A convergence theorem of nonlinear semigroups and its application to first order quasilinear equations

This PaPer deals with the convergence of nonlinear semigroups and the difference approximation for the Cauchy problem, $(CP)$ , for the scalar quasi- linear equation (DE) uₜ+∑ᵢ₌₁ᵈ(φᵢ(u))ₓ=0 for $t>0,$ x=(x₁, x₂, ⋯ , xd)∈ Rᵈ from the viewpoint of the aPproximation theory for nonlinear semigroups. This investigation is motivated by the works of Kru kov $[13, 14]$ , Crandall Kru kov treats this problem over the space L∞(Rᵈ) and discusses the existence and uniqueness of the generalized solution of $(CP)$ under the assumption that φᵢ∈ C¹(R¹) for all i . His proof is based on the so-called method of vanishing viscosity and generalizes (DE) to allow the