Abstract hyperbolic differential equation

Consider the Cauchy problem for the symmetric hyperbolic system (cf. also Hyperbolic partial differential equation)

with the boundary conditions

Here, is a bounded domain with smooth boundary (when , no boundary conditions are necessary), and , , and are smooth functions on with as values real matrices in , the being symmetric. It is assumed that the boundary matrix , , is non-singular, where is the unit outward normal vector to . Also, denotes the maximal non-negative subspace of with respect to , i.e. , , and is not a proper subset of any other subspace of with this property. The function is the unknown function.

One can handle this problem as the Cauchy problem for an evolution equation in a Banach space (cf. also Linear differential equation in a Banach space). Indeed, let be the smallest closed extension in of the operator defined by

with domain

Then is the negative generator of a semi-group on (cf. [a1], [a2]; see also Semi-group of operators). Hence, the Hille–Yoshida theorem proves the existence of a unique solution to the Cauchy problem

which is given in the form

Next to this idea of an abstract formulation for hyperbolic systems, the study of the linear evolution equation

was originated by T. Kato, and was developed by him and many others (cf. [a3], Chap. 7). Here, denotes a given function with values in the space of closed linear operators acting in a Banach space ; and are the initial data, and is the unknown function with values in .

Among others, Kato's theorem in [a4] is fundamental: Suppose that

I) is a stable family on , in the sense that

for any and any with some fixed and .

II) There is a second Banach space, , such that , and is a continuous function of with values in .

III) There is an isomorphism from onto such that , with a strongly continuous function of with values in . Then there is a unique solution , and it is given by

, , where is a unique evolution operator. It is easily seen that III) implies, in particular, the stability of on . When and are Hilbert spaces, III) can be replaced by the simpler condition [a5]:

III') There exists a positive-definite self-adjoint operator on with such that for any , with some constants .

The Cauchy problem for the quasi-linear differential equation

has been studied by several mathematicians on the basis of results for linear problems, [a7]. Here, depends also on the unknown function . In [a6], [a7], , defined for , where is a bounded open set, is assumed to satisfy conditions similar to I)–III) and a Lipschitz condition with respect to . Under such conditions, the existence and uniqueness of a local solution, continuous dependence on the initial data and applications to quasi-linear hyperbolic systems have been given.

References

[a1]	E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)
[a2]	K. Yoshida, "Functional analysis" , Springer (1957)
[a3]	H. Tanabe, "Functional analytic methods for partial differential equations" , M. Dekker (1997)
[a4]	T. Kato, "Linear evolution equations of "hyperbolic" type" J. Fac. Sci. Univ. Tokyo , 17 (1970) pp. 241–248
[a5]	N. Okazawa, "Remarks on linear evolution equations of hyperbolic type in Hilbert space" Adv. Math. Sci. Appl. , 8 (1998) pp. 399–423
[a6]	K. Kobayashi, N. Sanekata, "A method of iterations for quasi-linear evolution equations in nonreflexive Banach spaces" Hiroshima Math. J. , 19 (1989) pp. 521–540
[a7]	T. Kato, "Abstract evolution equations, linear and quasilinear, revisited" J. Komatsu (ed.) , Funct. Anal. and Rel. Topics. Proc. Conf. in Memory of K. Yoshida (RIMS, 1991) , Lecture Notes Math. , 1540 , Springer (1991) pp. 103–125