Critical Exponents For Semilinear Tricomi-Type Equations

In this thesis, we consider the Cauchy problem of semilinear Tricomi equation where t≥ 0, x ∈ Rn, n≥ 3 and m ∈ N, we have proven there exists a critic exponent Pcrit= Pcrit(m, n),such that if 1< p< pcrit, then the local solution will generally blow up in finite time. On the other hand, if p> pcrit, then the global existence can be established for small data.Tricomi equation is a kind of degenerate hyperbolic equation, which is not only an interesting topic in mathematics, but also plays an important role in the research of Physics. From the point of mathematical view, the Tricomi equation is a generalization of wave equation, with a curved characteristic cone and a degenerate line at time t= 0. From the point of physical view, the Tricomi equation is closely connected to the study of gas dynamics with transonic speeds. Namely, the Tricomi equation describes the transition from the subsonic flow (t> 0, elliptic region) to the supersonic flow (t> 0, hyperbolic region) in a de Laval nozzle, which is one of the most interesting problems in fluid dynamics.Recently the linear and the semilinear Tricomi equations became the focus of interests of many authors, there are extensive results concerning the Cauchy problem for both linear and semilinear generalized Tricomi equations. For instance, for the linear generalized Tricomi equation, the authors in [1], [37] and [39] have computed its fundamental solution explicitly. More recently, the authors in [26-29] established the local existence as well as the singularity structures of low regularity solutions to the semilinear equation (?)t2u - tm△u= f(t,x,u) in the degenerate hyperbolic region and the elliptic-hyperbolic mixed region, respectively, where f is a C1 function and has compact support with respect to the variable x. By establishing some classes of Lp-Lq estimates for the solution v of linear equation (?)t2v - tm△v= F(t,x), the author in [38] obtained a series of interesting results about the global existence or the blowup of solutions to problem (0.0.2) when the exponent p belongs to a certain range, however, there was a gap between the global existence interval and the blowup interval; moreover, the critical exponent pcrit{m,n) was not determined there.In this thesis, we consider the Cauchy problem (0.0.2) and assume that ui ∈ C0∞(B(0,M)) (i= 0,1), where B(0, M)={x:|x|< M}, and M> 0. Then we determine the critical exponent pcrit(m, n) as the positive root of the algebraic equation For 1< p< Pcrit, we use the test function method together with some crucial techniques for the modified Bessel function as in [15,28] to derive the blowup result. While for p> Pcrit, motivated by [11,18], where basic Strichartz estimates were obtained for the linear wave operator, we are required to establish weighted and unweighted Strichartz estimates for the generalized Tricomi operator (?)t2 - tm△. Based on the resulting inequalities and the contraction mapping principle, we eventually complete the proof of global existence.