The semigroup approach to first order quasilinear equations in several space variables

The Cauchy problem for u _t + Σ _{i = 1} ⁿ (φ _i (u)) _xi = 0 is treated via the theory of semigroups of nonlinear transformations. This treatment requires the development of results concerning the time-independent equation u + Σ _{i = 1} ⁿ (φ _i (u)) _xi = h for h∈L ¹(Rⁿ), which in turn is studied via the regularized equation $$ u + sumnolimits_{i = 1}^n {left( {phi _i left( u right)} right)} _{xi} - varepsilon Delta u = h $$ .