Existence of solutions for a higher order non-local equation appearing in crack dynamics

In this paper, we prove the existence of non-negative solutions for a non-local higher order degenerate parabolic equation arising in the modeling of hydraulic fractures. The equation is similar to the well-known thin film equation, but the Laplace operator is replaced by a Dirichlet-to-Neumann operator, corresponding to the square root of the Laplace operator on a bounded domain with Neumann boundary conditions (which can also be defined using the periodic Hilbert transform). In our study, we have to deal with the usual difficulty associated to higher order equations (e.g. lack of maximum principle). However, there are important differences with, for instance, the thin film equation: First, our equation is nonlocal; Also the natural energy estimate is not as good as in the case of the thin film equation, and does not yields, for instance, boundedness and continuity of the solutions (our case is critical in dimension $1$ in that respect)

Choosing of the topical anesthesia method for awake blind tracheal intubation of obese patients

В роботі проведено порівняння двох методів анестезії дихальних шляхів під час виконання сліпої інтубації трахеї через надгортанні повітряводи (SAD) у хворих з ожирінням при свідомості:трансларингеальної ін’єкції (TLI) місцевого анестетика та техніки непрямої ларингоскопії (IL). Виявлено, що інцидентність кашлю та позовів до блювання під час інтубації, кардіопульмональна стабільність пацієнтів, післяопераційний біль в горлі, охриплість та затруднене ковтання, а також задоволеність пацієнтів/дослідників якістю відчуття/виконання процедури достовірно не відрізнялися між групами TLI та IL. Однак, в групі TLI частіше спостерігались ознаки кровотечі зі слизової оболонки трахеї, ніж в групі IL (11 (79%) випадків проти 2 (17%) випадків відповідно, p30 kg/m2 with suspected or known difficult airways divided into 2 groups: TLI-group (n=14) and IL-group (n=12). Before the procedure patients got the same premedication: atropine 0.6 mg, diphenhydramine 10 mg, dexamethasone 4 mg, fentanyl 0.1 mg, diazepam 2.5 mg. For topical anesthesia of the oral cavity and the laryngopharynx 10% lidocaine spray solution was used in both groups. For anesthesia of the vocal cords and mucous membrane of the trachea in the TLI-group 4 ml of a 4% lidocaine solution was injected through the cricothyroid membrane, in the IL-group 2 ml of a 4% lidocaine solution was injected to the vocal cords under control of indirect laryngoscopy. The endpoints of the study were: incidence of coughing and gagging during intubation, cardiopulmonary stability, tracheal mucosa bleeding, post-operative sore throat, hoarseness and difficulty of swallowing, satisfaction of patients and researchers. Statistical analysis was provided with a program Statistica for Windows version 6.0. Demographic and clinical data of the patients did not significantly differ between two groups (p>0.05). Intubation through SAD was performed in all patients (100%). Herewith for successful SAD insertion in all cases only one attempt was required and for successful intubation of the trachea - 1-2 attempts in both TLI-group and IL-group (p>0.05). The duration of SAD fixing and the time of tracheal intubation through SAD did not significantly differ between the groups and amounted to 20 (15-30) sec and 10 (5-20) sec accordingly (p>0.05). There was not found a significant difference between groups neither in the incidence of coughing and gagging during insertion of the tube into the trachea, nor in the satisfaction of the procedure performer with the patient’s reaction to it, nor in sedation level of patients during intubation (p>0.05). But as it was expected the signs of bleeding from the tracheal mucosa were more frequent in the TLI-group than in the IL-group (11 (79%) cases vs. 2 (17%) cases accordingly, p 0.05). Regarding to postoperative audit patients showed the absence of significant differences in the level of pain during the procedure and memories after it and almost identical incidence of hoarseness and difficulty during swallowing (p> 0.05). So we determined that for anesthesia of the respiratory tract during awake blind tracheal intubation through the supraglottic airway devices in patients with obesity the method of choice is topical anesthesia IL-technique because of its safety, high efficiency and less invasiveness in comparison to TLI-technique