Asymptotic expansions of the solutions of the Cauchy problem for nonlinear parabolic equations

Let $u$ be a solution of the Cauchy problem for the nonlinear parabolic equation $$\partial_t u=\Delta u+F(x,t,u,\nabla u) \quad in \quad{\bf R}^N\times(0,\infty), \quad u(x,0)=\varphi(x)\quad in \quad{\bf R}^N,$$ and assume that the solution $u$ behaves like the Gauss kernel as $t\to\infty$. In this paper, under suitable assumptions of the reaction term $F$ and the initial function $\varphi$, we establish the method of obtaining higher order asymptotic expansions of the solution $u$ as $t\to\infty$. This paper is a generalization of our previous paper, and our arguments are applicable to the large class of nonlinear parabolic equations

Asymptotic behaviour of a semilinear elliptic system with a large exponent

Consider the problem \begin{eqnarray*} -\Delta u &=& v^{\frac 2{N-2}},\quad v>0\quad {in}\quad \Omega, -\Delta v &=& u^{p},\:\:\:\quad u>0\quad {in}\quad \Omega, u&=&v\:\:=\:\:0 \quad {on}\quad \partial \Omega, \end{eqnarray*} where $\Omega$ is a bounded convex domain in $\R^N,$ $N>2,$ with smooth boundary $\partial \Omega.$ We study the asymptotic behaviour of the least energy solutions of this system as $p\to \infty.$ We show that the solution remain bounded for $p$ large and have one or two peaks away form the boundary. When one peak occurs we characterize its location.Comment: 16 pages, submmited for publicatio

Longitudinal impact of demographic and clinical variables on Health-Related Quality of Life in Cystic Fibrosis

Objectives: The insights that people with cystic fibrosis have concerning their health are important given that aspects of health-related quality of life (HRQoL) are independent predictors of survival and a decrease in lung function is associated with a decrease in HRQoL over time. Cross-sectional data suggest that key variables, other than lung function, are also associated with HRQoL - although study results are equivocal. This work evaluates the relationship between these key demographic and clinical variables and HRQoL longitudinally. Design: Longitudinal observational study. Observations were obtained at seven time points: approximately every two years over a twelve year period. Setting: Large Adult Cystic Fibrosis Centre in the UK. Participants: 234 participants aged 14-48 years at recruitment. Outcome measure: Nine domains of HRQoL (Cystic Fibrosis Quality of Life Questionnaire) in relation to demographic (age, gender) and clinical measures (FEV1% predicted, BMI, cystic fibrosis related diabetes, B. cepacia complex, totally implantable vascular access device, nutritional and transplant status). Results: A total of 770 patient assessments were obtained for 234 patients. The results of random coefficients modelling indicated that demographic and clinical variables were identified as being significant for HRQoL over time. In addition to lung function, transplant status, age, having a totally implantable vascular access device, cystic fibrosis related diabetes, BMI and B. cepacia complex impacted on many HRQoL domains longitudinally. Gender was important for the domain of Body image. Conclusion: Demographic and changes in clinical variables were independently associated with a change in health-related quality of life over time. Compared with these longitudinal data, cross-sectional data are inadequate when evaluating the relationships between HRQoL domains and key demographic and clinical variables, as they fail to recognise the full impact of the CF disease trajectory and its treatments on quality of life

Development and initial validation of the bronchiectasis exacerbation and symptom tool (BEST)

BACKGROUND: Recurrent bronchiectasis exacerbations are related to deterioration of lung function, progression of the disease, impairment of quality of life, and to an increased mortality. Improved detection of exacerbations has been accomplished in chronic obstructive pulmonary disease through the use of patient completed diaries. These tools may enhance exacerbation reporting and identification. The aim of this study was to develop a novel symptom diary for bronchiectasis symptom burden and detection of exacerbations, named the BEST diary. METHODS: Prospective observational study of patients with bronchiectasis conducted at Ninewells Hospital, Dundee. We included patients with confirmed bronchiectasis by computed tomography, who were symptomatic and had at least 1 documented exacerbation of bronchiectasis in the previous 12\u2009months to participate. Symptoms were recorded daily in a diary incorporating cough, sputum volume, sputum colour, dyspnoea, fatigue and systemic disturbance scored from 0 to 26. RESULTS: Twenty-one patients were included in the study. We identified 29 reported (treated exacerbations) and 23 unreported (untreated) exacerbations over 6-month follow-up. The BEST diary score showed a good correlation with the established and validated questionnaires and measures of health status (COPD Assessment Test, r =\u20090.61, p =\u20090.0037, Leicester Cough Questionnaire, r =\u2009-\u20090.52,p =\u20090.0015, St Georges Respiratory Questionnaire, r =\u20090.61,p <\u20090.0001 and 6\u2009min walk test, r =\u2009-\u20090.46,p =\u20090.037). The mean BEST score at baseline was 7.1 points (SD 2.2). The peak symptom score during exacerbation was a mean of 16.4 (3.1), and the change from baseline to exacerbation was a mean of 9.1 points (SD 2.5). Mean duration of exacerbations based on time for a return to baseline symptoms was 15.3\u2009days (SD 5.7). A minimum clinically important difference of 4 points is proposed. CONCLUSIONS: The BEST symptom diary has shown concurrent validity with current health questionnaires and is responsive at onset and recovery from exacerbation. The BEST diary may be useful to detect and characterise exacerbations in bronchiectasis clinical trials

Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach

Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1, \quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For the semilinear heat equation $u_t= \Delta u+ u^p$, various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure

"The missing ingredient":The patient perspective of health related quality of life in bronchiectasis: a qualitative study

Abstract Background Bronchiectasis is a heterogeneous disease which affects quality of life. Measuring symptoms and quality of life has proved challenging and research is limited by extrapolation of questionnaires and treatments from other diseases. The objective of this study was to identify the major contributors to quality of life in bronchiectasis and to evaluate existing health related quality of life questionnaires in bronchiectasis. Methods Eight adults with bronchiectasis participated in one to one semi-structured interviews. These were recorded and transcribed verbatim. Thematic analysis was used to identify core themes relevant to disease burden and impact. Participant views on current health related quality of life questionnaires were also surveyed. Results Bronchiectasis symptoms are highly individual. Core themes identified were symptom burden, symptom variation, personal measurement, quality of life and control of symptoms. Themes contributing to quality of life were: social embarrassment, sleep disturbance, anxiety and modification of daily and future activities. Evaluation of 4 existing questionnaires established their individual strengths and weaknesses. A synthesis of the participants’ perspective identified desirable characteristics to guide future tool development. Conclusions: This qualitative study has identified core themes associated with symptoms and quality of life in bronchiectasis. Current treatments and quality of life tools do not fully address or capture the burden of disease in bronchiectasis from the patients’ perspective

Case Report: Dual nebulised antibiotics among adults with cystic fibrosis and chronic Pseudomonas infection [version 2; referees: 1 approved, 2 approved with reservations]

Pulmonary exacerbations in adults with cystic fibrosis (CF) and chronic Pseudomonas aeruginosa (Psae) infection are usually treated with dual intravenous antibiotics for 14 days, despite the lack of evidence for best practice. Intravenous antibiotics are commonly associated with various systemic adverse effects, including renal failure and ototoxicity. Inhaled antibiotics are less likely to cause systematic adverse effects, yet can achieve airway concentrations well above conventional minimum inhibitory concentrations. Typically one inhaled antibiotic is used at a time, but dual inhaled antibiotics (i.e. concomitant use of two different inhaled antibiotics) may have synergistic effect and achieve better results in the treatment of exacerbations. We presented anecdotal evidence for the use of dual inhaled antibiotics as an acute treatment for exacerbations, in the form of a case report. A female in her early thirties with CF and chronic Psae infection improved her FEV1 by 5% and 2% with two courses of dual inhaled antibiotics to treat exacerbations in 2016. In contrast, her FEV1 changed by 2%, –2%, 0% and 2%, respectively, with four courses of dual intravenous antibiotics in 2016. Baseline FEV1 was similar prior to all six courses of treatments. The greater FEV1 improvements with dual inhaled antibiotics compared to dual intravenous antibiotics suggest the potential role of using dual inhaled antibiotics to treat exacerbations among adults with CF and chronic Psae infection, especially since a greater choice of inhaled anti-pseudomonal antibiotics is now available. A previous study in 1985 has looked at the concomitant administration of inhaled tobramycin and carbenicillin, by reconstituting antibiotics designed for parenteral administration. To our knowledge, this is the first literature to describe the concomitant use of two different antibiotics specifically developed for delivery via the inhaled route

A treatment evaluator tool to monitor the real-world effectiveness of inhaled aztreonam lysine in cystic fibrosis

Background: Studies are required that evaluate real-world outcomes of inhaled aztreonam lysine in patients with cystic fibrosis (CF). Methods: Our treatment-evaluator tool assessed the effectiveness of inhaled aztreonam in routine practice in 117 CF patients across four time periods (6–12 (P2) and 0–6 months (P1) pre-initiation, and 0–6 (T1) and 6–12 months (T2) post-initiation). Outcomes were: changes in %-predicted forced expiratory volume in 1 s (FEV1), body-mass index (BMI), hospitalisation days and intravenous antibiotic usage. Results: Median FEV1% predicted for each 6-month period was 38.9%, 34.6%, 37.1% and 36.5%; median change was − 2.0% between P2 and P1, increasing to + 0.6% (p < 0.001) between P1 and T1. Annualised hospital bed-days was reduced (p = 0.05) post-initiation, as was intravenous antibiotics days (p = 0.001). BMI increased over 6 months post-initiation (p ≤ 0.001). Conclusions: In patients with CF in routine practice, inhaled aztreonam lysine is associated with improved lung function and weight, and reduced hospitalisation and intravenous antibiotic use

A phase 3, open-label, randomized trial to evaluate the safety and efficacy of levofloxacin inhalation solution (APT-1026) versus tobramycin inhalation solution in stable cystic fibrosis patients

Background: Inhaled antibiotics are standard of care for persons with cystic fibrosis (CF) and chronic Pseudomonas aeruginosa airway infection. APT-1026 (levofloxacin inhalation solution, LIS) is fluoroquinolone in development. We compared the safety and efficacy of LIS to tobramycin inhalation solution (TIS) in persons ≥12 years old with CF and chronic P. aeruginosa infection. Methods: This multinational, randomized (2:1), non-inferiority study compared LIS and TIS over three 28-day on/off cycles. Day 28 FEV1 % predicted change was the primary endpoint. Time to exacerbation and patient-reported quality of life superiority were among secondary endpoints. Results: Baseline demographics for 282 subjects were comparable. Non-inferiority was demonstrated (1.86% predicted mean FEV1 difference [95% CI −0.66 to 4.39%]). LIS was well-tolerated, with dysguesia (taste distortion) the most frequent adverse event. Conclusions: LIS is a safe and effective therapy for the management of CF patients with chronic P. aeruginosa