Meyer J C, Needham D J
School of Mathematics, University of Birmingham , Watson Building, Edgbaston, Birmingham B15 2TT, UK.
Proc Math Phys Eng Sci. 2015 Mar 8;471(2175):20140632. doi: 10.1098/rspa.2014.0632.
In this paper, we examine a semi-linear parabolic Cauchy problem with non-Lipschitz nonlinearity which arises as a generic form in a significant number of applications. Specifically, we obtain a well-posedness result and examine the qualitative structure of the solution in detail. The standard classical approach to establishing well-posedness is precluded owing to the lack of Lipschitz continuity for the nonlinearity. Here, existence and uniqueness of solutions is established via the recently developed generic approach to this class of problem (Meyer & Needham 2015 . London Mathematical Society Lecture Note Series, vol. 419) which examines the difference of the maximal and minimal solutions to the problem. From this uniqueness result, the approach of Meyer & Needham allows for development of a comparison result which is then used to exhibit global continuous dependence of solutions to the problem on a suitable initial dataset. The comparison and continuous dependence results obtained here are novel to this class of problem. This class of problem arises specifically in the study of a one-step autocatalytic reaction, which is schematically given by → at rate (where and are the concentrations of and , respectively, with 0<,<1) and well-posedness for this problem has been lacking up to the present.
在本文中，我们研究了一个具有非利普希茨非线性的半线性抛物型柯西问题，它在大量应用中以一般形式出现。具体而言，我们得到了一个适定性结果，并详细研究了解的定性结构。由于非线性缺乏利普希茨连续性，建立适定性的标准经典方法不再适用。在此，通过最近针对此类问题发展起来的一般方法（迈耶&尼达姆，2015年。《伦敦数学学会讲义系列》，第419卷）建立了解的存在性和唯一性，该方法研究了该问题的最大解和最小解的差异。基于这个唯一性结果，迈耶&尼达姆的方法允许发展出一个比较结果，然后用它来展示该问题的解对合适初始数据集的全局连续依赖性。这里得到的比较和连续依赖性结果对于此类问题来说是新颖的。此类问题特别出现在一步自催化反应的研究中，该反应示意性地表示为→，反应速率为（其中和分别是和的浓度，且0 < < 1），并且到目前为止该问题一直缺乏适定性。
