Hyperbolic partial differential equation

at a given point $ M( x _ {1} \dots x _ {n} ) $

A partial differential equation for which the Cauchy problem is uniquely solvable for initial data specified in a neighbourhood of $ M $ on any non-characteristic surface (cf. Characteristic surface). In particular, a partial differential equation for which the normal cone has no imaginary zones is a hyperbolic partial differential equation. The differential equation

$$ \tag{* } L ( u) = H ( D _ {1} \dots D _ {n} ) u + F ( D _ {1} \dots D _ {n} ) u + G ( x) = 0, $$

where $ D _ {i} = \partial / \partial x _ {i} $( $ i = 1 \dots n $), $ H( D _ {1} \dots D _ {n} ) $ is a homogeneous polynomial of degree $ m $, while the polynomial $ F $ is of lower degree than $ m $, is a hyperbolic partial differential equation if its characteristic equation

$$ Q ( \xi _ {1} \dots \xi _ {n} ) = H ( \xi _ {1} \dots \xi _ {n} ) = 0 $$

has $ m $ different real solutions with respect to one of the variables $ \xi _ {1} \dots \xi _ {n} $, the remaining ones being fixed. Any equation (*) of the first order $ ( m = 1 ) $ with real coefficients is a hyperbolic partial differential equation. A second-order equation

$$ L ( u) = u _ {tt} - \sum _ {i, j = 1 } ^ { n } a _ {ij} D _ {i} D _ {j} u + Fu + G = 0 $$

is hyperbolic if the quadratic form

$$ \sum _ {i, j = 1 } ^ { n } a _ {ij} \xi _ {i} \xi _ {j} $$

is positive definite.

Comments

The special variable among the $ \xi _ {1} \dots \xi _ {n} $ such that $ H ( \xi _ {1} \dots \xi _ {n} ) $ has $ m $ different real solutions for each set of fixed values of the other $ n - 1 $ is often taken to be $ t $( time). One speaks then of a (strictly) hyperbolic equation or an equation of (strictly) hyperbolic type with respect to the $ t $- direction. More generally one considers hyperbolicity with respect to a vector $ N $[a1].

A polynomial $ P $ of degree $ m $ with principal part $ P _ {m} $ is called hyperbolic with respect to the real vector $ N $ if $ P _ {m} ( N) \neq 0 $ and there exists a number $ \tau _ {0} > 0 $ such that

$$ P ( \xi + i \tau N) \neq 0 \ \ \textrm{ if } \ \xi \in \mathbf R ^ {n} ,\ \tau < \tau _ {0} . $$

If $ P _ {m} $ is such that $ P _ {m} ( N) \neq 0 $ and $ P _ {m} ( \xi + \tau N) $ has only simple real roots for every real $ \xi \neq 0 $, then $ P $ is said to be strictly hyperbolic or hyperbolic in the sense of Petrovskii.

The Cauchy problem for a constant-coefficient differential operator $ P $ with data on a non-characteristic plane is well posed for arbitrary lower-order terms if and only if $ P $ is strictly hyperbolic. For a discussion of similar matters for polynomials $ P $ with variable coefficients cf. [a2].

For a system of higher-order linear partial differential equations

$$ \sum _ {j = 1 } ^ { l } \sum _ {| \alpha | \leq N _ {j} } a _ \alpha ^ {ij} ( x) \frac{\partial ^ \alpha }{\partial x ^ \alpha } u _ {j} = 0,\ \ i = 1 \dots l , $$

where $ \alpha = ( \alpha _ {0} \dots \alpha _ {n} ) $, is a hyperbolic system of partial differential equations in the sense of Petrovskii if the determinant

$$ \mathop{\rm det} \ \left ( \sum _ {| \alpha | \leq N _ {j} } a _ \alpha ^ {ij} \frac{\partial ^ \alpha }{\partial x ^ \alpha } \right ) $$

calculated in the ring of differential operators is a hyperbolic polynomial in the sense of Petrovskii (as a polynomial of degree $ N = \sum N _ {j} $). The Cauchy problem for a system that is hyperbolic in this sense is well posed [a3], [a4].

Instead of strictly hyperbolic one also finds the term strongly hyperbolic and instead of hyperbolic also weakly hyperbolic (which is therefore the case in which the lower-order terms of $ P $ do matter).

References

[a1]	L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983) pp. Chapt. XII MR0717035 MR0705278 Zbl 0521.35002 Zbl 0521.35001
[a2]	L.V. Hörmander, "The analysis of linear partial differential operators" , III , Springer (1985) pp. Chapt. XXIII MR1540773 MR0781537 MR0781536 Zbl 0612.35001 Zbl 0601.35001
[a3]	I.G. Petrovskii, "Ueber das Cauchysche Problem für Systeme von partiellen Differentialgleichungen" Mat. Sb. (N.S.) , 2(44) (1937) pp. 815–870
[a4]	S. Mizohata, "The theory of partial differential equations" , Cambridge Univ. Press (1973) (Translated from Japanese) MR0599580 Zbl 0263.35001
[a5]	J. Chaillou, "Hyperbolic differential polynomials" , Reidel (1979) MR0557901 Zbl 0424.35055
[a6]	J. Chazarain, "Opérateurs hyperboliques à characteristique de multiplicité constante" Ann. Inst. Fourier , 24 (1974) pp. 173–202
[a7]	L. Gårding, "Linear hyperbolic equations with constant coefficients" Acta Math. , 85 (1951) pp. 1–62 MR41336
[a8]	O.A. Oleinik, "On the Cauchy problem for weakly hyperbolic equations" Comm. Pure Appl. Math. , 23 (1970) pp. 569–586 MR0264227