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Fritz John pp 335-352

# Algebraic Conditions for Hyperbolicity of Systems of Partial Differential Equations

• Fritz John
Chapter
Part of the Contemporary Mathematicians book series (CM)

## Abstract

Partial differential equations or systems of such equations are classified according to “type”, such as “elliptic”, “parabolic”, “hyperbolic”, etc. Hyperbolic systems (with respect to the initial plane t = 0) are those for which the initial value problem is well posed in the sense of Hadamard (see [8]). General conditions that are both necessary and sufficient for hyperbolicity are complicated (see [10], [11], [2], [3], [14], [15], [16], [17]). The situation, however, is rather simple for an m-th order linear homogeneous system of equations with constant coefficients:
$$P\left( {\frac{\partial }{{\partial t}},\frac{\partial }{{\partial {x_1}}},...,\frac{\partial }{{\partial {x_n}}}} \right)u = 0$$
(1)
for a vector u = u(t, x 1 • • •, x n ) = u(t, x) with N components. Here P is an N x N square matrix whose elements p ik are m-th degree forms in their n + 1 arguments. We associate with (1) the matrix (the “symbol” of (1))
$$P(\lambda ,\xi ) = P(\lambda ,{\xi _1} \cdot \cdot \cdot ,{\xi _n}) = (Pik(\lambda ,{\xi _1} \cdot \cdot \cdot ,{\xi _n}))$$
(2)
and the characteristic form
$$Q(\lambda ,\xi ) = Q(\lambda ,{\xi _1} \cdot \cdot \cdot ,{\xi _n}) = \det P(\lambda ,{\xi _1} \cdot \cdot \cdot ,{\xi _n})$$
(3)
of degree mN in its arguments.

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## Copyright information

© Springer Science+Business Media New York 1985

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• Fritz John

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