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

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 e^SA (s 0), and for u, v, w(r) Z_r(Z_n is a ball in ZX),; 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 u_o 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) with- 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.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 127, pp. 181–200, 1983.In conclusion, the author would like to express his sincere gratitude to O. A. Ladyzhenskaya for her interest in this paper.