On the global solution problem for semilinear generalized Tricomi equations, I

By Daoyin He, Ingo Witt, Huicheng Yin. 2015

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In this paper, we are concerned with the global Cauchy problem for the semilinear generalized Tricomi equation $$\partial _t^2 u-t^m \Delta u=|u|^p$$∂t2u-tmΔu=|u|p with initial data $$(u(0,\cdot ), \partial _t u(0,\cdot ))=(u_0, u_1)$$(u(0,·),∂tu(0,·))=(u0,u1), where $$t\ge 0$$t≥0, $$x\in \mathbb {R}^n$$x∈Rn ($$n\ge 2$$n≥2), $$m\in \mathbb {N}$$m∈N, $$p>1$$p>1, and $$u_i\in C_0^{\infty }(\mathbb {R}^n)$$ui∈C0∞(Rn) ($$i=0,1$$i=0,1). We show that there exists a critical exponent $$p_{\text {crit}}(m,n)>1$$pcrit(m,n)>1 such that the solution u, in general, blows up in finite time when $$1<p<p_{\text {crit}}(m,n)$$1<p<pcrit(m,n). We further show that there exists a conformal exponent $$p_{\text {conf}}(m,n)> p_{\text {crit}}(m,n)$$pconf(m,n)>pcrit(m,n) such that the solution u exists globally when $$p\ge p_{\text {conf}}(m,n)$$p≥pconf(m,n) provided that the initial data is small enough. In case $$p_{\text {crit}}(m,n)<p< p_{\text {conf}}(m,n)$$pcrit(m,n)<p<pconf(m,n), we will establish global existence of small data solutions u in a subsequent paper (He et al. 2015).

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/572e3af9d34161603a059fa5ffc74c952df451cb

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