Related papers: Positive periodic solutions for abstract evolution…
We study the asymptotic stability of traveling fronts and front's velocity selection problem for the time-delayed monostable equation $(*)$ $u_{t}(t,x) = u_{xx}(t,x) - u(t,x) + g(u(t-h,x)),\ x \in \mathbb{R},\ t >0$, considered with…
In this note we provide a self-contained proof of an existence and uniqueness result for a class of Banach space valued evolution equations with an additive forcing term. The framework of our abstract result includes, for example, finite…
We study a class of stochastic evolution equations in a Banach space $E$ driven by cylindrical Wiener process. Three different concept of solutions: generalised strong, weak and mild are defined and the conditions under which they are…
We consider a time-fractional semilinear parabolic abstract Cauchy problem for a time-dependent sectorial operator $A(t)$ which satisfies the Acquistapace-Terreni conditions. We first prove local existence results for the mild solution of…
This paper addresses the problem of wellposedness of non-autonomous linear evolution equations $\dot x = A(t)x$ in uniformly convex Banach spaces. We assume that $A(t):D \subset X\to X$, for each $t$ is the generator of a quasi-contractive…
Solutions of the differential equation $f''+Af=0$ are considered assuming that $A$ is analytic in the unit disc $\mathbb{D}$ and satisfies \begin{equation} \label{eq:dag} \sup_{z\in\mathbb{D}} \, |A(z)| (1-|z|^2)^2 \log\frac{e}{1-|z|} <…
We consider the equation $\Delta_x u+u_{yy}+f(u)=0,\ x=(x_1,\dots,x_N)\in\mathbb{R}^N,\ y\in \mathbb{R},$ where $N\geq 2$ and $f$ is a sufficiently smooth function satisfying $f(0)=0$, $f'(0)<0$, and some natural additional conditions. We…
In this work, we consider time-fractional Navier-Stokes equations (NSE) with the external forces involving finite delay. Equations are considered on a bounded domain in 3-D space having sufficiently smooth boundary. We transform the system…
It is well-known that wave-type equations with memory, under appropriate assumptions on the memory kernel, are uniformly exponentially stable. On the other hand, time delay effects may destroy this behavior. Here, we consider the…
We consider a linear non-autonomous evolutionary Cauchy problem \begin{equation} \dot{u} (t)+A(t)u(t)=f(t) \hbox{ for }\ \hbox{a.e. t}\in [0,T],\quad u(0)=u_0, \end{equation} where the operator $A(t)$ arises from a time depending…
We extend a contraction mapping argument for ordinary state-dependent delay differential equations to evolutionary partial differential equations in the sense of R. Picard, that is, to equations of the form $\bigl(\partial_{t}…
This paper is devoted to the study of periodic solutions for a semilinear Euler-Bernoulli beam equation with variable coefficients. Such mathematical model may be described the infinitesimal, free, undamped in-plane bending vibrations of a…
We prove existence and conditional energetic stability of solitary-wave solutions for the two classes of pseudodifferential equations $ u_t+\left(f(u)\right)_x-\left(L u\right)_x=0 $ and $ u_t+\left(f(u)\right)_x+\left(L u\right)_t=0, $…
In this paper we study the longtime dynamics of mild solutions to retarded stochastic evolution systems driven by a Hilbert-valued Brownian motion. As a preparation for this purpose we have to show the existence and uniqueness of a cocycle…
This paper is concerned with a parabolic evolution equation of the form $A(u_t) + B(u) = f$, settled in a smooth bounded domain of ${\bf R}^d$, $d \geq 1$, and complemented with the initial conditions and with (for simplicity) homogeneous…
In this paper, we prove a theorem of linearized asymptotic stability for fractional differential equations with a time delay. More precisely, using the method of linearization of a nonlinear equation along an orbit (Lyapunov's first…
We study the asymptotic behavior of the solutions of the time-delayed higher-order dispersive nonlinear differential equation \begin{equation*} u_t(x,t)+Au(x,t) +\lambda_0(x) u(x,t)+\lambda(x) u(x,t-\tau )=0 \end{equation*} where…
Assuming the existence of a general nonuniform dichotomy for the evolution operator of a non-autonomous ordinary linear differential equation in a Banach space, we establish the existence of invariant stable manifolds for the semiflow…
In this paper, we study regularity of solutions to linear evolution equations of the form $dX+AXdt=F(t)dt$ in a Banach space $H$, where $A$ is a sectorial operator in $H$ and $A^{-\alpha} F \, (\alpha>0)$ belongs to a weighted H\"{o}lder…
We discuss linear autonomous evolution equations on function spaces which have the property that a positive initial value leads to a solution which initially changes sign, but then becomes - and stays - positive again for sufficiently large…