Blow-up phenomena for a class of fourth-order parabolic problems
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Bibliographic record
Abstract
This paper deals with some qualitative properties of solutions to a class of semilinear fourth-order parabolic problems. It is shown that under certain conditions on the data, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis bold x comma t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">x</mml:mtext> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">u(\textbf {x}, t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> cannot exist for all time and an upper bound for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t Superscript star"> <mml:semantics> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ⋆ </mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">t^{\star }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is derived where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 comma t Superscript star Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ⋆ </mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(0,t^{\star })</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the interval of existence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis bold x comma t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">x</mml:mtext> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">u(\textbf {x}, t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Moreover we construct (under certain conditions on the data) a lower bound for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t Superscript star"> <mml:semantics> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ⋆ </mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">t^{\star }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when blow-up occurs. This last result is based on some Sobolev type inequality established at the end of the paper.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it