On semilinear Tricomi equations in one space dimension

2018/10/27 by Daoyin He, He, Daoyin, Ingo Witt +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.1810.12748

Abstract

For 1-D semilinear Tricomi equation ∂_t² u-t∂_x²u=|u|^p with initial data (u(0,x), ∂_t u(0,x)) =(u₀(x), u₁(x)), where t≥ 0, x∈ℝ, p>1, and u_i∈ C₀^∞(ℝ) (i=0,1), we shall prove that there exists a critical exponent p_{\rm crit}=5 such that the small data weak solution u exists globally when p>p_{\rm crit}; on the other hand, the weak solution u, in general, blows up in finite time when 11. By this paper and \citeHWYin1-\citeHWYin3, we have given a systematic study on the blowup or global existence of small data solution u to the equation ∂_t² u-tΔu=|u|^p for all space dimensions. One of the main ingredients in the paper is to establish a crucial weighted Strichartz-type inequality for 1-D linear degenerate equation ∂_t² w-t∂_x² w=F(t,x) with (w(0,x), ∂_t w(0,x))=(0,0), i.e., an inequality with the weight ((4)/(9)t³-|x|²)^α between the solution w and the function F is derived for some real numbers α.

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• Lifespan of semilinear generalized Tricomi equation with Strauss type exponent

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