Investigations on Type I Blow-up of Nonlinear Heat Systems With Potential Term
Bibliographic record
Abstract
In this paper, we are concerned with the following initial-boundary value problem: (P) \left\{% \begin{array}{ll} \hbox{$u_t(x,t)- \Delta u(x,t)- G(x)|u|^{p-1}u  =0, \quad x\in  \Omega, t\in(0,T)$,} \hbox{$u(x,t)=0 \quad x \in \partial \Omega,  t\in(0,T)$,} \hbox{$u(x,0)=u_{0}(x), \quad x \in \Omega,$} \\ \end{array}% \right. where $ p \geq p_s :=\dfrac{ d + 2}{d-2} $, $ u_0 \in L^\infty(D_\mu) $, and $ G(r) \in C^1([0, \mu]), $  $0 <  \underline{C} \leq G(r) \leq \overline{C}  < \infty,$ $G^{'}(r) \leq 0 $. We study the initial value problem and boundary conditions for a nonlinear heat equation incorporating a potential term. Particularly, we focus on the asymptotic behavior of solutions during blow-ups. We extend existing results on this phenomenon, specifically in the case where the potential term is constant, based on the works of Matano-Merle (Matano  \& Merle, 2004). We  show that when $p_s \leq p < p^* $, the radial solutions of this problem always exhibit Type I blow-up. This result generalizes previous results for the case where $G \equiv 1 $, and its achievement is non-trivial due to the presence of the potential term $ G $. We use the contraction mapping principle to  show the existence of singular stationary solutions to an associated elliptic equation with a potential. Furthermore, our analysis of the properties of the zeros of the solutions lead to the nonexistence of type II singularity for the problem. We also delve into the study of critical solutions for a class of nonlinear  parabolic equations in a bounded domain, focusing on the construction of appropriate approximate solutions.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
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 | 0.005 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.000 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.002 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".