JournalsprimsVol. 48, No. 4pp. 899–917

Existence and Uniqueness Theorem for a Class of Singular Nonlinear Partial Differential Equations

  • Dennis B. Bacani

    Sophia University, Tokyo, Japan
  • Hidetoshi Tahara

    Sophia University, Tokyo, Japan
Existence and Uniqueness Theorem for a Class of Singular Nonlinear Partial Differential Equations cover
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Abstract

This paper deals with singular nonlinear partial differential equations of the form tu/t=F(t,x,u,u/x)t\partial u/\partial t =F\left(t,x,u,\partial u/\partial x\right), with independent variables (t,x)R×C(t,x)\in \mathbb{R}\times\mathbb{C}, and where F(t,x,u,v)F(t,x,u,v) is a function continuous in tt and holomorphic in the other variables. Using the Banach fixed point theorem (also known as the contraction mapping principle), we show that a unique solution u(t,x)u(t,x) exists under the condition that F(0,x,0,0)=0F(0,x,0,0)=0, Fu(0,x,0,0)=0F_u(0,x,0,0)=0 and Fv(0,x,0,0)=xγ(x)F_v(0,x,0,0)=x \,\gamma(x) with \mboxReγ(0)<0\mbox{Re}\, \gamma(0)<0.

Cite this article

Dennis B. Bacani, Hidetoshi Tahara, Existence and Uniqueness Theorem for a Class of Singular Nonlinear Partial Differential Equations. Publ. Res. Inst. Math. Sci. 48 (2012), no. 4, pp. 899–917

DOI 10.2977/PRIMS/90