{"id":"3c273052-bedd-4c57-8f77-21c035c1c554","arxiv_id":"2505.14966","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For nonlinear heat and Schrödinger equations with non-algebraic power nonlinearities, the paper identifies the sharp Sobolev thresholds, s < p+2+1/q and s < p+5/2 respectively, with strong ill-posedness at the endpoint.","lead":"This paper proves the exact smoothness threshold for two basic nonlinear equations with rough power nonlinearities, showing well-posedness below a sharp exponent and non-existence at and above it. The result settles a long-standing open problem and shows that some Schrödinger equations are ill-posed in every Sobolev space.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 6.4's even-integer endpoint is left to the reader, yet p=2,4,... are inside the hypothesis p-1∉2N; the claimed optimal ill-posedness threshold is therefore not proven for any even power, including the headline case p=2.","rationale":"The reader's conditional verdict is appropriate. The central claim—sharp well-posedness thresholds—depends on two large blocks: the nonlinear estimates of Theorem 3.1 and Sections 4-5, and the one-dimensional non-existence mechanism of Lemma 6.4. I agree with the reader that the ill-posedness block is the most load-bearing weak point. In particular, the even-integer endpoint of Lemma 6.4 is explicitly deferred to the reader, and those exponents are included in the statement of Theorems 6.1 and 6.2. The proof for non-integer p uses a q* enlargement argument that cannot reach p=n+1, so the endpoint requires a separate argument. Because p=2 is one of the motivating examples in the introduction, this is not a harmless omission. I do not see evidence that the main well-posedness construction is wrong; the concern is a missing proof at a set of values covered by the theorem statement. This keeps the verdict at CONDITIONAL rather than ACCEPT, and does not justify REJECT. I also note the copy-paste NLS identity in the proof of Theorem 6.1, but the even-integer endpoint is the sharper and more serious gap. A complete proof of Lemma 6.4 for p=2, or a direct verification that the W^{p+1/q,q} norm of the nonlinear term diverges at t>0, would settle the issue.","tokens_in":64131,"tokens_out":22396,"duration_ms":198917,"concrete_test":"Write out Lemma 6.4 for the simplest even integer p=2, q=2, with w0=δχx and h∈L1([0,T];W^{5/2,2}(R)) satisfying h(t,0)=0. Compute the second spatial derivative of ∫0t λ|w|^{p-1}w ds and verify that it acquires a jump of size proportional to t at x=0, so the W^{5/2,2} norm of w(t) is infinite; then confirm that no admissible h can cancel this jump because W^{1/2,2}(R) does not contain sign-type discontinuities. If this computation can be made rigorous, the endpoint gap is repairable; if not, the ill-posedness theorems fail for p=2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The entire ill-posedness argument for both (NLH) and (NLS) reduces to the one-dimensional Lemma 6.4, which is then applied via the Fubini-type Theorem 6.6. As written, Lemma 6.4 proves the non-existence statement only for p∈(n,n+1): the initial argument handles p∈(n,n+1-1/q), and the extension to all p∈(n,n+1) is made by choosing q* large. That q*-trick cannot work when p=n+1, i.e. when p is an integer. The text then says 'a similar result holds when p is an even integer' and leaves the modifications to the reader. These even-integer values are not excluded by the theorem's hypothesis p-1∉2N: p=2,4,6,... are all covered by the stated theorems. The endpoint mechanism is genuinely different: for p=2, the leading singularity in ∫|w|^{p-1}w is a jump in the second derivative, not a fractional-power singularity, so the Hölder-type estimate Lemma 6.5 does not transfer verbatim. Since p=2 is explicitly highlighted in the introduction and in Corollary 1.9, this omitted proof is load-bearing rather than cosmetic. The separate copy-paste in the proof of Theorem 6.1, where the function f is defined using the NLS integral identity while the theorem under proof is (NLH), compounds the issue, but the even-integer endpoint is the deeper gap: without it the claimed sharp threshold for even p is unsupported as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general heuristic for the maximal Sobolev regularity of semilinear evolution equations with rough power nonlinearities and implements it for the nonlinear heat equation (NLH) and the nonlinear Schrödinger equation (NLS). The main positive results are the nonlinear estimate Theorem 3.1, the well-posedness theorems for NLH (Theorem 1.13/4.1) and NLS (Theorem 1.12/5.1), the one-dimensional improvement Theorem 1.14/5.10, and the non-existence theorems Theorem 1.5/6.2 for NLS and Theorem 1.6/6.1 for NLH. The ill-posedness arguments rest on a one-dimensional Lemma 6.4, which is then lifted to higher dimensions by a Fubini-type theorem (Theorem 6.6). The paper also draws corollaries such as the existence of NLS equations ill-posed in every Sobolev space (Corollary 1.9).","tokens_in":64457,"tokens_out":6644,"duration_ms":59673,"significance":"If the stated theorems are correct, the paper resolves a longstanding open problem by identifying the optimal Sobolev threshold for a non-algebraic power nonlinearity, and it introduces a new nonlinear estimate for complex-valued functions that is likely to be influential. The paper is commendable for deriving thresholds from estimates rather than fitting parameters, for explicitly separating heuristics from proofs, and for including detailed a priori bounds. However, the sharpness claims for even integer powers are not proven as written, and the proof of Theorem 6.1 contains a sign/copy-paste error that affects the derivation of the one-dimensional integral equation. These issues are load-bearing for the central claims, so the paper requires a major revision rather than acceptance in its current form.","major_comments":[{"comment":"The proof of Lemma 6.4 is carried out for p in (n, n+1-1/q) and then extended to all p in (n, n+1) by choosing q* large enough; this extension collapses when p = n+1, i.e., when p is an integer. Since the hypotheses of Theorems 6.1 and 6.2 are p-1 not in 2N, the values p = 2, 4, 6, ... are covered by the stated theorems, and p = 2 is explicitly featured in Corollary 1.9 and in the discussion of the questions from [16,17]. The sentence 'a similar result holds when p is an even integer' leaves the entire endpoint argument to the reader, and the endpoint mechanism is genuinely different because the leading singularity of |w|^{p-1}w at p = 2 is a jump in the second derivative rather than a fractional-power singularity. This is a load-bearing gap: without a proof for even integer p, the claimed sharp ill-posedness threshold for even p is not established.","section":"Section 6, Lemma 6.4"},{"comment":"In the proof of Theorems 6.1 and 6.2, the function f is defined by f(t,x) = u - u0 + i ∫ |u|^{p-1}u ds + i ∫ Δu ds and the text says 'since u is a solution to (NLS)'. Theorem 6.1 is the nonlinear heat equation, for which the correct identity is f = u - u0 - ∫ (|u|^{p-1}u + Δu) ds, with no factor i. As written, the displayed identity is consistent only with (NLS), so the proof does not establish Theorem 6.1. This is not a cosmetic typo: the reduction to the one-dimensional integral equation (6.1) with h = Δu depends on the sign and the factor i in this identity. The proof of Theorem 6.2 should also be written for the Schrödinger equation explicitly rather than being asserted by 'identical reasoning'.","section":"Proof of Theorem 6.1 (Section 6.1)"},{"comment":"Proposition 3.4(i) is a key technical input in the proof of the main nonlinear estimate Theorem 3.1; it is used, for example, in summing the terms J^1_jk and the analogous estimates on the sets B^2_k. Its proof is only an outline that refers to arguments on pages 64-65 of [49] and to [31, Lemma 3.9]. Given that Theorem 3.1 underpins all well-posedness theorems in the paper, the proof of Proposition 3.4(i) should either be written out in full or the precise quoted result should be reproduced, together with a verification that the hypotheses are satisfied in the complex-valued setting used here.","section":"Section 3, Proposition 3.4(i)"}],"minor_comments":[{"comment":"The statement of Lemma 6.5 concludes |I(t,x+h)-I(t,x)| ≈ t|x|^{p-n}, but the final line of the proof says '≈ t|x|^{p-1}'; the exponent should be p-n.","section":"Lemma 6.5"},{"comment":"The assertion that Δu is odd in x1 should be justified, since the reduction to Lemma 6.4 uses the condition h(t,0,x') = 0; a one-line argument using uniqueness and the oddness of the initial datum would suffice.","section":"Section 6.1"},{"comment":"In Section 2.2.1, the text attributes a result to 'Killip and Visan from [65, Appendix A]', but reference [65] is Visan's thesis; please check whether the intended citation is [65] or a different work by Killip and Visan.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a significant open problem and the positive results appear carefully executed. My main substantive reservation is the even-integer endpoint in Lemma 6.4, which is load-bearing for the headline case p = 2; this is fixable but requires real work and cannot be left to the reader. The Section 6.1 sign/copy-paste error is embarrassing but straightforward to correct. I recommend major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this paper is a serious, technically deep advance, and the positive well-posedness half is in good shape. But the ill-posedness half is not ready as written: the even-integer endpoint of Lemma 6.4 is left to the reader, and that gap cuts into the stated theorems, not just a remark. Also, the proof of Theorem 6.1 for the heat equation is literally the NLS proof, with i∂t and an appeal to \"a solution to (NLS)\".\n\nCredit where it is due: the main nonlinear estimate, Theorem 3.1, is new and seems to break the logjam on complex-valued rough nonlinearities. The well-posedness theorems are a big improvement over the s < p+2 barriers, and the proof architecture—frequency envelopes, time truncation, elliptic analysis—is coherent and laid out in detail. If the ill-posedness gaps are fixed, this will be a landmark paper.\n\nNow the problems. First, Lemma 6.4 is the load-bearing one-dimensional nonexistence result, and its proof covers p∈(n,n+1-1/q) then extends by a q* argument. The q* argument cannot reach p=n+1, and p even integer is exactly p=n+1 with n odd. The text says \"a similar result holds when p is an even integer\" and leaves the modifications to the reader. For p=2 the leading singularity is a jump in the second derivative, not a fractional power, so the Hölder estimate in Lemma 6.5 does not transfer verbatim. Since p=2 is used to answer an explicit open question and appears in Corollary 1.9, this is a load-bearing omission. Second, the proof of Theorem 6.1 asserts the NLH result using the NLS integral identity; the function f has i factors and the text says \"since u is a solution to (NLS)\". That is not NLH. So the NLH ill-posedness proof is currently absent.\n\nThese are fixable, and I would guess the results are true. But a referee should demand the missing endpoint and the correct NLH argument before the paper is accepted.\n\nWho it is for: anyone working on semilinear Schrödinger or heat equations with rough powers. The well-posedness part is already valuable. I would send it to a serious referee, but the referee should be told to focus on the ill-posedness section.","headline":"Deep and likely mostly correct results on optimal Sobolev thresholds, but the ill-posedness half has a real gap for even p and the NLH proof is a copy-paste of NLS.","tokens_in":65038,"tokens_out":6999,"would_cite":true,"duration_ms":43606,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A01","35B65","35K58","35Q55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims to settle the sharp Sobolev regularity boundary for rough nonlinear heat and Schrödinger equations.","keywords":["nonlinear Schrödinger equation","nonlinear heat equation","well-posedness","ill-posedness","rough nonlinearity","Sobolev threshold","composition estimates","power nonlinearity"],"falsifier":"Choose a non-integer power such as $p=1.5$, set $q=2$, take the one-dimensional integral equation (6.1) with data $w_0=\\delta\\chi(x)x$ and $h=0$, and compute the leading finite difference $I(t,x+h)-I(t,x)$ at small time, $x=1$, and $h=x/2$; the proof requires the ratio $|I(t,x+h)-I(t,x)|/(t|x|^{p-n})$ to be comparable to 1 with a strictly smaller remainder. If the ratio is not of that form, the load-bearing lemma fails. Alternatively, test the even-integer endpoint $p=2$: a solution in $C([0,T];W^{3,2}(\\mathbb{R}))$ for this data would disprove the claimed 'similar result' for even $p$.","tokens_in":63923,"feed_emoji":"🎯","tokens_out":18799,"duration_ms":153819,"temperature":0.7,"pith_summary":"For evolution equations with the rough power nonlinearity $|u|^{p-1}u$, the central question is: in which Sobolev spaces do solutions exist and depend continuously on the data? This paper claims to settle that question for the nonlinear heat equation and the nonlinear Schrödinger equation, two model cases that had resisted a sharp answer for non-algebraic $p>1$. It proves that the heat equation is locally well-posed in $W^{s,q}$ exactly when $\\max\\{0,s_c\\}<s<p+2+\\frac{1}{q}$, and that for $p-1$ not an even integer it is strongly ill-posed at and above $s=\\max\\{s_c,p+2+\\frac{1}{q}\\}$. For the Schrödinger equation it proves well-posedness in $H^s$ for $\\max\\{0,s_c\\}<s<\\min\\{p+\\frac52,2p+1\\}$, ill-posedness for $s\\ge\\max\\{s_c,p+\\frac52\\}$, and a one-dimensional improvement up to $\\min\\{3p,p+\\frac52\\}$. The reason the classical thresholds were wrong is that the nonlinearity gains roughly an extra $1/q$ derivative near its zeros, a gain the paper captures with new composition estimates for complex-valued functions.","feed_headline":"Exact regularity limit found for rough heat and Schrödinger equations","feed_subtitle":"Beyond the sharp thresholds (p+2+1/q and p+5/2), even smooth small data can produce no solution.","key_machinery":"The engine of the paper is a sharp nonlinear estimate for complex-valued functions, proved as Theorem 3.1: for $p\\le s<p+\\frac{1}{q}$, there holds $\\|\\,|u|^{p-1}u\\,\\|_{W^{s,q}} \\lesssim \\|u\\|^{p-1}_{W^{1/q+\\varepsilon,q}}\\|u\\|_{W^{s,q}}$. The proof replaces $u$ by dyadic piecewise-linear approximations, so that the only dangerous contribution to the $W^{s,q}$ norm of $|u|^{p-1}u$ comes from intervals where $u$ crosses zero; on those intervals finite-difference comparisons and a precise power-type estimate in the change of variables show the loss of one full $1/q$ derivative that the classical chain rule misses. The non-existence half rests on a one-dimensional integral-equation lemma: for data proportional to $\\chi(x)x$, the differentiated nonlinearity develops a finite difference of size $t|x|^{p-n}$ at the origin, which cannot belong to $W^{p+1/q-n,q}$; a slicing theorem then carries this obstruction to every dimension.","core_discovery":"The paper's central discovery is that the true Sobolev regularity of the power nonlinearity $u\\mapsto|u|^{p-1}u$ is governed by the zero set of $u$ rather than by the pointwise smoothness of the function $z\\mapsto|z|^{p-1}z$, and that this extra derivative gain moves the optimal well-posedness boundary for the heat equation to $s< p+2+\\frac{1}{q}$ and for the Schrödinger equation to $s< \\min\\{p+\\frac52,2p+1\\}$. Concretely, the heat equation is locally well-posed in $W^{s,q}(\\mathbb{R}^d)$ for $\\max\\{0,s_c\\}<s<p+2+\\frac{1}{q}$, and for $p-1\\notin 2\\mathbb{N}$ there is smooth, small, compactly supported data for which no solution exists in $C([0,T];W^{s,q})$ at any $s\\ge\\max\\{s_c,p+2+\\frac{1}{q}\\}$. The Schrödinger equation is locally well-posed in $H^s(\\mathbb{R}^d)$ for $\\max\\{0,s_c\\}<s<\\min\\{p+\\frac52,2p+1\\}$, and for $p-1\\notin 2\\mathbb{N}$ the same non-existence holds at $s\\ge\\max\\{s_c,p+\\frac52\\}$; a separate one-dimensional argument improves the upper bound to $\\min\\{3p,p+\\frac52\\}$. Because the ill-posedness boundary is dimension-independent, the paper concludes that for $d\\gg p$ there are nonlinear Schrödinger equations ill-posed in every Sobolev space $H^s(\\mathbb{R}^d)$.","pith_inferences":["Beyond the paper: the same one-dimensional zero-crossing mechanism should trigger non-existence at the boundary for any smooth datum with a simple transverse zero, not only the modeled cutoff data $\\chi(x)x$.","Beyond the paper: completing the asserted even-integer endpoint of the one-dimensional lemma would extend the ill-posedness theorems to algebraic powers such as $p=2$, closing the last excluded range.","Beyond the paper: the composition estimate should transfer to derivative nonlinearities such as $|u|^{2\\sigma}\\partial_x u$, where the same one-derivative gain predicts sharp high-regularity thresholds of the form $2\\sigma+\\frac52$ and $4\\sigma+1$.","Beyond the paper: the endpoint shift by $1/q$ can be read as a dimensional-reduction effect, since in large dimension the boundary $p+2+1/q$ approaches $p+2$; this predicts that the same one-dimensional cancellation should control the sharp threshold for more general semilinear parabolic systems."],"forward_implications":["For the nonlinear heat equation with $1<p<\\infty$, local well-posedness holds in $W^{s,q}$ for every $s$ between the scaling threshold and $p+2+\\frac{1}{q}$, and fails for every $s$ at or above that boundary when $p-1$ is not an even integer.","For the nonlinear Schrödinger equation, once $p\\ge\\frac{3}{2}$ the sharp high-regularity boundary is $p+\\frac52$; above it, smooth small data fail to yield any solution in $H^s$.","For $1<p<\\frac{3}{2}$, the paper still proves well-posedness up to $\\min\\{p+\\frac52,2p+1\\}$, and in one dimension the boundary improves to $\\min\\{3p,p+\\frac52\\}$, showing that Strichartz integrability can improve high-regularity results.","Because the ill-posedness threshold is dimension-independent while the scaling threshold $s_c$ grows with dimension, there are nonlinear Schrödinger equations that are ill-posed in every Sobolev space $H^s$.","The same refined nonlinear estimate is expected to set sharp thresholds for other equations with order-$\\alpha$ dispersion and limited-regularity nonlinearities: well-posedness up to $s<\\alpha+\\mu$, and in dispersive cases up to $\\min\\{\\alpha\\mu,\\alpha+\\mu\\}$."],"supporting_citations":[{"why":"Supplies the piecewise-linear approximation characterization of Besov norms used to prove the sharp nonlinear estimate (Proposition 3.4).","marker":"[49]"},{"why":"Supplies the first-order finite-difference summation estimate used inside the proof of the sharp composition inequality.","marker":"[31]"},{"why":"Gives the finite-difference characterization of W^{s,q} and the slicing theorem used to lift the one-dimensional obstruction to all dimensions.","marker":"[55]"},{"why":"Points out the time-truncation inconsistency in previous fractional-time well-posedness proofs; the paper's truncation scheme is designed to fix it.","marker":"[67]"},{"why":"Establishes the previous high-regularity ill-posedness results and poses the question of the optimal threshold that this paper answers.","marker":"[13]"},{"why":"Provides the fractional composition estimate used as a tool in the high-regularity NLS estimates.","marker":"[65]"},{"why":"Provides the fractional Leibniz inequality used to distribute derivatives across products in the high-regularity estimates.","marker":"[25]"}],"fun_headline_variants":["Sharp regularity barrier found for heat and Schrödinger equations","Optimal Sobolev threshold identified for rough nonlinear evolutions","Rough nonlinearities: exact well-posedness boundary found","Sobolev threshold exposed: ill-posed beyond sharp heat and Schrödinger limits","Dimension-independent ill-posedness: every Sobolev space fails"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that, for data that is a small multiple of a cutoff function times $x$, the dominant change in the differentiated nonlinearity across a small shift is exactly proportional to $t|x|^{p-n}$, with all remainders strictly smaller; if this one-dimensional estimate fails for any $p$, the non-existence results in all dimensions collapse.","fun_headline_variants_meta":{"raw":{"variants":["Sharp regularity barrier found for heat and Schrödinger equations","Optimal Sobolev threshold identified for rough nonlinear evolutions","Rough nonlinearities: exact well-posedness boundary found","Sobolev threshold exposed: ill-posed beyond sharp heat and Schrödinger limits","Dimension-independent ill-posedness: every Sobolev space fails"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000676,"raw_usage":{"total_tokens":3280,"prompt_tokens":1357,"completion_tokens":1923,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":973,"completion_tokens_details":{"reasoning_tokens":1833}},"tokens_in":973,"tokens_out":1923,"duration_ms":17738,"temperature":1.0,"reasoning_tokens":1833,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:26:37.022225+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose a non-integer power such as $p=1.5$, set $q=2$, take the one-dimensional integral equation (6.1) with data $w_0=\\delta\\chi(x)x$ and $h=0$, and compute the leading finite difference $I(t,x+h)-I(t,x)$ at small time, $x=1$, and $h=x/2$; the proof requires the ratio $|I(t,x+h)-I(t,x)|/(t|x|^{p-n})$ to be comparable to 1 with a strictly smaller remainder. If the ratio is not of that form, the load-bearing lemma fails. Alternatively, test the even-integer endpoint $p=2$: a solution in $C([0,T];W^{3,2}(\\mathbb{R}))$ for this data would disprove the claimed 'similar result' for even $p$.","supporting_citations":[{"cited_title":"On the boundedness of the mapping f → |f | in Besov spaces","cited_arxiv_id":null,"evidence_quote":"Supplies the piecewise-linear approximation characterization of Besov norms used to prove the sharp nonlinear estimate (Proposition 3.4)."},{"cited_title":"On the boundedness of the mapping f ↦→ |f |µ , µ > 1 on Besov spaces","cited_arxiv_id":null,"evidence_quote":"Supplies the first-order finite-difference summation estimate used inside the proof of the sharp composition inequality."},{"cited_title":"Strichartz","cited_arxiv_id":null,"evidence_quote":"Gives the finite-difference characterization of W^{s,q} and the slicing theorem used to lift the one-dimensional obstruction to all dimensions."},{"cited_title":"A remark on local well-posedness for nonlinear S chr¨ odinger equations with power nonlinearity— an alternative approach","cited_arxiv_id":null,"evidence_quote":"Points out the time-truncation inconsistency in previous fractional-time well-posedness proofs; the paper's truncation scheme is designed to fix it."},{"cited_title":"Weissler","cited_arxiv_id":null,"evidence_quote":"Establishes the previous high-regularity ill-posedness results and poses the question of the optimal threshold that this paper answers."},{"cited_title":"The defocusing energy-critical nonlinear Schroedinger eq uation in dimensions five and higher","cited_arxiv_id":null,"evidence_quote":"Provides the fractional composition estimate used as a tool in the high-regularity NLS estimates."},{"cited_title":"The Kato-Ponce inequality","cited_arxiv_id":null,"evidence_quote":"Provides the fractional Leibniz inequality used to distribute derivatives across products in the high-regularity estimates."}],"review_version":1}