A general local existence theorem for quasi-linear evolution equations in nonreflexive banach spaces

Abstract

KATO’S [4] general existence theorem for quasilinear evolution equations in reflexive Banach spaces is local in time, and so are the theorems obtained in papers [l] and [3] for nonreflexive Banach spaces. However, it turns out that using the method developed in both papers mentioned above as well as the simplified approach to linear evolution equations presented in paper [2] one can prove an existence theorem for quasilinear evolution equations with simpler time estimate. It was also shown in paper [3] that it suffices to utilize the approximate evolution operators constructed in paper [2] in order to solve the auxiliary linear equation for each iteration step. This is also the case here and we have to take advantage of the corresponding evolution operator while solving the appropriate linear equation at each iteration step. Let X, Y, Z be Banach spaces and denote by 11. I/x, I/. IIy, 11. I/= the norms in X, Y, Z, respectively. We assume that the embedding Z C Y C X is continuous and Y, Z are dense in X, Y, respectively. Let W,J be an open ball in Y with center x0 = 0 E Y and radius Y > 0, and denote by W its closure in Y, and let Z0 be an open ball in Z with the same center x0 = 0 and radius R > 0. Given 0 < b, denote by C(0, b; X), C(0, b; Y), C(0, b; Z) the spaces of all continuous functions x = x(t), y = y(t), z = z(t) defined on the interval [0, b] with values in X, Y, Z and norms