On semilinear Tricomi equations with critical exponents or in two space dimensions

[摘要] This paper is a complement of our recent works on the semilinear Tricomi equations in [9] and [10]. For the semilinear Tricomi equation partial derivative(2)(t)u - t Delta u = vertical bar u vertical bar(p) with initial data (u(0,.), partial derivative(t)u = (u(0), u(1)), where t >= 0, x is an element of R-n (n >= 3), p > 1, and u(i) is an element of C-0(infinity) (R-n) (i = 0, 1), we have shown in [9] and [10] that there exists a critical exponent p(crit)(n) > 1 such that the solution u, in general, blows up in finite time when 1 < p < P-crit(n), and there is a global small solution for p > p(crit)(n). In the present paper, firstly, we prove that the solution of partial derivative(2)(t)u - t Delta u = vertical bar u vertical bar(p) will generally blow up for the critical exponent p = p(crit)(n) and n > 2, secondly, we establish the global existence of small data solution to qu tAu = 1W for P > P-crit (n) and n = 2. Thus, we have given a systematic study on the blowup or global existence of small data solution u to the equation partial derivative(2)(t)u - t Delta u = vertical bar u vertical bar(p) for n >= 2. (C) 2017 Elsevier Inc. All rights reserved.

[发布日期] 2017-12-15

[关键词] Tricomi equation;Critical exponent;Blowup;Global existence;Strichartz estimate