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

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• 作者：Daoyin He ; Ingo Witt ; Huicheng Yin
• 关键词：Mathematics Subject Classification35L70 ; 35L65 ; 35L67
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：56
• 期：2
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-0835
• 卷排序：56

文摘

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