2012/11/02 by Zhuoping Ruan, Ruan, Zhuoping, Ingo Witt +3
Mathematics · #35L65 #35L67 #35L70 #76N15 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #advanced mathematical theories #math.AP #msc:35L65 #msc:35L67 #msc:35L70 #msc:76N15
paper · pdf · doi:10.48550/arxiv.1211.0334
37 pages
arxiv created 2012/11/02 · openalex publication_date 2012/11/02 · arxiv updated 2012/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we are concerned with the local existence and singularity structure of low regularity solutions to the semilinear generalized Tricomi equation \pt2u-tmΔu=f(t,x,u) with typical discontinuous initial data (u(0,x), \ptu(0,x))=(0, \vp(x)); here m∈\Bbb N, x=(x1, ..., xn), n≥ 2, and f(t,x,u) is C∞ smooth in its arguments. When the initial data \vp(x) is a homogeneous function of degree zero or a piecewise smooth function singular along the hyperplane t=x1=0, it is shown that the local solution u(t,x)∈ L∞([0,T]×\Bbb Rn) exists and is C∞ away from the forward cuspidal cone Γ0=(t,x)\colon t>0, |x|2=\ds\f4tm+2(m+2)2 and the characteristic cuspidal wedge \G1±=(t,x)\colon t>0, x1=± \ds\f2t^\fm2+1m+2, respectively. On the other hand, for n=2 and piecewise smooth initial data \vp(x) singular along the two straight lines t=x1=0 and t=x2=0, we establish the local existence of a solution u(t,x)∈ L∞([0,T]×\Bbb R2)∩ C([0, T], H^\fm+62(m+2)-(\Bbb R2)) and show further that u(t,x)\not∈ C2((0,T]×\Bbb R2∖(\G0∪\G1±∪\G2±)) in general due to the degenerate character of the equation under study; here \G2±=(t,x)\colon t>0, x2=±\ds\f2t^\fm2+1m+2. This is an essential difference to the well-known result for solutions v(t,x)∈ C∞(\Bbb R+×\Bbb R2∖ (Σ0∪Σ1±∪ Σ2±)) to the 2-D semilinear wave equation \pt2v-Δv=f(t,x,v) with (v(0,x), \ptv(0,x))=(0, \vp(x)), where Σ0=t=|x|, Σ1±=t=± x1, and Σ2±=t=± x2.