Search

In this paper, we investigate a class of the linear evolution process with memory in Banach space by a different approach. Suppose that the linear evolution process is well posed, we introduce a family pair of bounded linear operators, {(G(t),F(t)),t≥0}$\lbrace (G(t), F(t)),t\ge 0\rbrace$, that is, called the resolvent family for the linear evolution process with memory, the F(t)$F(t)$ is called the memory effect family. In this paper, we prove that the families G(t)$G(t)$ and F(t)$F(t)$ are exponentially bounded, and the family (G(t),F(t))$(G(t),F(t))$ associate with an operator pair (A,L)$(A, L)$ that is called generator of the resolvent family. Using (A,L)$(A,L)$, we derive associated differential equation with memory and representation of F(t)$F(t)$ via L. These results give necessary conditions of the well‐posed linear evolution process with memory. To apply the resolvent family to differential equation with memory, we present a generation theorem of the resolvent family under some restrictions on (A,L)$(A,L)$. The obtained results can be directly applied to linear delay differential equation, integro‐differential equation and functional differential equations. [ABSTRACT FROM AUTHOR]

In this paper we give characterizations of the existence and uniqueness of Hölder continuous solutions of certain abstract integro‐differential equation with memory in terms of a resolvent operator. Moreover, we give necessary conditions in order to ensure the existence and uniqueness of mild solutions on the real line. [ABSTRACT FROM AUTHOR]

In this paper, we consider the bifurcation problem for the fractional Laplace equation [ABSTRACT FROM AUTHOR]

In this paper we study local existence, uniqueness, and continuous dependence of an abstract integrodifferential equation. We also present a result on unique continuation and a blow-up alternative for mild solutions of the integrodifferential equation. Finally, we apply our results to an interesting strongly damped plate equation with memory. [ABSTRACT FROM AUTHOR]

A table of contents of the issue Mathematische Nachrichten is presented.

We study the well‐posedness of the fractional degenerate integro‐differential equations (Pα):Dα(Mu)(t)=Au(t)+∫−∞ta(t−s)Au(s)ds+∫−∞tb(t−s)Bu(s)ds+f(t),(t∈T:=[0,2π]),in Lebesgue–Bochner spaces Lp(T;X) and Besov spaces Bp,qs(T;X), where A, B and M are closed linear operators on a Banach space X satisfying D(A)∩D(B)⊂D(M), D(A)∩D(B)≠{0}, α>0 and a,b∈L1(R+). We completely characterize the well‐posedness of (Pα) in the above vector‐valued function spaces on T by using operator‐valued Fourier multiplier. We also give an example that our abstract results may be applied. [ABSTRACT FROM AUTHOR]

For ν∈(0,1), we investigate the nonlinear integro‐differential equation on a multidimensional space domain Ω⊂RnDtνu−L1u−∫0tK(t−s)L2u(·,s)ds=f(x,t,u)+g(x,t)where Dtν is the Caputo fractional derivative and L1 and L2 are uniform elliptic operators with smooth coefficients depending on time. Under suitable conditions on the nonlinearity, the global existence and uniqueness of the classical solution to the related initial and boundary value problems are established. [ABSTRACT FROM AUTHOR]

Linear and nonlinear elliptic complex partial differential equations of higher‐order are considered under Schwarz conditions in the upper‐half plane. Firstly, using the integral representations for the solutions of the inhomogeneous polyanalytic equation with Schwarz conditions, a class of integral operators is introduced together with some of their properties. Then, these operators are used to transform the problem for linear equations into singular integral equations. In the case of nonlinear equations such a transformation yields a system of integro‐differential equations. Existence of the solutions of the relevant boundary value problems for linear and nonlinear equations are discussed via Fredholm theory and fixed point theorems, respectively. [ABSTRACT FROM AUTHOR]

We study the well‐posedness of the fractional differential equations with infinite delay (Pα):Dαu(t)=Au(t)+∫−∞ta(t−s)Au(s)ds+∫−∞tb(t−s)Bu(s)ds+f(t),(0≤t≤2π),on Lebesgue–Bochner spaces Lp(T;X) and Besov spaces Bp,qs(T;X), where A and B are closed linear operators on a Banach space X satisfying D(A)∩D(B)≠{0},  α>0 and a,b∈L1(R+). Under suitable assumptions on the kernels a and b, we completely characterize the well‐posedness of (Pα) in the above vector‐valued function spaces on T by using known operator‐valued Fourier multiplier theorems. We also give concrete examples where our abstract results may be applied. [ABSTRACT FROM AUTHOR]

In this work we construct a Markov family of martingale solutions for 3D stochastic Navier–Stokes equations (SNSE) perturbed by Lévy noise with periodic boundary conditions. Using the Kolmogorov equations of integrodifferential type associated with the SNSE perturbed by Lévy noise, we construct a transition semigroup and establish the existence of a unique invariant measure. We also show that it is ergodic and strongly mixing. [ABSTRACT FROM AUTHOR]

We study the well-posedness of the second order degenerate differential equations with infinite delay: with periodic boundary conditions , where and M are closed linear operators in a Banach space satisfying , . Using operator-valued Fourier multiplier techniques, we give necessary and sufficient conditions for the well-posedness of this problem in Lebesgue-Bochner spaces , periodic Besov spaces and periodic Triebel-Lizorkin spaces . [ABSTRACT FROM AUTHOR]