Nonlinear and quasilinear evolution equations: Existence, uniqueness, and comparison of solutions; the rate of convergence of the difference method

Abstract

In the Banach space X one investigates the Cauchy problem where [u](t)=u|[o, t], f ∈ L1 (0, T; X); for fixed t, w, the nonlinear operator A(t, w)=A is a pseudogenerating operator of the semigroup eSA (s ≥ 0), and for u, v, w(r) ∈ Zr(Zn is a ball in Z⊂X), ; the conditions on the dependence of A(t, w) on w admit the occurrence of w in the “leading” terms. One proves local and global theorems of existence and uniqueness of the limit-difference solution of the Cauchy problem, one investigates its differentiability and its dependence on uo and f. Similar results of Crandall-Pazy, Benilan, Crandall-Evans, Evans, Oharu, Pavel, etc. for the equations du(t)/dt=A(t)u(t)+f (t) withω- dissipative operators are special cases of ours. In the quasilinear case, our results complement and generalize T. Kato's well-known theorem. In addition, one obtains estimates for the convergence rates of the difference method and estimates for the norm of the difference of the solutions of Cauchy problems with different operators A(t, w); these results are new also for the equations with dissipative operators.