Local existence of solutions to dissipative nonlinear evolution equations with mixed types

Abstract

In this paper, we consider the local existence of solutions to the Cauchy problems for the following nonlinear evolution equations with mixed types { ψ t = − ( 1 − α ) ψ − θ x + α ψ x x , θ t = − ( 1 − α ) θ + γ ψ x + 2 ψ θ x + α θ x x , with initial data ( ψ , θ ) ( x , 0 ) = ( ψ 0 ( x ) , θ 0 ( x ) ) → ( ψ ± , θ ± ) , as x → ± ∞ , where α and γ are positive constants satisfying α < 1 , γ < α ( 1 − α ) . Through constructing an approximation solution sequence, we obtain the local existence by using the contraction mapping principle.