Volterra Equations In Space Research Articles

Lévy-driven Volterra Equations in Space and Time

We investigate nonlinear stochastic Volterra equations in space and time that are driven by L\'evy bases. Under a Lipschitz condition on the nonlinear term, we give existence and uniqueness criteria in weighted function spaces that depend on integrability properties of the kernel and the characteristics of the L\'evy basis. Particular attention is devoted to equations with stationary solutions, or more generally, to equations with infinite memory, that is, where the time domain of integration starts at minus infinity. Here, in contrast to the case where time is positive, the usual integrability conditions on the kernel are no longer sufficient for the existence and uniqueness of solutions, but we have to impose additional size conditions on the kernel and the L\'evy characteristics. Furthermore, once the existence of a solution is guaranteed, we analyse its asymptotic stability, that is, whether its moments remain bounded when time goes to infinity. Stability is proved whenever kernel and characteristics are small enough, or the nonlinearity of the equation exhibits a fractional growth of order strictly smaller than one. The results are applied to the stochastic heat equation for illustration.

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-