New York, Basel: Marcel Dekker, 2003. - 374 p.
The papers collected in this volume are conceed with hyperbolic problems, or problems the methods of which are related to hyperbolic techniques.
T. NISHITANI introduces a notion of nondegenerate characteristic for systems of linear partial differential equations of general order. He shows that nondegenerate characteristics are stable under hyperbolic perturbations, and he proves that if the coefficients of the system are real analytic and all characteristics are nondegenerate then the Cauchy problem for the system is well posed in the class of smooth functions.
K. KAJITANI studies a class of operators that generalize the linear hyperbolic operators, introducing the notion of time function, and proving the well-posedness of the Cauchy problem in the class of C functions.
The Cauchy problem is also the subject of the paper by A. BOVE and C. BERNARDI; they state some results for a class of linear hyperbolic operators with double characteristics, not effectively hyperbolic. In particular they prove well-posedness in the C class under a geometric condition and a Levi condition, and well-posedness in the Gevrey class under more general assumptions.
For a linear system whose principal part is hyperbolic and whose coefficients depend only on time, H. YAMAHARA establishes necessary and sufficient conditions for well-posedness in the Gevrey class, whatever the lower order terms are.
L. MENCHERINI and S. SPAGNOLO consider a first order hyperbolic system in two variables whose coefficients depend only on time; they define the notion of pseudosymmetry for matrix symbols of order zero, and determine the Gevrey class where the Cauchy problem is well-posed, according to the type of pseudosymmetry of the principal matrix symbol.
The 2-phase Goursat problem has been solved by means of Bessel functions; here J. CARVALHO E SILVA considers the 3-phase Goursat problem, using instead some hyper-geometric functions in four variables. He also discusses the general problem, pointing out that the main difficulties are due to the lack of results on special functions.
The Stricharz inequality for the classical linear wave equation has been generalized by M. REISSIG and Y.G. WANG to the case of time-dependent coefficients: the coefficient is the product of an increasing factor and an oscillatory factor. The interaction was studied by the authors; in the present paper they extend the result to a one-dimensional system describing thermoelasticity.
The noncharacteristic, nonlinear Cauchy problem is the subject of the paper by M. TSUJI. The classical solution has some singularities, so that the problem arises of studying the extension of the solution beyond the singularities. The author constructs a Lagrangian solution in the cotangent bundle, getting a multivalued classical solution; then he explores how to construct a reasonable univalued solution with singularities.
Y. CHOQUET considers the Einstein equations. By a suitable choice of the gauge, (for instance, an a priori hypothesis on coordinates choice) she obtains a hyperbolic system in the sense of Leray-Ohya, well-posed in the Gevrey class of index
2. She studies old and new cases where the system is strictly hyperbolic and well-posed in the C°° class.
Necessary and sufficient conditions for the Cauchy-Kowalevskaya theorem on systems of linear PDEs were given by Matsumoto and Yamahara; on the other hand, Nagumo constructed a local solution, unique, for a higher order scalar Kowalevskian operator, whose coefficients are analyitic in x and continuous in t.
The papers collected in this volume are conceed with hyperbolic problems, or problems the methods of which are related to hyperbolic techniques.
T. NISHITANI introduces a notion of nondegenerate characteristic for systems of linear partial differential equations of general order. He shows that nondegenerate characteristics are stable under hyperbolic perturbations, and he proves that if the coefficients of the system are real analytic and all characteristics are nondegenerate then the Cauchy problem for the system is well posed in the class of smooth functions.
K. KAJITANI studies a class of operators that generalize the linear hyperbolic operators, introducing the notion of time function, and proving the well-posedness of the Cauchy problem in the class of C functions.
The Cauchy problem is also the subject of the paper by A. BOVE and C. BERNARDI; they state some results for a class of linear hyperbolic operators with double characteristics, not effectively hyperbolic. In particular they prove well-posedness in the C class under a geometric condition and a Levi condition, and well-posedness in the Gevrey class under more general assumptions.
For a linear system whose principal part is hyperbolic and whose coefficients depend only on time, H. YAMAHARA establishes necessary and sufficient conditions for well-posedness in the Gevrey class, whatever the lower order terms are.
L. MENCHERINI and S. SPAGNOLO consider a first order hyperbolic system in two variables whose coefficients depend only on time; they define the notion of pseudosymmetry for matrix symbols of order zero, and determine the Gevrey class where the Cauchy problem is well-posed, according to the type of pseudosymmetry of the principal matrix symbol.
The 2-phase Goursat problem has been solved by means of Bessel functions; here J. CARVALHO E SILVA considers the 3-phase Goursat problem, using instead some hyper-geometric functions in four variables. He also discusses the general problem, pointing out that the main difficulties are due to the lack of results on special functions.
The Stricharz inequality for the classical linear wave equation has been generalized by M. REISSIG and Y.G. WANG to the case of time-dependent coefficients: the coefficient is the product of an increasing factor and an oscillatory factor. The interaction was studied by the authors; in the present paper they extend the result to a one-dimensional system describing thermoelasticity.
The noncharacteristic, nonlinear Cauchy problem is the subject of the paper by M. TSUJI. The classical solution has some singularities, so that the problem arises of studying the extension of the solution beyond the singularities. The author constructs a Lagrangian solution in the cotangent bundle, getting a multivalued classical solution; then he explores how to construct a reasonable univalued solution with singularities.
Y. CHOQUET considers the Einstein equations. By a suitable choice of the gauge, (for instance, an a priori hypothesis on coordinates choice) she obtains a hyperbolic system in the sense of Leray-Ohya, well-posed in the Gevrey class of index
2. She studies old and new cases where the system is strictly hyperbolic and well-posed in the C°° class.
Necessary and sufficient conditions for the Cauchy-Kowalevskaya theorem on systems of linear PDEs were given by Matsumoto and Yamahara; on the other hand, Nagumo constructed a local solution, unique, for a higher order scalar Kowalevskian operator, whose coefficients are analyitic in x and continuous in t.