In this paper we consider the Cauchy problem for the semilinear damped wave equationHere n is the space dimension and µ is a modulus of continuity. Our goal is to obtain sharp conditions on µ to obtain a threshold between global (in time) existence of small data solutions (stability of the zero solution) and blow-up behavior even of small data solutions.
In this work, we prove the existence of global (in time) small data solutions for wave equations with two dissipative terms and with power nonlinearity $$|u|^p$$ | u | p or nonlinearity of derivative type $$|u_t|^p$$ | u t | p , in any space dimension $$n\geqslant 1$$ n ⩾ 1 , for supercritical powers $$p>{\bar{p}}$$ p > p ¯ . The presence of two dissipative terms strongly influences the nature of the problem, allowing us to derive $$L^r-L^q$$ L r - L q long time decay estimates for the solution in the full range $$1\leqslant r\leqslant q\leqslant \infty $$ 1 ⩽ r ⩽ q ⩽ ∞ . The optimality of the critical exponents is guaranteed by a nonexistence result for subcritical powers $$p<{\bar{p}}$$ p < p ¯ .
Objective: There have been plenty of articles published in recent decades on patient care in the form of case management (CM), but conclusions regarding health outcomes and costs have often been discordant. The objective of this study was to examine previous systematic reviews and meta-analyses with a view to assessing and pooling the overwhelming amount of data available on CM-based health outcomes and resource usage. Methods: We conducted a review of reviews of secondary studies (meta-analyses and systematic reviews) addressing the effectiveness of CM compared with usual care (or other organizational models) in adult (18+) with long-term conditions. PubMed, the Cochrane Database of Systematic Reviews, and the Database of Abstracts of Reviews of Effects (DARE) were searched from 2000 to the end of December 2017. The outcomes of interest are related to process of care, health measures, and resource usage. Results: Twenty-two articles were ultimately considered: 4 meta-analyses and 18 systematic reviews. There is strong evidence of CM increasing adherence to treatment guidelines and improving patient satisfaction, but none of the secondary studies considered demonstrated any effect on patient survival. Based on the available literature, there is contrasting evidence regarding all the other health outcomes, such as quality of life (QOL), clinical outcomes, and functional status. Good-quality secondary studies consistently found nothing to indicate that CM prompts any reduction in the use of hospital resources. Conclusion: The source of variability in the literature on the consistency of the evidence for most outcomes is unclear. It may stem from the heterogeneity of CM programs in terms of what their intervention entails, the populations targeted, and the tools used to measure the results. That said, there was consistently strong evidence of CM being associated with a greater adherence to treatment guidelines and higher patient satisfaction, but not with a longer survival or better use of hospital resources.
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