For semilinear parabolic equations on metric graphs, ordered initial data yield ordered solutions, and solutions between ordered stationary sub/supersolutions converge monotonically to stationary limits.
Mugnolo, Semigroup Methods for Evolution Equations on Networks (Springer, 2014)
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Monotonicity results for semilinear parabolic equations on metric graphs
For semilinear parabolic equations on metric graphs, ordered initial data yield ordered solutions, and solutions between ordered stationary sub/supersolutions converge monotonically to stationary limits.