On the global solution problem for semilinear generalized Tricomi equations, I

Abstract

In this paper, we are concerned with the global Cauchy problem for the semilinear generalized Tricomi equation partial derivative(2)(t) u - t(m) Delta u = vertical bar u vertical bar(p) with initial data (u(0, .), partial derivative(t)u(0, .)) = (u(0), u(1)), where t >= 0, x is an element of R-n (n >= 2), m is an element of N, p > 1, and u(i) is an element of C-0(infinity) (R-n) (i = 0, 1). We show that there exists a critical exponent pcrit(m, n) > 1 such that the solution u, in general, blows up in finite time when 1 < p < p(crit)(m, n). We further show that there exists a conformal exponent p(conf)(m, n) > p(crit)(m, n) such that the solution u exists globally when p >= p(conf)(m, n) provided that the initial data is small enough. In case p(crit)(m, n) < p < p(conf)(m, n), we will establish global existence of small data solutions u in a subsequent paper (He et al. 2015).