Asymptotic profile of solutions to a non‐linear dissipative evolution system with conservation
Bibliographic record
Abstract
Abstract We investigate the asymptotic profile to the Cauchy problem for a non‐linear dissipative evolution system with conservational form provided that the initial data are small, where constants α, ν are positive satisfying ν2<4α(1 − α), α<1. In (J. Phys. A 2005; 38:10955–10969), the global existence and optimal decay rates of the solution to this problem have been obtained. The aim of this paper is to apply the heat kernel to examine more precise behaviour of the solution by finding out the asymptotic profile. Precisely speaking, we show that, when time t → ∞ the solution $ {\psi \rightarrow De^{-(1-\alpha-\nu^{2}/{4 \alpha})t}G(t, x) \cos ((\nu / 2{\alpha})x + {\Pi / 4} + \beta)}$ and solution ${\theta \rightarrow - D \nu e^{-(1-\alpha-\nu^{2}/{ 4\alpha}) t}G(t, x)\sin((\nu/2{\alpha})x + {\Pi / 4} + \beta)}$ in the Lp sense, where G(t, x) denotes the heat kernel and $D=2\sqrt{2(\vartheta_{+}^{2}+\vartheta_{-}^{2})}$ is determined by the initial data and the solution to a reformulated problem obtained in Section 3, β is related to ϑ+ and ϑ− which are determined by (41) in Section 4. The numerical simulation is presented in the end. The motivation of this work thanks to Nishihara (Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity. Z. Angew Math Phys 2006; 57: 604–614). Copyright © 2007 John Wiley & Sons, Ltd.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.007 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".