Abstract wave equation
Here, is a bounded domain with smooth boundary , and are smooth real functions on such that for all , with some fixed ; is the unit outward normal vector to . Also, , , are given functions. The function is the unknown function.
One can state this problem in the abstract form
on the space , see [a1], where (respectively ) when the boundary conditions are Dirichlet (respectively, Neumann), by the relation if and only if for all . There are several ways to handle this abstract problem.
Let be a Banach space. A strongly continuous function of with values in is called a cosine function if it satisfies , , and . Its infinitesimal generator is defined by , with . The theory of cosine functions, which is very similar to the theory of semi-groups, was originated by S. Kurera [a2] and was developed by H.O. Fattorini [a3] and others.
A necessary and sufficient condition for a closed linear operator to be the generator of a cosine family is known. The operator determined by (a2) is easily shown to generate a cosine function which provides a fundamental solution for (a1).
Suppose one sets in (a1). Then one obtains the equivalent problem
with its domain is the negative generator of a semi-group. The theory of semi-groups of abstract evolution equations provides the existence of a unique solution of (a1) for and , .
This method is also available for a non-autonomous equation
In the case of Neumann boundary conditions, the difficulty arises that the domain of
may change with . One way to avoid this is to introduce the extension of defined by for all . Since is a bounded operator from into , the operator
acting in , has constant domain.
Another way is to reduce (a3) to
by setting , , under the assumption that is strongly differentiable with values in . Obviously, the linear operator of the coefficient has constant domain . Differentiability of the square root was studied in [a6], [a7].
In order to consider in (a1) the case when , one has to use the Lions–Magenes variational formulation. In this, one is concerned with the solution of the problem
The existence of a unique solution has been proved if and , ; see [a8], Chap. 5.
This method is also available for a non-autonomous equation (a3).
The variational method enables one to take from a wide class, an advantage that is very useful in, e.g., the study of optimal control problems. On the other hand, the semi-group method provides regular solutions, which is often important in applications to non-linear problems. Using these approaches, many papers have been devoted to non-linear wave equations.
|[a1]||J.-L. Lions, "Espaces d'interpolation et domaines de puissances fractionnaires d'opérateurs" J. Math. Soc. Japan , 14 (1962) pp. 233–241|
|[a2]||S. Kurepa, "A cosine functional equation in Hilbert spaces" Canad. J. Math. , 12 (1960) pp. 45–50|
|[a3]||H.O. Fattorini, "Ordinary differential equations in linear topological spaces II" J. Diff. Eq. , 6 (1969) pp. 50–70|
|[a4]||E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)|
|[a5]||K. Yoshida, "Functional analysis" , Springer (1957)|
|[a6]||A. McIntosh, "Square roots of elliptic operators" J. Funct. Anal. , 61 (1985) pp. 307–327|
|[a7]||A. Yagi, "Applications of the purely imaginary powers of operators in Hilbert spaces" J. Funct. Anal. , 73 (1987) pp. 216–231|
|[a8]||J.-L. Lions, E. Magenes, "Problèmes aux limites non homogènes et applications" , 1–2 , Dunod (1968)|
Abstract wave equation. A. Yagi (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Abstract_wave_equation&oldid=17290