Abstract hyperbolic differential equation

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

\begin{equation*} \left. \begin{cases} { \frac { \partial u } { \partial t } + \sum _ { j = 1 } ^ { m } a _ { j } ( x ) \frac { \partial u } { \partial x _ { j } } + c ( x ) u = f ( x , t ) }, \\ { ( x , t ) \in \Omega \times [ 0 , T ] }, \\ { u ( x , 0 ) = u _ { 0 } ( x ) , \quad x \in \Omega, } \end{cases} \right. \end{equation*}

with the boundary conditions

\begin{equation*} u ( x , t ) \in P ( x ) , \quad ( x , t ) \in \partial \Omega \times [ 0 , T ]. \end{equation*}

Here, $\Omega \subset \mathbf{R} ^ { m }$ is a bounded domain with smooth boundary $\partial \Omega$ (when $\Omega = \mathbf{R} ^ { m }$, no boundary conditions are necessary), and $a_{j} ( x )$, $j = 1 , \ldots , m$, and $c ( x )$ are smooth functions on $\Omega$ with as values real matrices in $L ( \mathbf{R} ^ { p } )$, the $a_{j} ( x )$ being symmetric. It is assumed that the boundary matrix $b ( x ) = \sum _ { j = 1 } ^ { m } n _ { j } ( x ) a _ { j } ( x )$, $x \in \partial \Omega$, is non-singular, where $n = ( n _{1} , \ldots , n _ { m } )$ is the unit outward normal vector to $\partial \Omega$. Also, $P ( x )$ denotes the maximal non-negative subspace of $\mathbf{R} ^ { p }$ with respect to $b ( x )$, i.e. $( b ( x ) u , u ) \geq 0$, $u \in P ( x )$, and $P ( x )$ is not a proper subset of any other subspace of $\mathbf{R} ^ { p }$ with this property. The function $u = ( u _ { 1 } , \ldots , u _ { p } )$ 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 $A$ be the smallest closed extension in $X = [ L ^ { 2 } ( \Omega ) ] ^ { p }$ of the operator $\mathcal{A}$ defined by

\begin{equation*} {\cal A} u = \sum _ { j = 1 } ^ { m } a _ { j } ( x ) \frac { \partial u } { \partial x _ { j } } + c ( x ) u \end{equation*}

with domain

\begin{equation*} D ( \mathcal{A} ) = \left\{ u \in [ H ^ { 1 } ( \Omega ] ^ { p } : u ( x ) \in P ( x ) \text { a.e. on } \partial \Omega \right\}. \end{equation*}

Then $A$ is the negative generator of a $C _ { 0 }$ semi-group on $X$ (cf. [a1], [a2]; see also Semi-group of operators). Hence, the Hille–Yoshida theorem proves the existence of a unique solution $u \in C ( [ 0 , T ] ; D ( \mathcal{A} ) ) \cap C ^ { 1 } ( [ 0 , T ] ; X )$ to the Cauchy problem

\begin{equation*} \frac { d u } { d t } + A u = f ( t ) , t \in [ 0 , T ], \end{equation*}

which is given in the form

\begin{equation*} u ( t ) = e ^ { - t A } u _ { 0 } + \int _ { 0 } ^ { t } e ^ { - ( t - s ) A } f ( s ) d s, \end{equation*}

\begin{equation*} u ( 0 ) = u _ { 0 } \in D ( \mathcal{A} ) , f \in C ( [ 0 , T ] ; D ( A ) ). \end{equation*}

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

\begin{equation*} \left\{ \begin{array} { l l } { \frac { d u } { d t } + A ( t ) u = f ( t ) , } & { t \in [ 0 , T ], } \\ { u ( 0 ) = u _ { 0, } } \end{array} \right. \end{equation*}

was originated by T. Kato, and was developed by him and many others (cf. [a3], Chap. 7). Here, $A ( t )$ denotes a given function with values in the space of closed linear operators acting in a Banach space $X$; $f ( t )$ and $u_{0}$ are the initial data, and $u = u ( t )$ is the unknown function with values in $X$.

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

I) $A ( t )$ is a stable family on $X$, in the sense that

\begin{equation*} \left\| ( \lambda + A ( t _ { k } ) ) ^ { - 1 } \ldots ( \lambda + A ( t _ { 1 } ) ) ^ { - 1 } \right\| _ { L ( X ) } \leq \frac { M } { ( \lambda - \beta ) ^ { k } } \end{equation*}

for any $0 \leq t _ { 1 } \leq t _ { k } \leq T$ and any $\lambda > \beta$ with some fixed $M$ and $\beta$.

II) There is a second Banach space, $Y$, such that $Y \subset D ( A ( t ) )$, and $A ( t )$ is a continuous function of $t$ with values in $L ( X , Y )$.

III) There is an isomorphism $S$ from $Y$ onto $X$ such that $S A ( t ) S ^ { - 1 } = A ( t ) + B ( t )$, with $B ( t )$ a strongly continuous function of $t$ with values in $L ( X )$. Then there is a unique solution $u \in C ( [ 0 , T ] ; Y ) \cap C ^ { 1 } ( [ 0 , T ] ; X )$, and it is given by

\begin{equation*} u ( t ) = U ( t , 0 ) u _ { 0 } + \int _ { 0 } ^ { t } U ( t , s ) f ( s ) d s, \end{equation*}

$u _ { 0 } \in Y$, $f \in C ( [ 0 , T ] ; Y )$, where $U ( t , s )$ is a unique evolution operator. It is easily seen that III) implies, in particular, the stability of $A ( t )$ on $Y$. When $X$ and $Y$ are Hilbert spaces, III) can be replaced by the simpler condition [a5]:

III') There exists a positive-definite self-adjoint operator $S$ on $X$ with $D ( S ) = Y$ such that $| \operatorname { Re } ( A ( t ) u , S ^ { 2 } u ) _ { X } | \leq \gamma \| S u \| _ { X } ^ { 2 }$ for any $u \in D ( S ^ { 2 } )$, with some constants $\gamma$.

The Cauchy problem for the quasi-linear differential equation

\begin{equation*} \left\{ \begin{array} { l } { \frac { d u } { d t } + A ( t , u ) u = f ( t , u ) , \quad t \in [ 0 , T ], } \\ { u ( 0 ) = u _ { 0 }, } \end{array} \right. \end{equation*}

has been studied by several mathematicians on the basis of results for linear problems, [a7]. Here, $A ( t , u )$ depends also on the unknown function $u$. In [a6], [a7], $A ( t , u )$, defined for $( t , u ) \in [ 0 , T ] \times W$, where $W \subset Y$ is a bounded open set, is assumed to satisfy conditions similar to I)–III) and a Lipschitz condition $\| A ( t , u ) - A ( t , u ^ { \prime } ) \| _ { L ( Y , X ) } \leq \mu \| u - u ^ { \prime } \| _ { X }$ with respect to $u$. 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