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Title: Exact solutions of Cauchy problem for partial differential equations with double characteristics and singular coefficients (English)
Author: Lu, Zhu-Jia
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 121
Issue: 1
Year: 1996
Pages: 9-24
Summary lang: English
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Category: math
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Summary: Let $ L_{a, b}\eq(\pa_x-ax^k\pa_t)(\pa_x-bx^k\pa_t)+kbx^{k-1}\pa_t -kx\pa_x $ be a family of operators with double characteristics and singular coefficients, where $a$, $b$ are reals with $ab\ne0$ and $a\ne b$, $k>0$ is an odd integer. Let $\ome$ be the first quadrant in the plane and $H_+$ the upper half-plane. Consider Cauchy problems \cases{L_{a,b}u=0 & in $\ome or H_+$, \cr u(x, 0)=\vp_0(x), u_t(x, 0)=\vp_1(x) \quad& for $x \in\ov{{{\Bbb R}}_+} or x \in{{\Bbb R}}$ \cr} \eqno(P_1) for $a>0$, $b>0$, and initial-boundary value problems \cases{L_{a, b}u=0 & in $\ome or H_+$, \cr u(x, 0)=\vp_0(x), u_t(x, 0)=\vp_1(x) \quad& for $x \in{\ov{{{\Bbb R}}_+}} or x\in{{\Bbb R}}$, \cr u(0, t)=\psi_0(t) & for $t\in\ov{{{\Bbb R}}_+}$,\cr} \eqno(P_2) \cases{L_{a, b}u=0 & in $\ome or H_+$, \cr u(x, 0)=\vp_0(x), u_t(x, 0)=\vp_1(x) \quad& for $x \in{\ov{{{\Bbb R}}_+}} or x \in{{\Bbb R}}$, \cr\displaystyle\lim_{{(x, \tau) \to(0, t), x \ne0} \atop{(x, \tau)\in\ome or H_+}} \d{u_x(x, \tau)}{x^k}=\psi_1(t) & for $t \in\ov{{{\Bbb R}}_+}$ \cr} \eqno(P_3) for $ab<0$ and \cases{L_{a, b}u=0 & in $\ome or H_+$, \cr u(x, 0)=\vp_0(x), u_t(x, 0)=\vp_1(x) \quad& for $x \in{\ov{{{\Bbb R}}_+}} or x\in{{\Bbb R}}$, \cr u(0, t)=\psi_0(t), \displaystyle\lim_{{(x, \tau)\to(0, t), x \ne0} \atop{(x, \tau)\in\ome or H_+}}\d{u_x(x, \tau)}{x^k} \! \! \! & = $\psi_1(t) \quad for t \in\ov{{{\Bbb R}}_+}$ \cr} \eqno(P_4) for $a<0$, $ b<0$. Under appropriate smoothness conditions on $\vp_0$, $ \vp_1$, $ \psi_0$ and $\psi_1$, we obtain different sufficient and necessary conditions for each problem to have classical solutions. Moreover, we obtain also explicit expressions of solutions in each case. (English)
Keyword: exact solutions
Keyword: Cauchy problem
Keyword: singular coefficients
Keyword: double characteristics
MSC: 35A05
MSC: 35C15
MSC: 35C99
MSC: 35L99
idZBL: Zbl 0863.35025
idMR: MR1388169
DOI: 10.21136/MB.1996.125949
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Date available: 2009-09-24T21:14:29Z
Last updated: 2020-07-29
Stable URL: http://hdl.handle.net/10338.dmlcz/125949
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Reference: [1] Z .J. Lu: Existence of solutions of the non-homogeneous Goursat problem for a class of partial differential equations with double characteгistics.J. Math. Res. & Expositions (1986), no. 2, 40-49. English edition.
Reference: [2] Z. J. Lu M. C. Mai, G. Y. Wang: Discrete phenomena in existence in the initial value problems.Scientia Sinica 22 (1979), no. 11, 1229-1237. Zbl 0424.35052, MR 0557330
Reference: [3] A. Menikoff: Uniqueness of the Cauchy problem for a class of partial differential equations with double characteristics.Indiana Univ. Math. J. 25 (1976), no. 1, 1-23. Zbl 0317.35078, MR 0399626, 10.1512/iumj.1976.25.25001
Reference: [4] F. Trèves: Discrete phenomena in uniqueness in the Cauchy problem.Proc. Amer. Math. Soc. 46 (1974), no. 2, 229-233. MR 0352679, 10.2307/2039900
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