The phrase "a sharp criterion of" is correct and usable in written English.
It can be used when discussing a clear and precise standard or measure for evaluating something.
Example: "The study provided a sharp criterion of success, allowing researchers to assess the effectiveness of the new treatment."
Alternatives: "a clear standard of" or "a precise measure of".
Moreover, we address a borderline problem in the regularity theory and give a sharp criterion for continuity of the gradient of a solution.
Using this result, we derive a sharp criterion for no blow-up.
A sharp criterion for the transition from a dimerized state to a long-range ordered state can be inferred from Figure 2(b), where the energy of the lowest singlet and the lowest triplet excitation is plotted.
In order to show the results of the spreading-vanishing dichotomy, the eigenvalue problems associated with model (1.3) are given in Section 3, and a sharp criterion for spreading and vanishing is established in Section 4.
We provide a sharp criterion in term of generalized principal eigenvalues for existence/nonexistence of positive solution of (Pβ) in suitable classes of functions.
Lastly, we apply these results to a special situation where two species are competing for the same resources and obtain a sharp criteria in term of certain variable parameters for all kinds of global dynamics.
Our primary aim of this paper is to obtain sharp criteria of the global threshold dynamics for the system.
Finally, applying the convexity of the energy inequality, we establish two types of invariant evolution flows, and we obtain the sharp energy criterion of blow-up and global existence of the solutions to the Cauchy problem (1.1 - 1.2 1.1 - 1.2ows.
However, one of the interesting features resulting from the breakdown of the scaling is that the sharp energy criterion of the blow-up solutions is not given by the ground state of the original nonlinear Klein-Gordon equation but that of a modified equation.
