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MONOTONICITY AND ASYMPTOTIC PROPERTIES OF SOLUTIONS FOR PARABOLIC EQUATIONS VIA A GIVEN INITIAL VALUE CONDITION ON GRAPHS

In this paper, we establish several results involving the minimum and maximum principles and the comparison principles for elliptic equations and parabolic equations on finite graphs. The results are then used to prove the monotonicity and asymptotic properties of solutions for parabolic equations whose initial values are given by the equation [Formula: see text] with Dirichlet boundary conditions. Finally, an illustration with numerical experiments is provided to demonstrate our main results.