Linear Hyperbolic Equations

Abstract

The present survey is devoted to the linear hyperbolic equations and systems. The concept of a hyperbolic equation first appeared in the case of a second-order equation $$Pu = \sum\limits_{i,j = 0}^n {{a_{ij}}} {\partial _i}{\partial _j}u = 0$$ (0.1) with constant coefficients. It implied that the quadratic form \(a\left( \xi \right) = \sum {_{i,j = 1}^n} {a_{ij}}{\xi _i}{\xi _j}\) is hyperbolic, that is, its positive subspaces are of dimension 1 and the negative subspaces of dimension n − 1 (or vice-versa). But now such equations are referred to as strictly hyperbolic; a hyperbolic equation may have its quadratic form a(·) degenerate (parabolically degenerate, in old terminology). In this case, the equation (0.1) has solutions of the type of moving (plane) waves u = v(〈x, ξ〉), where \(\left\langle {x,\xi } \right\rangle = {x_0}{\kern 1pt} {\xi _0} + \cdot \cdot \cdot + {x_n}{\xi _n},{\kern 1pt} {\kern 1pt} a\left( \xi \right) = 0\) and v is an arbitrary function of a single variable. If n = 1, the general solution of (0.1) is a linear combination of two moving waves, and on the basis of this fact we can easily solve the Cauchy problem with data prescribed on any non-characteristic curve (that is, on a curve γ such that a(N) ≠ 0, where N is the normal to γ), and even suitable mixed problems. Moreover, the Cauchy problem is uniquely solvable and there exists a triangle of dependence; the same is true for non-homogeneous equations. When lower terms are present, the Cauchy problem and the mixed problem can be solved, in principle at least, by the method of successive approximation, just as it is done in the case of the Cauchy problem for ordinary differential equations. Similar construction is available for the case of variable coefficients also, and again the non-characteristic Cauchy problem is uniquely solvable and there is a characteristic triangle of dependence (now curvilinear). These statements do not hold for equations of other types. Therefore, apart from the algebraic definition of hyperbolicity, it is possible to give a meaningful and close analytical definition: a given equation (system) is hyperbolic if some non-characteristic Cauchy problem for it is uniquely solvable for any smooth right-hand sides and initial data and if there exists a cone of dependence. This fact was already recognised in the last century, and from that time the algebraic definition of hyperbolicity was extended to systems of first order, to equations and systems of higher orders and with a large number of independent variables in such a way that it remained meaningfully close to the analytical definition.