We prove the existence of Kolmogorov-Petrovsky-Piskunov (KPP) type traveling fronts in space-time periodic and mean zero incompressible advection, and establish a variational (minimization) formula for the minimal speeds. We approach the existence by considering limit of a sequence of front solutions to a regularized traveling front equation where the nonlinearity is combustion type with ignition cut-off. The limiting front equation is degenerate parabolic and does not permit strong solutions, however, the necessary compactness follows from monotonicity of fronts and degenerate regularity. We apply a dynamic argument to justify that the constructed KPP traveling fronts propagate at minimal speeds, and derive the speed variational formula. The dynamic method avoids the degeneracy in traveling front equations, and utilizes the parabolic maximum principle of the governing reaction-diffusion-advection equation. The dynamic method does not rely on existence of traveling fronts.
Research QuestionWhat is the impact of the duration of cough monitoring on its accuracy in detecting changes in the cough frequency?Materials and MethodsThis is a statistical analysis of a prospective cohort study. Participants were recruited in the city of Pamplona (Northern Spain) and their cough frequency was passively monitored using smartphone-based acoustic artificial intelligence software. Differences in cough frequency were compared using a one-tailed Mann-Whitney U test and a randomisation routine to simulate 24-h monitoring.Results616 participants were monitored for an aggregated duration of over 9 person-years and registered 62 325 coughs. This empiric analysis found that an individual's cough patterns are stochastic, following a binomial distribution. When compared to continuous monitoring, limiting observation to 24 h can lead to inaccurate estimates of change in cough frequency, particularly in persons with low or small changes in rate.InterpretationDetecting changes in an individual's rate of coughing is complicated by significant stochastic variability within and between days. Assessing change based solely on intermittent sampling, including 24-h, can be misleading. This is particularly problematic in detecting small changes in individuals who have a low rate and/or high variance in cough pattern.
This paper constitutes a short survey of existence methods for variational inequalities of parabolic type. After discussing several illustrative examples in detail, we discuss some of the common methods for proving existence of solutions to such problems: the translation method, Rothe's method, and the penalty method. As these methods rely on existence results for elliptic variational inequalities, we also provide a summary of basic results and techniques for static problems.
Generalizing the well-known mean-value property of harmonic functions, we prove that a p-harmonic function of two variables satisfies, in a viscosity sense, two asymptotic formulas involving its local statistics. Moreover, we show that these asymptotic formulas characterize pharmonic functions when 1 < p < ∞. An example demonstrates that, in general, these formulas do not hold in a non-asymptotic sense.
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