Boundary value problem for a nonlinear equation of mixed type

Abstract In this paper, we consider the boundary value problem for a nonlinear Lavrentiev–Bitsadze equation of mixed type ∂ 2 u ˆ ∂ x 2 + ( sgn y ) ( 1 + u ˆ x 2 ) ∂ 2 u ˆ ∂ y 2 = 0 , whose coefficients depend on the first order derivatives of unknown function. Above y = 0 and below y = 0 , the equation becomes the nonlinear elliptic equation and nonlinear hyperbolic equation respectively, this is different from the equation studied in Chen (2011) [9] . We prove the existence of solution to the problem by the method which can be used to study more difficult problems for nonlinear equations of mixed type arising in gas dynamics.

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