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

A particle system with explosions: law of large numbers for the density of particles and the blow-up time

Consider a system of independent random walks in the discrete torus with creation-annihilation of particles and possible explosion of the total number of particles in finite time. Rescaling space and rates for diffusion/creation/annihilation of particles, we obtain a stong law of large numbers for the density of particles in the supremum norm. The limiting object is a classical solution to the semilinear heat equation u_t =u_{xx} + f(u). If f(u)=u^p, 1<p \le 3, we also obtain a law of large numbers for the explosion time

A quality-of-life measure for adults with primary ciliary dyskinesia: QOL-PCD

Primary ciliary dyskinesia (PCD) is characterised by chronic suppurative lung disease, rhino-sinusitis, hearing impairment and sub-fertility. We have developed the first multidimensional measure to assess health-related quality of life (HRQoL) in adults with PCD (QOL-PCD). Following a literature review and expert panel meeting, open-ended interviews with patients investigated the impact of PCD on HRQoL in the UK and North America (n=21). Transcripts were content analysed to derive saturation matrices. Items were rated for relevance by patients (n=49). Saturation matrices, relevance scores, literature review, evaluation of existing measures, and expert opinion contributed to development of a preliminary questionnaire. The questionnaire was refined following cognitive interviews (n=18). Open-ended interviews identified a spectrum of issues unique to adults with PCD. Saturation matrices confirmed comprehensive coverage of content. QOL-PCD includes 48 items covering the following seven domains: Physical Functioning, Emotional Functioning, Treatment Burden, Respiratory and Sinus Symptoms, Ears and Hearing, Social Functioning, and Vitality and Health Perceptions. Cognitive testing confirmed that content was comprehensive and the items were well-understood by respondents. Content validity and cognitive testing supported the items and structure. QOL-PCD has been translated into other languages and is awaiting psychometric testing

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

An Early Health Economic Analysis of the Potential Cost Effectiveness of an Adherence Intervention to Improve Outcomes for Patients with Cystic Fibrosis

Background Cystic fibrosis (CF) negatively impacts upon health-related quality of life and survival. Adherence to nebulised treatments is low; improving adherence is hypothesised to reduce rates of exacerbation requiring intravenous antibiotics and lung function decline. Objective A state transition model was developed to assess the cost effectiveness of an intervention aimed at increasing patient adherence to nebulised and inhaled antibiotics compared with current CF care, in advance of the forthcoming CFHealthHub randomised controlled trial (RCT). Methods The model estimated the costs and health outcomes for each option from the perspective of the UK National Health Service and Personal Social Services over a lifetime horizon. Health gains were valued in terms of quality-adjusted life-years (QALYs) gained. Forced expiratory volume in 1 second (FEV1) trajectories were predicted over three lung function strata: (1) FEV1 ≥70%, (2) FEV1 40–69% and (3) FEV1 <40%. Additional states were included to represent ‘post-lung transplantation’ and ‘dead’. The model was populated using CF Registry data, literature and expert opinion. Costs were presented at 2016 values. Uncertainty was assessed using deterministic and probabilistic sensitivity analyses. Results If effective, the adherence intervention is expected to produce an additional 0.19 QALYs and cost savings of £64,078 per patient. Across all analyses, the intervention dominated current care. Over a 5-year period, the intervention is expected to generate cost savings of £49.5 million for the estimated 2979 patients with CF with Pseudomonas aeruginosa currently aged ≥16 years in the UK. If applied to a broader population of adult patients with CF receiving any nebulised therapy, the expected savings could be considerably greater. Conclusions If effective, the adherence intervention is expected to produce additional health gains at a lower cost than current CF care. However, the economic analysis should be revisited upon completion of the full RCT. More generally, the analysis suggests that considerable gains could be accrued through the implementation of adherence interventions that shift care from expensive hospital-based rescue to community-based prevention

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

Assessing disease disclosure in adults with cystic fibrosis: the Adult Data for Understanding Lifestyle and Transitions (ADULT) survey Disclosure of disease in adults with cystic fibrosis

<p>Abstract</p> <p>Background</p> <p>As more patients with cystic fibrosis (CF) reach adulthood and participate in age-appropriate activities (e.g. employment, dating), disclosure of medical status becomes more important. This study assessed rates of disclosure and its perceived impact on relationships using the Adult Data for Understanding Lifestyle and Transitions (ADULT) online survey.</p> <p>Methods</p> <p>Adults with CF participated in the survey via the United States national network of CF Centers. Descriptive and inferential statistics were utilized.</p> <p>Results</p> <p>Participants (n = 865) were more likely to disclose to relatives (94%) and close friends (81%) than to dating partners (73%), bosses/supervisors/teachers (51%) or co-workers (39%). Respondents generally reported a neutral/positive effect on relationships following disclosure. Negative effects of disclosure were infrequent, but more likely with dating partners or bosses/supervisors/teachers. Results also indicated that disclosure may be influenced by severity of lung disease and gender, with those having normal/mild lung disease less likely to disclose their diagnosis to both co-workers (p < 0.01) and bosses/supervisors/teachers (p < 0.01), and women being more likely to disclose to close friends (p < 0.0001) and dating partners (p < 0.05) than men.</p> <p>Conclusions</p> <p>Most adults with CF disclosed their disease to relatives and close friends. Individuals with severe CF lung disease were more likely to disclose their diagnosis to coworkers and supervisors/teachers. It may be helpful to provide support for disclosure of disease in situations such as employment and dating.</p