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This paper is concerned with the Cauchy problem of a multivalued ordinarydifferential equation governed by the hypergraph Laplacian, which describes thediffusion of ``heat'' or ``particles'' on the vertices of hypergraph. Weconsider the case where the heat on several vertices are manipulated internallyby the observer, namely, are fixed by some given functions. This situation canbe reduced to a nonlinear evolution equation associated with a time-dependentsubdifferential operator, whose solvability has been investigated in numerousprevious researches. In this paper, however, we give an alternative proof ofthe solvability in order to avoid some complicated calculations arising fromthe chain rule for the time-dependent subdifferential. As for results whichcannot be assured by the known abstract theory, we also discuss the continuousdependence of solution on the given data and the time-global behavior ofsolution.

Heat equation on the hypergraph containing vertices with given data