Semilinear integrodifferential equations in Banach space

Such equations have physical application; for example, they arise in problems concerned with heat flow in materials with memory. Linear versions of (1.2) are treated in [6], [ll], [24], [25], and [26] ; the nonlinear version is analysed in [14], [15], and [20] ; similar equations are also treated by [12] and [13]. If we replace u,(x, t) by u,,(x, t) we obtain an equation arising in the theory of viscoelasticity [4] and [5]. There is a growing recent literature concerning Volterra equations in abstract spaces. In [lo] Friedman and Shinbrot study a linear version of (1.1); they investigate the existence, boundedness and asymptotic behavior of solutions. R. Miller [23] associates a linear semigroup with solutions and he obtains information concerning the continuous dependence of solutions on initial data. Barbu [l] considers Volterra equations in Hilbert space; Crandall, Londen, and Nohel [2] and Crandall and Nohel [3] apply the theory of multivalued accretive operators to study such equations; Webb in [36] studies a version with A 3 0 and examines the semilinear equation in [33] and [34]. F or other related work the reader is referred to [7], [8], [30], [31], [32]. In this study we are motivated to apply the techniques of the theory of abstract quasi-linear equations of parabolic type found in Friedman [9] and Sobolevskii [29] to the class of semi-