ON A FAMILY OF CONVEX SOLUTIONS TO A HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION

摘要

A family of convex solutions of Φ_(xx) - f(x)Φ_(yy) = 0, for x > 0 and y ? R, where f is positive and continuously differentiable in (0, oo), is discussed. It consists of all convex solutions of that equation which are of the form Φ(x, y) = p(x)q(y). The separation of variables is an easy task to perform. In particular, it results in an explicit form of q(y). Imposing convexity conditions requires however more insight. It is observed that a nonlinear part of those conditions, in case of f' ≤ 0, is related to an asymptotic behavior of p(x) and p'(x) as x → ∞. Then, under an additional assumption that lim_(x→∞) f(x) > 0, a satisfactory description of the set of all the functions p(x), which determines convex Φ(x, y) via the formula Φ(x, y) = p(x)q(y), is obtained. So constructed functions Φ(x,y) are convex entropies for the corresponding p-system. Finally two nontrivial examples, involving a modified Bessel and hypergeometric equation are provided.