Partial differential equations with discrete (concentrated) state-dependent delays in the space of continuous functions are investigated. In general, the corresponding initial value problem is not well posed, so we find an additional assumption on the state-dependent delay function to guarantee the well posedness. For the constructed dynamical system we study the long-time asymptotic behavior and prove the existence of a compact global attractor.
We establish the uniqueness of semi-wavefront solution for a non-local delayed reaction-diffusion equation. This result is obtained by using a generalization of the Diekman-Kaper theory for a nonlinear convolution equation. Several applications to the systems of non-local reaction-diffusion equations with distributed time delay are also considered.
In this paper we give a necessary and suffcient conditions for the existence and uniqueness of periodic solutions of functional differential equations with n delay d dt x(t) = Ax(t) + n j=1 Bx(t -- r j) + f (t). The conditions are obtained in terms of R-boundedness of operator valued Fourier multipliers.
A modified version of the three dimensional Navier-Stokes equations is considered with periodic boundary conditions. A bounded constant delay is introduced into the convective term, that produces a regularizing effect on the solution. In fact, by assuming appropriate regularity on the initial data, the solutions of the delayed equations are proved to be regular and, as a consequence, existence and also uniqueness of a global weak solution is obtained. Moreover, the associated flow is constructed and the continuity of the semigroup is proved. Finally, we investigate the passage to the limit on …
We consider positive travelling fronts of the time-delayed reaction-diffusion equation with the monostable birth function. Our main result says that for every fixed and sufficiently large velocity c, the positive travelling front is unique (modulo translations). To prove the uniqueness, we introduce a small parameter 1/c and realize the Lyapunov-Schmidt reduction in a scale of Banach spaces.
In this paper, we present a general framework for solving stochastic functional differential equations in infinite dimensions in the sense of martingale solutions, which can be applied to a large class of SPDE with finite delays, e.g. $d$-dimensional stochastic fractional Navier-Stokes equations with delays, $d$-dimensional stochastic reaction-diffusion equations with delays, $d$-dimensional stochastic porous media equations with delays. Moreover, under local monotonicity conditions for the nonlinear term we obtain the existence and uniqueness of strong solutions to SPDE with delays.
This work is the first attempt to treat partial differential equations with discrete (concentrated) state-dependent delay. The main idea is to approximate the discrete delay term by a sequence of distributed delay terms (all with state-dependent delays). We study local existence and long-time asymptotic behavior of solutions and prove that the model with distributed delay has a global attractor while the one with discrete delay possesses the trajectory attractor.
Parabolic differential equations with discrete state-dependent delay are studied. The approach, based on an additional condition on the delay function introduced in [A.V. Rezounenko, Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions, Nonlinear Analysis: Theory, Methods and Applications, 70 (11) (2009), 3978-3986] is developed. We propose and study a state-dependent analogue of the condition which is sufficient for the well-posedness of the corresponding initial value problem on the whole space of continuous functions …
We investigate a class of non-linear partial differential equations with discrete state-dependent delays. The existence and uniqueness of strong solutions for initial functions from a Banach space are proved. To get the well-posed initial value problem we restrict our study to a smaller metric space, construct the dynamical system and prove the existence of a compact global attractor.
A wide class of non-autonomous nonlinear parabolic partial differential equations with delay is studied. We allow in our investigations different types of delays such as constant, time-dependent, state-dependent (both discrete and distributed) to be presented simultaneously. The main difficulties arise due to the presence of discrete state-dependent delays since the nonlinear delay term is not Lipschitz on the space of continuous functions. We find conditions for the local existence, uniqueness and study the invariance principle.
Partial differential equations with discrete (concentrated) state-dependent delays are studied. The existence and uniqueness of solutions with initial data from a wider linear space is proven first and then a subset of the space of continuously differentiable (with respect to an appropriate norm) functions is used to construct a dynamical system. This subset is an analogue of \textit{the solution manifold} proposed for ordinary equations in [H.-O. Walther, The solution manifold and $C\sp 1$-smoothness for differential equations with state-dependent delay, J. Differential Equations, {195}(1), (…
We propose a new approach for proving uniqueness of semi-wavefronts in generally non-monotone monostable reaction-diffusion equations with distributed delay. This allows to solve an open problem concerning the uniqueness of non-monotone (hence, slowly oscillating) semi-wavefronts to the KPP-Fisher equation with delay. Similarly, a broad family of the Mackey-Glass type diffusive equations is shown to possess a unique (up to translation) semi-wavefront for each admissible speed.
Using Carleman estimates, we give a lower bound for solutions to the discrete Schrödinger equation in both dynamic and stationary settings that allows us to prove uniqueness results, under some assumptions on the decay of the solutions.
In this work we prove uniqueness result for an implicit discrete system defined on connected graphs. Our discrete system is motivated from a certain class of spatial segregation of reaction-diffusion equations.
We prove that if a solution of the discrete time-dependent Schrödinger equation with bounded real potential decays fast at two distinct times then the solution is trivial. For the free Shrödinger operator and for operators with compactly supported time-independent potentials a sharp analog of the Hardy uncertainty principle is obtained, using an argument based on the theory of entire functions. Logarithmic convexity of weighted norms is employed in the case of general real-valued time-dependent bounded potentials. In the latter case the result is not optimal.
We study the existence and uniqueness of wavefronts to the scalar reaction-diffusion equations $u_{t}(t,x) = Δu(t,x) - u(t,x) + g(u(t-h,x)),$ with monotone delayed reaction term $g: \R_+ \to \R_+$ and $h >0$. We are mostly interested in the situation when the graph of $g$ is not dominated by its tangent line at zero, i.e. when the condition $g(x) \leq g'(0)x,$ $x \geq 0$, is not satisfied. It is well known that, in such a case, a special type of rapidly decreasing wavefronts (pushed fronts) can appear in non-delayed equations (i.e. with $h=0$). One of our main goals here is to establish a simi…
Classical solutions to PDEs with discrete state-dependent delay are studied. We prove the well-posedness in a set $X_F$ which is an analogous to the solution manifold used for ordinary differential equations with state-dependent delay. We prove that the evolution operators are $C^1$-smooth on the solution manifold.
We study abstract linear and nonlinear evolutionary systems with single or multiple delay feedbacks, illustrated by several concrete examples. In particular, we assume that the operator associated with the undelayed part of the system generates an exponentially stable semigroup and that the delay damping coefficients are locally integrable in time. A step by step procedure combined with Gronwall's inequality allows us to prove the existence and uniqueness of solutions. Furthermore, under appropriate conditions we obtain exponential decay estimates.
By introducing a kind of special functions namely exponent-like function, cosine-like function and sine-like function, we obtain explicitly the basic structures of solutions of initial value problem at the original point for this kind of linear pantograph equations. In particular, we get the complete results on the existence, uniqueness and non-uniqueness of the initial value problems at a general point for the kind of linear pantograph equations.
In this paper we study a Hegselmann-Krause opinion formation model with distributed time delay and positive influence functions. Through a Lyapunov functional approach, we provide a consensus result under a smallness assumption on the initial delay. Furthermore, we analyze a transport equation, obtained as mean--field limit of the particle one. We prove global existence and uniqueness of the measure-valued solution for the delayed transport equation and its convergence to consensus under a smallness assumption on the delay, using a priori estimates which are uniform with respect to the number …
We study a McKendrick-von Foerster type equation with renewal. This model is represented by a single equation which describes one species which produces young individuals. The renewal condition is linear, but takes into account some history of the population. This model addresses nonlocal interactions between individuals structured by age. The vast majority of size-structured models are also treatable. Our model generalizes a number of earlier models with delays and integrals. The existence and uniqueness is proved through a fixed-point approach to an equivalent integral problem in $L^{\infty}…
In this paper we study a nonlinear size-structured population model with distributed delay in the recruitment. The delayed problem is reduced into an abstract initial value problem of an ordinary differential equation in the Banach space by using the delay semigroup techniques. The local existence and uniqueness of solution as well as the continuous dependence on initial conditions are obtained by using the general theory of quasi-linear evolution equations in nonreflexive Banach spaces, while the global existence of solution is obtained by the estimates of the solution and the extension theor…
We derive a simple criterion that ensures uniqueness, Lipschitz stability and global convergence of Newton's method for the finite dimensional zero-finding problem of a continuously differentiable, pointwise convex and monotonic function. Our criterion merely requires to evaluate the directional derivative of the forward function at finitely many evaluation points and for finitely many directions. We then demonstrate that this result can be used to prove uniqueness, stability and global convergence for an inverse coefficient problem with finitely many measurements. We consider the problem of d…
In this paper, we investigate the global existence, uniqueness and asymptotic stability of time $ω$-periodic classical solution for a class of extended Fisher-Kolmogorov equations with delays and general nonlinear term. We establish a general framework to find time $ω$-periodic solutions for nonlinear extended Fisher-Kolmogorov equations with delays and general nonlinear function, which will provide an effective way to deal with such kinds of problems. The discussion is based on the theory of compact and analytic operator semigroups and maximal regularization method.
This paper is concerned with system of magnetic effected piezoelectric beams with interior time-varying delay and time-dependent weights, in which the beam is clamped at the two side points subject to a single distributed state feedback controller with a time-varying delay. By combining the semigroup theory with Kato's variable norm technique, we obtain the existence and uniqueness of solution. By imposing appropriate assumptions on the time-varying delay term and time-dependent weights, we introduce suitable perturbed Lyapunov functional to obtain exponential stability estimates.
We show novel types of uniqueness and rigidity results for Schrödinger equations in either the nonlinear case or in the presence of a complex-valued potential. As our main result we obtain that the trivial solution $u=0$ is the only solution for which the assumptions $u(t=0)\vert_{D}=0, u(t=T)\vert_{D}=0$ hold, where $D\subset \mathbb{R}^d$ are certain subsets of codimension one. In particular, $D$ is discrete for dimension $d=1$. Our main theorem can be seen as a nonlinear analogue of discrete Fourier uniqueness pairs such as the celebrated Radchenko--Viazovska formula, and the uniqueness res…
We prove an existence and uniqueness result for the infinitely delayed stochastic evolution equation $$dU(t) = &\big(AU(t) + F(t,U_t)\big) dt + B(t,U_t)dW_H(t), t\in[0,T_0]$$ where $A$ is the generator of an analytic semigroup on a UMD space $E$, $F$ and $B$ satisfy Lipschitz conditions and $\mathscr{B}$ is a weighted $L^p$ history space. This paper is based on recent work of van Neerven \emph{et al.}~which developed the theory of abstract stochastic evolution equations in UMD spaces.
