On the existence of low regularity solutions to semilinear generalized Tricomi equations in mixed type domains

摘要

In [20,21], we have established the existence and singularity structures of low regularity solutions to the semilinear generalized Tricomi equations in the degenerate hyperbolic regions and to the higher order degenerate hyperbolic equations, respectively. In the present paper, we shall be concerned with the low regularity solution problem for the semilinear mixed type equation partial derivative(2)(t)u - t(2l-1) Delta u = f (t, x, u) with an initial data u (0, x) = phi(x) is an element of H-s(R-n) (0 <= s < n/2), where (t, x) is an element of R X R-n, n >= 2, l is an element of N, f (t, x, u) is C-1 smooth in its arguments and has compact support with respect to the variable x. Under the assumption of the subcritical growth of f (t, x, u) on u, we will show the existence and regularity of the considered solution in the mixed type domain [-T-0, T-0] x R-n for some fixed constant T-0 > 0. (C) 2015 Elsevier Inc. All rights reserved.