Unipolar Euler–Poisson equations with time-dependent damping: blow-up and global existence
Bibliographic record
Abstract
This paper is concerned with the Cauchy problem for one-dimensional unipolar Euler-Poisson equations with time-dependent damping, where the time-asymptotically degenerate damping in the form of -µ (1+t) λ ρu for λ > 0 with µ > 0 plays a crucial role for the structure of solutions.The main issue of the paper is to investigate the critical case with λ = 1.We first prove that, for all cases with λ > 0 and µ > 0 (including the critical case of λ = 1), once the initial data is steep at a point, then the solutions are locally bounded but their derivatives will blow up in finite time, by means of the method of Riemann invariants and the technical convex analysis.Secondly, for the critical case of λ = 1 with µ > 7/3, we prove that there exists a unique global solution, once the initial perturbation around the constant steady-state is sufficiently small.In particular, we derive the algebraic convergence rates of the solution to the constant steady-state, which are piecewise, related to the parameter µ for 7/3 < µ ≤ 3, 3 < µ ≤ 4 and µ > 4. The adopted method of proof in this critical case is the technical time-weighted energy method and the time-weight depends on the parameter µ.Finally, we carry out some numerical simulations in two cases for blow-up and global existence, respectively, which numerically confirm our theoretical results.
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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".