Abstract
This paper deals with a parabolic-elliptic-parabolic system arising from ion transport networks originally proposed by Albi et al. (Anal Appl 14(1):185–206, 2016). Our investigation demonstrates that the local existence of the weak solution can always be achieved. Moreover, by establishing the uniform a-priori decay estimates of solution, the local solution can be extended to a global one provided that the initial data is suitably small. In addition, the time-decay rate of the weak solution is obtained. To our best knowledge, this seems to be the first analytical work on this system indeed.
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