We prove existence and uniqueness of weak solutions to certain abstract evolutionary integro-differential equations in Hilbert spaces, including evolution equations of fractional order less than 1. Our results apply, e.g., to parabolic partial integro-differential equations in divergence form with merely bounded and measurable coefficients.
We are concerned with a forest kinetic model with memory. In 1994 Kuznetsov et al. [6] has presented a mathematical model describing the kinetics of forest given by a system of ordinary and diffusion equations. This paper handles a generalized model of them which includes a memory effect in the process of establishment of seeds in the air. We shall prove the global existence and shall construct a dynamical system determined from the generalized model. It is also shown that the dynamical system possesses a bounded absorbing set. These results are then generalization of those obtained in [1, 8] for the case of no memory.
In this paper, we will study a property of solutions of q-Painlevé equation of type A7(1). We propose the notion of decomposable extension and then prove the equation has no solution in any decomposable extension of C(t).
An irregularity of an ordinary differential equation with an irregular singular point is defined by the maximal rate of exponential growth of all solutions of the associated homogeneous equation. We shall study first order singular systems of Poincaré rank p (≥ 0) and characterize its irregularity from two different viewpoints; the first one is from the order of zeros of coefficient matrices, and the second one is from the comparison of indices of the system on formal Gevrey spaces.
A class of special solutions of Painlevé/Garnier systems arising as the Bäcklund or Schlesinger transformations of the Riccati solutions is known. In the past several years, the corresponding τ-functions have been explicitly computed and expressed as certain specialization of the Schur functions with rectangle shape partitions. In this note, we will give a simple and direct derivation of these solutions. Our method is based on the Padé approximation and its intrinsic relation to iso-monodromy deformations.
This paper is concerned with positive stationary solutions for a Lotka-Volterra cooperative model with a density-dependent diffusion term of a fractional type. The existence of stationary patterns is proven by the presence of density-dependent diffusion. Our proof is based on the Leray-Schauder degree theory and some a priori estimates. We also derive a certain limiting system which positive stationary solutions satisfy.
This work is concerned with implicit second order abstract differential equations with nonlocal conditions. Assuming that the involved operators satisfy some compactness properties, we establish the existence of local mild solutions, the existence of global mild solutions and the existence of asymptotically almost periodic solutions.