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We study the initial-value problem for semilinear hyperbolic systems in one space dimension. When the power of nonlinearity is even, we have global existence of solution which is related with the repulsive Liouville equation. When  n  = 2 k −1, we may have a finite time blow-up solution which is related with the attractive Liouville equation.

Solvability of certain one-dimensional semilinear hyperbolic systems and hyperbolic Liouville equation