Fujita Type Research Articles

Critical curves for fast diffusive polytropic filtration equations coupled through boundary

This article deals with the critical curve of a fast diffusive polytropic filtration system coupled at the boundary condition. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve. The critical curve of Fujita type is conjectured with the aid of some new results.

Large time behaviour of solutions to a class of quasilinear parabolic equations with convection terms

In this paper, we investigate the large time behaviour of solutions to the exterior problems of a class of quasilinear parabolic equations with convection terms. We establish the critical Fujita exponents pc and blow-up theorems of the Fujita type for both homogeneous Neumann and Dirichlet problems. In particular, it is shown that the critical p = pc belongs to the blow-up case under any nontrivial initial data. An interesting phenomenon is exploited that the critical Fujita exponent pc could even be infinite for the considered model because of the nonlinear convection.

Critical curves for a fast diffusive polytropic filtration system coupled via nonlinear boundary flux

This paper deals with the critical curves of the fast diffusive polytropic filtration equations coupled via nonlinear boundary flux. By constructing self-similar super and sub solutions we obtain the critical global existence curve. The critical curve of Fujita type is conjectured with the aid of some new results.

Algebraic criteria for global existence or blow-up for a boundary coupled system of nonlinear diffusion equations

This article deals with the critical curves for a degenerate parabolic system coupled via nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve. The critical curve of Fujita type is conjectured with the aid of some new results.

Critical Fujita exponents for a class of quasilinear equations with homogeneous Neumann boundary data

In this paper, we establish the blow-up theorems of the Fujita type for a class of homogeneous Neumann problems of quasilinear equations with convection terms. The critical Fujita exponents are determined and it is shown that the exponents belong to the blow-up case under any nontrivial initial data. An interesting phenomenon is exploited such that the critical Fujita exponent even could be infinite for the model considered in the paper owing to the effect of convection.

Critical curves for degenerate parabolic equations coupled via non-linear boundary flux

This paper deals with the critical curve for a degenerate parabolic system coupled via non-linear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve. The critical curve of Fujita type is conjectured with the aid of some new results. The possible further extensions are also presented.

Blowup behaviors for degenerate parabolic equations coupled via nonlinear boundary flux

This paper deals with the blow-up properties of solutions to adegenerate parabolic system coupled via nonlinear boundary flux.Firstly, we construct the self-similar supersolution andsubsolution to obtain the critical global existence curve. Secondly, we establish the precise blow-up rate estimates for solutions which blow up in a finite time. Finally, we investigate the localization of blow-up points. The critical curve of Fujita type is conjectured with the aid of some new results.

Bipolar radiofrequency induced thermotherapy of the tongue base: its complications, acceptance and effectiveness under local anesthesia

We assessed adverse events and complications of bipolar radiofrequency induced thermotherapy of the tongue base (RFTB) in patients with socially unacceptable snoring (SUS) or obstructive sleep apnea syndrome (OSAS) and determine its acceptance and effectiveness when conducted under local anesthesia. This investigation consisted of (1) a prospective, open-enrollment study of 24 consecutive patients with snoring and OSAS at the tongue base level only (Fujita III), assessed by sleep endoscopy. Polysomnography, questionnaires, and visual analog scales (VAS) were used to assess outcome. (2) In addition, a retrospective review of 83 patients, who underwent RFTB (in 59 cases as part of a multilevel treatment), was performed to evaluate adverse events and complications. Twenty-two of the 24 patients completed postoperative questionnaires and VAS, and ten patients had postoperative polysomnography. Reduction in snoring (P = 0.0003), hypersomnolence (P = 0.002), and globus (P = 0.031) was significant. A positive trend in AHI (P = 0.001, n = 3) is shown in patients with moderate to severe OSAS. Concerning postoperative adverse events and complications, only two patients had a mild and transient tongue deviation directly after the procedure, which resolved within an hour postoperatively (adverse event rate 1.8%). No postoperative complications such as infections, abscesses, hematomas, or ulcerations of the tongue base occurred. This study demonstrates that bipolar RFTB in patients with obstruction at the tongue base only (Fujita type III) as visualized by sleep endoscopy is a safe and simple procedure under local anesthesia and can be effective in patients with SUS. No complications during this study were observed. Its effect on OSAS has been shown by other authors, although long-term effects are not stable. The RFTB can be considered as first choice treatment in case of snoring and mild OSAS in Fujita type III obstruction. In the case of moderate to severe sleep apnea, RFTB can be considered as an additional treatment.

Critical exponents of Fujita type for certain evolution equations and systems with spatio-temporal fractional derivatives

This paper is concerned with establishing necessary or sufficient conditions for the existence of solutions to evolution equations with fractional derivatives in space and time. The Fujita exponent is determined. Then, these results are extended to systems of reaction–diffusion equations. Our new results shed lights on important practical questions.

Same-stage palatopharyngeal and hypopharyngeal surgery for severe obstructive sleep apnea

ObjectivesTo investigate surgical outcomes with two types of combined palatopharyngeal and hypopharyngeal surgery for the treatment of severe obstructive sleep apnea (OSA).Material and methodsTwelve consecutive OSA patients with a respiratory disturbance index (RDI) >30/h and Fujita type II anatomy were enrolled. Patients were divided into two groups according to their fiberscopic manifestations. Six patients with obstruction at the uvulopalatal complex and tongue base (Group 1) were selected for extended uvulopalatal flap (EUPF) and midline laser glossectomy (MLG). EUPF and laser lingual tonsillectomy were performed in another six patients shown to have obstruction at the uvulopalatal complex and lingual tonsil (Group 2). Polysomnographic parameters included the RDI and minimal oxygen saturation (MSAT). Surgical success was defined as a postoperative RDI of <20/h and a >50% reduction in the preoperative RDI.ResultsSix months postoperatively, 5 patients (83.3%) had responded successfully in Group 1 and none in Group 2. In Group 1 the mean RDI decreased from 50.7±12.6 to 8±14.3 (95% CI 23.0−62.7; p<0.01) and MSAT increased from 76.3%±11.6% to 88.8%±3.2% (95% CI −25.9–0.87; p=0.06). There was no improvement in sleep parameters in Group 2 patients. No persistent nasal regurgitation, swallowing disturbance or change in taste was noted at 1-year follow-up in either group.ConclusionsEUPF combined with MLG improves OSA in Fujita type II patients. The hypertrophic lingual tonsil, although obscure the laryngeal structure, did not contribute significantly to OSA.

Fujita type theorems for quasilinear parabolic equations with initial data slowly decaying to zero

This work deals with the Cauchy problem for a parabolic equation with a double non-linearity of the following type: where . Existence and non-existence results for global solutions of this problem with initial conditions that slowly decay to zero are established.

Preliminary findings from a prospective, randomized trial of two tongue-base surgeries for sleep-disordered breathing

Objectives This study compares the efficacy of 2 tongue-base surgical procedures in the treatment of patients with moderate to severe sleep-disordered breathing. Study design and setting We conducted a prospective, randomized crossover surgical trial at a university hospital. Methods Seventeen patients with moderate to severe sleep-disordered breathing and Fujita type II upper airway collapse for whom conservative treatment failed were enrolled into an institutional review board–approved surgical protocol. They were randomly assigned to undergo palatopharyngoplasty combined with either tongue advancement (mandibular osteotomy) or tongue suspension. Parameters assessed included severity of sleep-disordered breathing (polysomnography), sleepiness (Epworth Sleepiness Scale), and anatomic changes (upper airway endoscopy), as well as demographic factors. Patients not achieving satisfactory improvement in their condition were offered nonsurgical management or additional surgical treatment that varied based on the postoperative assessment but included crossing over to the other tongue surgical procedure. Results Nine of the 17 patients were randomized to the tongue suspension group, and 8 to the tongue advancement group. In the 9 tongue suspension patients, Epworth Sleepiness Scale scores fell from 12.1 to 4.1 ( P = 0.007). Airway collapse for all 9 patients measured on Müller maneuver improved, by a mean of 64% ( P = 0.0006) at the palate and 83% ( P = 0.0003) at the base of the tongue. In the 8 tongue advancement patients, Epworth Sleepiness Scale scores fell from a mean of 13.3 to 5.4 ( P = 0.004). Airway collapse for 5 of 8 patients measured on Müller maneuver improved by a mean of 31% ( P = 0.1) at the palate and 75% ( P = 0.03) at the base of the tongue. Conclusion Prospective, randomized trials of tongue-base surgery for sleep-disordered breathing are possible. Preliminary findings from the current protocol reveal a slight advantage of tongue suspension over tongue advancement.

Nonexistence of positive solutions to a quasilinear elliptic system and blow-up estimates for a non-Newtonian filtration system

The prior estimate and decay property of positive solutions are derived for a system of quasilinear elliptic differential equations first. Then the result of nonexistence for a differential equation system of radially nonincreasing positive solutions is implied. By using this nonexistence result, blow-up estimates for a class of quasilinear reaction-diffusion systems (non-Newtonian filtration systems) are established to extend the result of semilinear reaction-diffusion (Fujita type) systems.

Nonexistence of positive solutions to a quasilinear elliptic system and blow-up estimates for a quasilinear reaction–diffusion system

The prior estimate and decay property of positive solutions are derived for a system of quasilinear elliptic differential equations first. Then, the nonexistence result for radially nonincreasing positive solutions of the system is implied. By using this nonexistence result, blow-up estimates for a class of quasilinear reaction–diffusion systems (non-Newtonian filtration systems) are established to extend the result for semilinear reaction–diffusion systems (Fujita type).

Blow up for a completely coupled Fujita type reaction-diffusion system

Blow up for a completely coupled Fujita type reaction-diffusion system

The Absence of Solutions of Elliptic Systems with Dynamic Boundary Conditions

u =0 ; (t;x)2 (0;1)D; ut+ku = h(t;x;u); (t;x 0 )2 (0;1)@D; (P 1) u =0 ; (t;x)2 (0;1)D; utt+ku = f (u); (t;x 0 )2 (0;1)@D; (P 2) ut u = u 1+ ; (t;x)2 (0;1)D; ut+ku = u 1+ ; (t;x 0 )2 (0;1)@D; (P 3) where D is a bounded domain and k and are positive constants. Conditions under which a solution of a second-order degenerate parabolic equation with appropriate restrictions imposed on some nonlinearities blows up in nite time were given in [2]; moreover, criteria for the blow-up of the solution were obtained there on the basis of appropriate ap rioriestimates. A Fujita type result was recently obtained in [3] for the problem u =0 ; (t;x)2 (0;1) ; ut uxN = u q ; (t;x 0 )2 (0;1)@; u (0;x 0 )= ’ (x 0 ) ;x 0 2 @;

Non-existence of positive solution to a quasilinear elliptic system and blow-up estimates for a reaction-diffusion system

The non-existence result for radially nonincreasing positive solutions of the system is implied. By using this non-existence result, blow-up estimates for a class of quasilinear reaction-diffusion systems (non-Newtonian filtration systems) are established to extend the result for semilinear reaction-diffusion systems (Fujita type).

Combined Temperature-Controlled Radiofrequency Tongue Reduction and UPPP in Apnea Surgery

The objective of this study was to investigate the effects on outcomes and morbidity of combining temperature-controlled radiofrequency (TCRF) tongue reduction with uvulopalatopharyngoplasty (UPPP) as an initial site-directed approach to the surgical treatment of obstructive sleep apnea syndrome (OSAS). This investigation was a prospective, nonrandomized, open-enrollment study of 20 consecutive eligible patients with OSAS. Seven patients had a single-level velopharyngeal obstruction (Fujita type I or IIa), and they were assigned to undergo UPPP only (group 1). Thirteen patients had a multilevel velopharyngeal and retroglossal obstruction (Fujita type IIb), and they were assigned to undergo TCRF tongue reduction in addition to UPPP (group 2). Patients who had only a retroglossal obstruction (Fujita type III) were not included in this study. Following their initial operation, nine patients in group 2 underwent two subsequent in-office TCRF tongue treatments under local anesthesia. Three patients in group 2 were lost to followup, and one patient underwent only one TCRF procedure at the initial operation; data on the latter patient are included in some of the outcomes measures reported here where indicated, but no data are reported on the three who were lost to followup. The primary post-treatment outcomes measures were the results of comparative polysomnography and clinician and patient evaluations regarding morbidity and symptom improvement. Analysis of these data showed that there was no difference between the two groups in terms of postoperative pain or dysphagia following the initial operative session. The degree of symptom improvement in the two groups was similar. Overall success rates--as measured by the apnea/hypopnea index (group 1: 57.1% success; group 2: 50.0%) and by the apnea index (group 1: 71.4% success; group 2: 70.0%)--were statistically comparable, given the small size of the sample (figures are based on 10 patients in group 2). The author concludes that combining TCRF tongue reduction with UPPP in patients with multilevel obstruction improves response rates to a degree that is comparable to that seen with UPPP alone in patients with single-level velopharyngeal obstruction. Moreover, the combination treatment does not increase the risk of additional pain, morbidity, and complications compared with UPPP alone.

Blow-up estimates for a non-newtonian filtration system

The prior estimate and decay property of positive solutions are derived for a system of quasi-linear elliptic differential equations first. Hence, the result of non-existence for differential equation system of radially nonincreasing positive solutions is implied. By using this non-existence result, blow-up estimates for a class quasi-linear reaction-diffusion systems (non-Newtonian filtration systems) are established, which extends the result of semi-linear reaction-diffusion (Fujita type) systems.

Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions

Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions