A Blow-Up Result for a Generalized Tricomi Equation with Nonlinearity of Derivative Type

摘要

In this note, we prove a blow-up result for a semilinear generalized Tricomi equation with nonlinear term of derivative type, i.e., for the equation T(l)u = vertical bar partial derivative(t)u vertical bar(p), where T-l = partial derivative(2)(t) - t(2l)Delta. Smooth solutions blow up in finite time for positive Cauchy data when the exponent p of the nonlinear term is below Q/Q-2, where Q = (l + 1) n + 1 is the quasi-homogeneous dimension of the generalized Tricomi operator T-l. Furthermore, we get also an upper bound estimate for the lifespan.