A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:206

Article

Yagdjian, K

关键词: Tricomi-type equation;    fundamental solution;    L-p - L-q estimates;   

DOI  :  10.1016/j.jde.2004.07.028

来源: Elsevier

【 摘 要 】

Barros-Neto and Gelfand (Duke Math. J. 98 (3) (1999) 465; Duke Math. J. 117 (2) (2003) 561) constructed for the Tricomi operator ypartial derivative(x)(2) + partial derivative(y)(2) on the plane the fundamental solutions with the supports in the regions related to the geometry of the characteristics of the Tricomi operator. We give for the Tricomi-type operator partial derivative(t)(2) -t(m) Delta(x) a fundamental solution relative to an arbitrary point of Rn+1 with the support in the region tgreater than or equal to0, where the operator is hyperbolic. Our key observation is that the fundamental solution for the Tricomi-type operator can be written like an integral of the distributions generated by the fundamental solution of the Cauchy problem for the wave equation. The application of that fundamental solution to the L-P - L-q estimate for the forced Tricomi-type equation is given as well. (C) 2004 Elsevier Inc. All rights reserved.

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