_quote":"Standard monograph supplying the unweighted comparison principle and the Cauchy–Dirichlet construction used in Corollary 3.9 and in the local approximation of very weak solutions."}],"review_version":1}