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

Abstract

This paper is a complement of our recent works on the semilinear Tricomi equations in [9] and [10]. For the semilinear Tricomi equation ∂t2u−tΔu=|u|p with initial data (u(0,⋅),∂tu(0,⋅))=(u0,u1), where t≥0, x∈Rn (n≥3), p>1, and ui∈C0∞(Rn) (i=0,1), we have shown in [9] and [10] that there exists a critical exponent pcrit(n)>1 such that the solution u, in general, blows up in finite time when 1<p<pcrit(n), and there is a global small solution for p>pcrit(n). In the present paper, firstly, we prove that the solution of ∂t2u−tΔu=|u|p will generally blow up for the critical exponent p=pcrit(n) and n≥2, secondly, we establish the global existence of small data solution to ∂t2u−tΔu=|u|p for p>pcrit(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 ∂t2u−tΔu=|u|p for n≥2.