{"id":"e07865aa-a22d-41c2-b66c-803d171565b9","arxiv_id":"2412.08834","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Small data blow up before time C epsilon^{-2p(p-1)/((1+alpha)gamma)} for u_tt - a(t)^2 Delta u + b(t) u_t = |u|^p whenever p is below the generalized Strauss exponent.","lead":"This paper proves finite-time blowup for semilinear wave equations whose wave speed varies in time and whose damping decays fast enough, for a range of nonlinearity powers below a Strauss-type threshold. It supplies explicit upper bounds on the lifespan and unifies known cases including the classical wave, Tricomi, and expanding-spacetime equations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"For N≥9 the local-existence restriction (1.7) cuts off the top of the sub-Strauss interval; e.g., N=10, α=0 gives pHWY≈1.383 > (N−2)/(N−4)=4/3, so the advertised full sub-Strauss blowup is not proven.","rationale":"The reader identified exactly the weak point: Proposition 1.1 and (1.7) restrict p to (1,(N−2)/(N−4)] for N≥5, while the abstract and the claimed Strauss-type blowup phenomenon aim at the full sub-Strauss interval p<pHWY(N,α). For α=0 this is p<pS(N), and for N≥9 the upper endpoint (N−2)/(N−4) is strictly below pS(N); the gap grows for negative α. This is not a flaw in the proof of the stated theorem, but it is a real mismatch between the central advertised claim and the theorem as proven. I checked the main ingredients of the argument: the Liouville–Green approximation in Lemma 2.1, the conservative-quantity identity in Lemma 3.1, the cutoff estimates in Lemma 3.2, and the derivation of the lifespan estimate in Corollary 1.3 are coherent, and I found no independent error in the exponent. The load-bearing issue is therefore the solution-class restriction. Since the test-function method works at the level of finite-energy solutions and the missing p-range is covered by standard energy well-posedness, the appropriate verdict is CONDITIONAL: the authors should either extend local well-posedness to H^1×L^2 solutions for the full sub-Strauss range, or explicitly restrict the abstract and Corollary 1.3 to p satisfying (1.7).","tokens_in":15043,"tokens_out":18105,"duration_ms":182064,"concrete_test":"Compute pHWY(10,0)=(11+√193)/18 and (10−2)/(10−4)=4/3; since the former exceeds the latter, the gap is real. Then check whether the Liouville-transformed problem (A.2) admits local energy solutions for p≤1+4/(N−2), and verify pS(N)≤1+4/(N−2) for all N; if both hold, the gap is an artifact of the unnecessarily strong H^2 solution class and can be closed by replacing or augmenting Proposition A.1.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The theorem is internally consistent: Corollary 1.3 explicitly requires p to satisfy (1.7), so the estimate is true for the range stated. The load-bearing problem is that the advertised scope ('sub-Strauss exponent') exceeds that range in high dimensions. For the simplest case a≡1, b≡0 (α=0), pHWY(N,0)=pS(N). For N=10, pS(10)=(11+√193)/18≈1.383, whereas (1.7) permits only p≤(N−2)/(N−4)=4/3≈1.333. For N=9, pS(9)≈1.425>1.4, and for all N≥9 with α close to 0 or negative the same phenomenon occurs. Hence the central claim that arbitrary small data blow up for every p<pHWY is not established in the gap. The proof of Proposition A.1 constructs H^2 strong solutions only under (1.7), because |f|^p∈H^1 for general f∈H^2 fails for larger p. The test-function argument itself does not require H^2 regularity, so the gap is likely repairable by an energy-class local well-posedness theorem, but that extension is not present in the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper considers the Cauchy problem for semilinear wave equations ∂_t^2 u - a(t)^2 Δu + b(t)∂_t u = |u|^p with small nonnegative compactly supported data. Under assumptions that a(t) is positive and b(t) is nonnegative and integrable, it proves upper bounds for the lifespan of strong solutions via a test-function method based on conservative quantities for the linear equation. The test functions are constructed from a special solution of the separated linear equation using the Liouville–Green (WKB) approximation. For coefficients behaving like a(t)~(1+t)^α and b(t)~(1+t)^{-β}, the authors derive the Strauss/HWY-type exponent pHWY(N,α) and the lifespan estimate T_ε ≤ C ε^{-2p(p-1)/((1+α)γ(N,α;p))} for subcritical p.","tokens_in":15284,"tokens_out":19807,"duration_ms":187836,"significance":"The test-function framework is clean and the WKB construction is explicit. The paper gives a unified derivation of Strauss-type blow-up for a class of time-dependent speeds, recovering the classical wave and generalized Tricomi cases, and it provides explicit lifespan bounds. The proof of Theorem 1.2 is a chain of well-controlled estimates (Lemmas 3.1, 3.2, 3.4) with no free parameters or post-hoc exclusions. The main caveat is the local well-posedness restriction (1.7), which narrows the claimed sub-Strauss range in high dimensions; this is a correctness-risk concern for the advertised scope rather than an internal inconsistency.","major_comments":[{"comment":"The advertised sub-Strauss blow-up range is not fully covered for N≥9. Corollary 1.3 requires p to satisfy (1.7), i.e., p≤(N−2)/(N−4) for N≥5, while for a≡1, b≡0 it claims p<pHWY(N,0)=pS(N). For N=10, pS(10)=(11+√193)/18≈1.383 exceeds 4/3, and the same phenomenon occurs for all N≥9 when α is close to 0 or negative. The gap is not a defect in the test-function argument, which only uses integration by parts and L^p bounds, but it means the local well-posedness result in Proposition A.1, proved via the Liouville transform and contraction mapping in H^1×L^2, is the limiting step: |f|^p∈H^1 for f∈H^2 fails for p above (N−2)/(N−4). Since the abstract and Remark 1.2(iv) advertise the Strauss exponent as the outcome of the method, the paper needs either an extension of local well-posedness to weaker solution classes covering 1<p<pHWY(N,α), or a revised statement that explicitly restricts the sub-Strauss claim to the dimensions and α-ranges in which (1.7) is not binding.","section":"Proposition 1.1 and Eq. (1.7); Corollary 1.3; Remark 1.2(iv)"}],"minor_comments":[{"comment":"The line 'supp u1 ⊂ B(0, r0+∫_0^t a(r)dr)' should read 'supp u(t) ⊂ ...', since u1 denotes the initial velocity.","section":"Proposition A.1"},{"comment":"There is a typo in the sentence 'the the integral of e^{B(t)}(∂_t v Φ − v ∂_t Φ) is independent of t'; delete the duplicated article.","section":"Introduction"},{"comment":"The statement that 'the conditions (A1) and (A3) also give the boundedness of 1/((1+t)a(t))' is terse; a one-line proof (using (a^{-1})'→0 to get a^{-1}(t)=o(t)) would improve readability.","section":"Section 3, proof of the second estimate"},{"comment":"It would be helpful to state explicitly that the power-type estimates on a and b imply the assumptions (A2)–(A3) needed for Lemma 3.4 and Theorem 1.2.","section":"Corollary 1.3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is technically sound in the range it actually proves. The only serious issue is the mismatch between the advertised sub-Strauss range and the local well-posedness restriction (1.7). The gap is likely repairable by extending local well-posedness to weaker solution classes or by transparently restricting the claims in high dimensions; I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Core take: the general lifespan estimate in Theorem 1.2 is new and the proof is clean; but the advertised full sub-Strauss blowup is not actually proved in high dimensions because Proposition 1.1 only gives strong solutions for p up to (N−2)/(N−4).\n\nWhat's new: the estimate ε ≤ C T^{-(p^2+1)/(p(p-1))} A(T)^{N/p} (∫ a(t)^{p'/2}(1+A(t))^{N−1−(N−1)p'/2} dt)^{1/p'} with the A(t) weight, and the WKB-based construction of m* that turns the test-function method into a compact argument. This genuinely improves Tsutaya–Wakasugi's condition (1.6): for σ=1 their condition gives the Kato exponent pK(N), while Corollary 1.3 yields pS(N) (Strauss). Special cases (a≡1, b≡0; generalized Tricomi; a≡1 with integrable damping) are recovered as limits, and the critical case is honestly left out.\n\nSoft spots: the scope gap already mentioned. The theorem is internally consistent—Corollary 1.3 explicitly assumes (1.7)—but the abstract and several remarks overstate the range. For N=10, α=0, pHWY ≈ 1.383 while (1.7) stops at 4/3. So the headline claim 'blowup for every p < pHWY' is not established in the gap. The local well-posedness appendix is standard but only provides H^2 strong solutions; the test-function part doesn't need that regularity, so an energy-class local existence result would close the hole. I'd call it a presentational overreach rather than a mathematical error.\n\nMinor point: Lemma 3.2 uses b ∈ L^1, and the damping class is narrower than Wakasa–Yordanov, but that is stated. Citation pattern looks fair; the self-citation [10] is appropriate because the method is imported and credited.\n\nWho this is for: people working on blowup for semilinear wave equations with time-dependent speed and damping. A serious referee should see it. I'd send it to review and ask the authors to either extend local well-posedness for the missing p range or refine the statements to what (1.7) actually permits.","headline":"Genuinely new lifespan estimate via a clean WKB/test-function argument, but the abstract overclaims full sub-Strauss blowup in high dimensions where the local existence theorem doesn't reach.","tokens_in":15891,"tokens_out":2618,"would_cite":true,"duration_ms":26185,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35L71","35Q85"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that arbitrarily small nonnegative data blow up in finite time for semilinear wave equations with time-dependent speed and integrable damping, whenever the nonlinearity lies below the He–Witt–Yin/Strauss threshold.","keywords":["semilinear wave equation","time-dependent speed of propagation","scattering damping","blow-up","lifespan estimate","Strauss exponent","generalized Tricomi equation","Liouville-Green approximation"],"falsifier":"For a covered case such as $N=3$, $\\alpha=0$, $p=1.5$, with small nonnegative compact data, a numerical or rigorous construction of a global solution, or of a lifespan growing faster than $\\varepsilon^{-2p(p-1)/\\gamma_S(3,p)}$, would contradict the theorem. Alternatively, for $N=10$ and $p=1.36$ (between the local-existence cap $4/3$ and $p_S(10)\\approx1.383$), finding a global strong solution would show that the abstract's sweeping sub-Strauss statement needs the solution-class restriction.","tokens_in":14806,"feed_emoji":"⏳","tokens_out":10055,"duration_ms":94618,"temperature":0.7,"pith_summary":"This paper establishes that for the semilinear wave equation with time-dependent propagation speed $a(t)$ and scattering (integrable) damping $b(t)$, arbitrary small nonnegative compactly supported data still produce finite-time blowup whenever the nonlinear power $p$ lies below the He–Witt–Yin threshold $p_{\\mathrm{HWY}}(N,\\alpha)$, where $a(t)$ behaves like $(1+t)^\\alpha$ with $\\alpha>-1$. In the power-type case the lifespan obeys $T_\\varepsilon \\le C \\varepsilon^{-2p(p-1)/((1+\\alpha)\\gamma(N,\\alpha;p))}$. This unifies the classical Strauss blowup for the constant-speed wave equation ($\\alpha=0$), the generalized Tricomi blowup, and variable-speed models of FLRW type, and it improves earlier general blowup conditions that only reached the Kato-type exponent. The proof works through a conservative quantity for the linear wave equation, built from a decaying separation-of-variables solution, controlled by the Liouville–Green approximation.","feed_headline":"Variable speed still forces wave blowup at small data","feed_subtitle":"A Strauss-type exponent controls the lifetime even with time-dependent coefficients and scattering damping.","key_machinery":"The argument rests on a conservative quantity for the linear equation $\\partial_t(e^{B(t)}\\partial_t v)-e^{B(t)}a(t)^2\\Delta v=0$: for any solution $\\Phi$ of the linear equation, $e^{B(t)}\\int(\\partial_t v\\,\\Phi-v\\,\\partial_t\\Phi)\\,dx$ is constant in time. The paper chooses $\\Phi(x,t)=m(t)\\varphi(x)$, with $\\varphi(x)=\\int_{S^{N-1}}e^{x\\cdot\\omega}dS(\\omega)$ (so $\\Delta\\varphi=\\varphi>0$) and $m$ the decaying solution of $m''+b\\,m'=a^2m$. Lemma 3.4, via the Liouville–Green (WKB) approximation of Lemma 2.1, gives $m(t)\\asymp a(t)^{-1/2}e^{-A(t)}$ and $m'(t)\\asymp -a(t)^{1/2}e^{-A(t)}$, making $\\Phi$ concentrate near the light cone $|x|=A(t)$. Inserting this $\\Phi$ into the test-function identity of Lemma 3.1 and using a carefully cut-off test function $\\psi_R$ yields a lower bound on the weighted $L^p$ norm of $u$, which together with a first mass estimate gives the lifespan bound.","core_discovery":"The central claim is that the Strauss-type exponent governs blowup for the whole family $\\partial_t^2 u - a(t)^2\\Delta u + b(t)\\partial_t u = |u|^p$, not just for the constant-speed equation. Concretely, Theorem 1.2 shows that for strong solutions with nonnegative data, $\\varepsilon \\le C\\, T^{-(p^2+1)/(p(p-1))} A(T)^{N/p} \\left(\\int_0^T a(t)^{p'/2}(1+A(t))^{N-1-(N-1)p'/2}\\,dt\\right)^{1/p'}$. When $a(t)\\sim(1+t)^\\alpha$ and $b(t)\\sim(1+t)^{-\\beta}$ with $\\alpha>-1$ and $\\beta>1$, Corollary 1.3 converts this into $T_\\varepsilon\\le C\\,\\varepsilon^{-2p(p-1)/((1+\\alpha)\\gamma(N,\\alpha;p))}$ for all $1<p<p_{\\mathrm{HWY}}(N,\\alpha)$, where $\\gamma(N,\\alpha;p)=2+\\left(\\frac{N+1-3\\alpha}{1+\\alpha}\\right)p-\\left(\\frac{N-1+\\alpha}{1+\\alpha}\\right)p^2$. Thus the same exponent that appears for the generalized Tricomi equation appears for variable speeds, and for $\\alpha=0$ it reduces to the Strauss exponent $p_S(N)$, so the result contains the classical small-data blowup for sub-Strauss powers.","pith_inferences":["The authors explicitly leave the critical case $p=p_{\\mathrm{HWY}}(N,\\alpha)$ open; a natural extension of the same conservative-quantity estimate would give an exponential lifespan upper bound, which they indicate as forthcoming.","Because the proof only treats $H^2$ strong solutions with $p$ satisfying (1.7), in high dimensions (starting at $N=9$ for $\\alpha=0$) the theorem does not cover powers just below the Strauss exponent; extending local well-posedness or passing to weaker solutions would close that gap.","The same $\\Phi$-based conservative quantity may apply to weakly coupled systems of such equations, where the Strauss-type exponent is typically replaced by a curve in $(p,q)$.","One could test numerically whether the lifespan exponent matches the predicted $2p(p-1)/((1+\\alpha)\\gamma(N,\\alpha;p))$ for intermediate $\\alpha$, which is not yet covered by other methods."],"forward_implications":["For $a(t)\\sim(1+t)^\\alpha$ with $\\alpha>-1$ and integrable damping, the finite-time blowup interval is $1<p<p_{\\mathrm{HWY}}(N,\\alpha)$; the earlier condition based only on $\\sigma=1+\\alpha$ is sharpened to the Strauss-type threshold.","Setting $\\alpha=0$ recovers the classical sub-Strauss blowup for the constant-speed wave equation, and setting $b\\equiv 0$ recovers a slightly generalized version of the subcritical Tricomi lifespan estimate.","The lifespan upper bound has the same $\\varepsilon$-power as in the generalized Tricomi theory, so the time-dependent speed does not change the blowup rate.","The support of the data enters only through fixed constants, so the blowup mechanism is independent of the detailed shape of $f$ and $g$ as long as they are nonnegative and not identically zero."],"supporting_citations":[{"why":"Supplies the test-function method with conservative quantities that the paper adapts.","marker":"[10]"},{"why":"Provides the Liouville–Green approximation lemma used to control the decaying solution $m$.","marker":"[17]"},{"why":"Introduced the exponent $p_{\\mathrm{HWY}}$ for semilinear generalized Tricomi equations.","marker":"[5, 6, 7, 8]"},{"why":"Established lifespan blowup for semilinear generalized Tricomi equations that this paper extends to variable speed.","marker":"[29]"},{"why":"Gave the previous blowup condition based on $\\sigma$ that the present result improves.","marker":"[26]"},{"why":"Supplies the Yordanov–Zhang test function $\\varphi$ and the $L^{p'}$ estimate near the light cone.","marker":"[37]"},{"why":"Proved critical Strauss exponent blowup for integrable damping with $a\\equiv 1$, the constant-speed analogue.","marker":"[30]"}],"fun_headline_variants":["Strauss exponent governs blowup in variable-speed waves","Time-dependent speed doesn't dodge Strauss blowup bound","Strauss exponent survives variable speed in blowup","Variable speed waves obey Strauss-type blowup law"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the solution is a strong solution for which the nonlinear term $|u|^p$ is regular enough, meaning $p$ satisfies the local-well-posedness restriction (1.7); in high dimensions this range ends below the Strauss threshold, so the theorem's blowup conclusion is not asserted for the remaining sub-Strauss powers without an extra existence argument.","fun_headline_variants_meta":{"raw":{"variants":["Strauss exponent governs blowup in variable-speed waves","Time-dependent speed doesn't dodge Strauss blowup bound","Strauss exponent survives variable speed in blowup","Variable speed waves obey Strauss-type blowup law"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00058,"raw_usage":{"total_tokens":2731,"prompt_tokens":941,"completion_tokens":1790,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":1728}},"tokens_in":557,"tokens_out":1790,"duration_ms":11755,"temperature":1.0,"reasoning_tokens":1728,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T17:30:58.312377+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a covered case such as $N=3$, $\\alpha=0$, $p=1.5$, with small nonnegative compact data, a numerical or rigorous construction of a global solution, or of a lifespan growing faster than $\\varepsilon^{-2p(p-1)/\\gamma_S(3,p)}$, would contradict the theorem. Alternatively, for $N=10$ and $p=1.36$ (between the local-existence cap $4/3$ and $p_S(10)\\approx1.383$), finding a global strong solution would show that the abstract's sweeping sub-Strauss statement needs the solution-class restriction.","supporting_citations":[{"cited_title":"Ikeda, M","cited_arxiv_id":null,"evidence_quote":"Supplies the test-function method with conservative quantities that the paper adapts."},{"cited_title":"Asymptotics and special functions,","cited_arxiv_id":null,"evidence_quote":"Provides the Liouville–Green approximation lemma used to control the decaying solution $m$."},{"cited_title":"Tsutaya, Y.Wakasugi, Blow up of solutions of semilinear wave equations with time- dependent propagation speed and damping , to appear in: Advanced Studies in Pure Math- ematics","cited_arxiv_id":null,"evidence_quote":"Gave the previous blowup condition based on $\\sigma$ that the present result improves."},{"cited_title":"Yordanov, Q.S","cited_arxiv_id":null,"evidence_quote":"Supplies the Yordanov–Zhang test function $\\varphi$ and the $L^{p'}$ estimate near the light cone."},{"cited_title":"Wakasa, B","cited_arxiv_id":null,"evidence_quote":"Proved critical Strauss exponent blowup for integrable damping with $a\\equiv 1$, the constant-speed analogue."}],"review_version":1}