Existence and nonexistence results for a class of pseudo-parabolic equations with combined logarithmic nonlinearities
Bibliographic record
Abstract
Abstract In this paper, we study the existence and nonexistence results for a class of pseudo-parabolic equations with combined logarithmic nonlinearities γ | u | p- 2 u log | u | + u log | u |, where γ > 0 and p > 2 satisfy certain conditions. Employing the key ingredients in modified potential well method, we obtain the existence of a global weak solution and the finite time blow-up phenomenon for low initial energy and critical initial energy. We show that the global weak solution has exponential decay. An interesting observation is that the higher power term γ | u | p- 2 u log | u | ( p > 2) breaks the infinite time blow-up phenomenon of u log | u | with appropriate coefficient γ . Furthermore, an upper bound and a lower bound for the blow-up time are estimated. We give sufficient conditions for global existence and blow-up result of weak solutions in the case of high initial energy. Then we take account of the case when the initial energy is independent of the well depth. In addition, we show that the solution to the same pseudo-parabolic equation with only logarithmic nonlinearity u log | u | blows up at infinity.
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.002 |
| 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".