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We study the so-called expansion of positivity for doubly nonlinear parabolic type equations, including the porous medium and evolutionary p -Laplace equations. The expansion of positivity yields an alternative proof of the fundamental Hölder regularity for the evolutional p -Laplace equations, originally due to DiBenedetto.

We show a global existence for the Cauchy problem with large initial data for the p -harmonic flow between two smooth, compact Riemannian manifolds. We devise new monotonicity type formulas of a local scaled energy and establish a partial regularity for the solution. The partial regularity obtained is almost optimal, comparing with that of the corresponding stationary case. The p -harmonic flow obtained also converges to a p -harmonic map along a certain time sequence tending to infinity.

In this paper, assuming the initial-boundary datum belonging to suitable Sobolev and Lebesgue spaces, we prove the global existence result for a (possibly sign changing) weak solution to the Cauchy–Dirichlet problem for doubly nonlinear parabolic equations of the form ∂ t | u | q - 1 u - Δ p u = 0 in Ω ∞ , where p > 1 and q > 0 . This is a fair improvement of the preceding result by authors (Nonlinear Anal 175C :157–172, 2018). The key tools we employ are energy estimates for approximate equations of Rothe type and the integral strong convergence of gradients of approximate solutions.

In this article, we consider a fast diffusive type doubly nonlinear parabolic equation and study the extinction behavior of a solution at a finite time. We show the complete extinction of a weak solution with a nonnegative initial datum, that is, a weak solution is positive before a finite time and vanishes after it, and derive the optimal decay estimates of extinction. Our key ingredient of the proof is a nonlinear intrinsic scaling and the expansion of positivity.

We consider a volume constraint problem for the nonlocal doubly nonlinear parabolic equation, called the nonlocal$ p $ -Sobolev flow, and introduce a nonlinear intrinsic scaling, converting a prototype nonlocal doubly nonlinear parabolic equation into the nonlocal$ p $ -Sobolev flow. This paper is dedicated to Giuseppe Mingione on the occasion of his 50th birthday, who is a maestro in the regularity theory of PDEs.

Objectives Endoscopic submucosal dissection (ESD) is an effective procedure for the en bloc resection of colorectal neoplasms. However, risk factors for local recurrence after ESD have not been identified. This study aimed to evaluate such risk factors after ESD for colorectal neoplasms. Methods This retrospective study included 1344 patients with 1539 consecutive colorectal lesions who underwent ESD between September 2003 and December 2019. We investigated various factors associated with local recurrence in these patients. The main outcomes were the incidence of local recurrence and its relationship with clinicopathological factors during long‐term surveillance. Results The en bloc resection rate was 98.6%, the R0 resection rate was 97.2%, and the histologically complete resection rate was 92.7%. Local recurrence was observed in 7/1344 (0.5%) patients and the median follow‐up period was 72 months (range 4–195 months). The incidence of local recurrence was significantly higher in lesions ≥40 mm in diameter (hazard ratio [HR] 15.68 [1.88–130.5]; p = 0.011), piecemeal resection (HR 48.42 [10.7–218.7]; p < 0.001), non‐R0 resection (HR 41.05 [9.025–186.7]; p < 0.001), histologically incomplete resection (HR 16.23 [3.627–72.63]; p<0.001), and severe fibrosis (F2; HR 9.523 [1.14–79.3]; p = 0.037). Conclusions Five risk factors for local recurrence after ESD were identified. Patients with such factors should undergo careful surveillance colonoscopy.

Purpose Endoscopic submucosal dissection (ESD) is a minimally invasive treatment for early gastric cancer. However, perforations may happen and cause peritonitis during ESD. Thus, there is a potential demand for a computer-aided diagnosis system to support physicians in ESD. This paper presents a method to detect and localize perforations from colonoscopy videos to avoid perforation ignoring or enlarging by ESD physicians. Method We proposed a training method for YOLOv3 by using GIoU and Gaussian affinity losses for perforation detection and localization in colonoscopic images. In this method, the object functional contains the generalized intersection over Union loss and Gaussian affinity loss. We propose a training method for the architecture of YOLOv3 with the presented loss functional to detect and localize perforations precisely. Results To qualitatively and quantitatively evaluate the presented method, we created a dataset from 49 ESD videos. The results of the presented method on our dataset revealed a state-of-the-art performance of perforation detection and localization, which achieved 0.881 accuracy, 0.869 AUC, and 0.879 mean average precision. Furthermore, the presented method is able to detect a newly appeared perforation in 0.1 s. Conclusions The experimental results demonstrated that YOLOv3 trained by the presented loss functional were very effective in perforation detection and localization. The presented method can quickly and precisely remind physicians of perforation happening in ESD. We believe a future CAD system can be constructed for clinical applications with the proposed method.

Objectives This study evaluates risk factors for lymph node metastasis (LNM) in T2 colorectal cancer to refine patient selection for endoscopic resection. Methods We reviewed records from consecutive patients who had undergone curative surgical resection of T2 colorectal cancer at our institution in Japan between April 2001 and December 2021. Data on conventional clinicopathologic variables were retrieved from the pathology reports at the time of surgery. The clinicopathological features included patient age, sex, tumor diameter, morphology, tumor location, lymphatic invasion, vascular invasion, tumor differentiation, carcinoembryonic antigen and carbohydrate antigen 19‐9 levels, number of lymph node dissections, presence of adenoma component, and LNM. Results Among the patients (338 men, 320 women), 170 (25.8%) exhibited LNM. Multivariate logistic regression identified three independent risk factors for LNM: lymphatic invasion (odds ratio [OR], 32.6; 95% confidence interval [CI], 17.3–61.4; p < 0.0001), female sex (OR, 1.70; 95% CI, 1.10–2.62; p = 0.02), and elevated carcinoembryonic antigen levels (OR, 2.56; 95% CI, 1.10–5.96; p = 0.03). Conclusions Lymphatic invasion, female sex, and high carcinoembryonic antigen levels significantly increase the risk of LNM in T2 colorectal cancer.

Objectives Japanese guidelines include high‐grade (poorly differentiated) tumors as a risk factor for lymph node metastasis (LNM) in T1 colorectal cancer (CRC). However, whether the grading is based on the least or most predominant component when the lesion consists of two or more levels of differentiation varies among institutions. This study aimed to investigate which method is optimal for assessing the risk of LNM in T1 CRC. Methods We retrospectively evaluated 971 consecutive patients with T1 CRC who underwent initial or additional surgical resection from 2001 to 2021 at our institution. Tumor grading was divided into low‐grade (well‐ to moderately differentiated) and high‐grade based on the least or predominant differentiation analyses. We investigated the correlations between LNM and these two grading analyses. Results LNM was present in 9.8% of patients. High‐grade tumors, as determined by least differentiation analysis, accounted for 17.0%, compared to 0.8% identified by predominant differentiation analysis. A significant association with LNM was noted for the least differentiation method (p < 0.05), while no such association was found for predominant differentiation (p = 0.18). In multivariate logistic regression, grading based on least differentiation was an independent predictor of LNM (p = 0.04, odds ratio 1.68, 95% confidence interval 1.00–2.83). Sensitivity and specificity for detecting LNM were 27.4% and 84.1% for least differentiation, and 2.1% and 99.3% for predominant differentiation, respectively. Conclusions Tumor grading via least differentiation analysis proved to be a more reliable measure for assessing LNM risk in T1 CRC compared to grading by predominant differentiation.