Maximal regularity for non‐autonomous Robin boundary conditions

We consider a non‐autonomous Cauchy problem where is associated with the form , where V and H are Hilbert spaces such that V is continuously and densely embedded in H. We prove H‐maximal regularity, i.e., the weak solution u is actually in (if and ) under a new regularity condition on the form with respect to time; namely Hölder continuity with values in an interpolation space. This result is best suited to treat Robin boundary conditions. The maximal regularity allows one to use fixed point arguments to some non linear parabolic problems with Robin boundary conditions.     

Asymptotic behavior of solutions to a class of nonlocal non‐autonomous diffusion equations

In this paper, we consider the nonlocal non‐autonomous evolution problems where Ω is a bounded smooth domain in , N≥1, β is a positive constant, the coefficient a is a continuous bounded function on , and K is an integral operator with symmetric kernel , being J a non‐negative function continuously differentiable on and . We prove the existence of global pullback attractor, and we exhibit a functional to evolution process generated by this problem that decreases along of solutions. Assuming the parameter β is small enough, we show that the origin is locally pullback asymptotically stable. Copyright © 2014 John Wiley & Sons, Ltd.     

On non‐autonomous integro‐differential‐algebraic evolutionary problems

In this article, we show that a technique for showing well‐posedness results for evolutionary equations in the sense of Picard and McGhee [Picard, McGhee, Partial Differential Equations: A unified Hilbert Space Approach, DeGruyter, Berlin, 2011] established in [Picard, Trostorff, Wehowski, Waurick, On non‐autonomous evolutionary problems. J. Evol. Equ. 13:751‐776, 2013] applies to a broader class of non‐autonomous integro‐differential‐algebraic equations. Using the concept of evolutionary mappings, we prove that the respective solution operators do not depend on certain parameters describing the underlying spaces in which the well‐posedness results are established. Copyright © 2014 John Wiley & Sons, Ltd.     

Blowing‐up solutions to two‐times fractional differential equations

Nonexistence results for a class of two‐times differential equations with fractional derivatives of orders between zero and one are presented. Furthermore, the result is extended to a two‐times system of two differential equations with fractional derivatives of orders between zero and one.     

Exact boundary controllability for non‐autonomous quasilinear wave equations

In this paper, we first show that quite different from the autonomous case, the exact boundary controllability for non‐autonomous wave equations possesses various possibilities. Then we adopt a constructive method to establish the exact boundary controllability for one‐dimensional non‐autonomous quasilinear wave equations with various types of boundary conditions. Finally, we apply the results to multi‐dimensional quasilinear wave equation with rotation invariance. Copyright © 2007 John Wiley & Sons, Ltd.     

Attractors for autonomous double‐diffusive convection systems based on Brinkman–Forchheimer equations

In this paper, we consider the existence of global attractor and exponential attractor for some dynamical system generated by nonlinear parabolic equations in bounded domains with the dimension N≤4 which describe double‐diffusive convection phenomena in a porous medium. We deal with both of homogeneous Dirichlet and Neumann boundary condition cases. Especially, when Neumann condition is imposed, we need some assumptions and restrictions for the external forces and the average of initial data, since the mass conservation law holds. Copyright © 2015 John Wiley & Sons, Ltd.     

Uniform attractor for non‐autonomous suspension bridge equations with localized damping

In this paper, we prove the existence of a uniform attractor for non‐autonomous suspension bridge equations with localized damping. Copyright © 2010 John Wiley & Sons, Ltd.     

Non‐autonomous homogeneous rational difference equations of degree one: convergence and monotone solutions for second and third order case

Consider the non‐autonomous equations: where and also These are some non‐autonomous homogeneous rational difference equations of degree one. A reduction in order is consi‐dered. Convergence and monoton character of positive solutions are studied. Copyright © 2013 John Wiley & Sons, Ltd.     

Well‐posedness of non‐autonomous linear evolution equations in uniformly convex spaces

This paper addresses the problem of well‐posedness of non‐autonomous linear evolution equations in uniformly convex Banach spaces. We assume that for each t is the generator of a quasi‐contractive, strongly continuous group, where the domain D and the growth exponent are independent of t . Well‐posedness holds provided that is Lipschitz for all . Hölder continuity of degree is not sufficient and the assumption of uniform convexity cannot be dropped.     

On a class of bounded trajectories for some non‐autonomous systems

We prove, by variational arguments, the existence of a solution to the boundary value problem in the half line ((0.1)) where c ≥ 0 and a belongs to a certain class of positive functions. The existence of such a solution in the case c = 0 means that the system (0.1) behaves in significantly different way from its autonomous counterpart. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Critical spaces for quasilinear parabolic evolution equations and applications

We present a comprehensive theory of critical spaces for the broad class of quasilinear parabolic evolution equations. The approach is based on maximal $L_p$-regularity in time-weighted function spaces. It is shown that our notion of critical spaces coincides with the concept of scaling invariant spaces in case that the underlying partial differential equation enjoys a scaling invariance. Applications to the vorticity equations for the Navier–Stokes problem, convection–diffusion equations, the Nernst–Planck–Poisson equations in electro-chemistry, chemotaxis equations, the MHD equations, and some other well-known parabolic equations are given.     

A non‐autonomous epidemic model with time delay and vaccination

In this paper, a non‐autonomous SIRVS epidemic model with time delay and vaccination is investigated. We assume that the vaccinated have a constant immunity period. Some new threshold conditions are obtained. These threshold conditions govern the extinction and permanence of the disease. When the model degenerates into the periodic or autonomous case, the corresponding basic reproduction number can be derived from these threshold conditions. Copyright © 2009 John Wiley & Sons, Ltd.     

Impulsive non‐autonomous dynamical systems and impulsive cocycle attractors

In this work, we define the notions of ‘impulsive non‐autonomous dynamical systems’ and ‘impulsive cocycle attractors’. Such notions generalize (we will see that not in the most direct way) the notions of autonomous dynamical systems and impulsive global attractors in the current published literature. We also establish conditions to ensure the existence of an impulsive cocycle attractor for a given impulsive non‐autonomous dynamical system, which are analogous to the continuous case. Moreover, we prove the existence of such attractor for a non‐autonomous 2D Navier–Stokes equation with impulses, using energy estimates. Copyright © 2016 John Wiley & Sons, Ltd.     

Exterior Navier–Stokes flows for bounded data

We prove unique existence of mild solutions on for the Navier–Stokes equations in an exterior domain in , subject to the non‐slip boundary condition.     

Non‐uniform dependence and persistence properties for coupled Camassa–Holm equations

This paper deals with the non‐uniform dependence and persistence properties for a coupled Camassa–Holm equations. Using the method of approximate solutions in conjunction with well‐posedness estimate, it is proved that the solution map of the Cauchy problem for this coupled Camassa–Holm equation is not uniformly continuous in Sobolev spaces H^s with s > 3/2. On the other hand, the persistence properties in weighted L^p spaces for the solution of this coupled Camassa–Holm system are considered. Copyright © 2016 John Wiley & Sons, Ltd.     

On the sufficient condition for regularity of a boundary point for singular parabolic equations with non‐standard growth

We investigate the continuity of solutions for general nonlinear parabolic equations with non‐standard growth near a nonsmooth boundary of a cylindrical domain. We prove a sufficient condition for regularity of a boundary point.     

Non‐rainbow colorings of 3‐, 4‐ and 5‐connected plane graphs

We study vertex‐colorings of plane graphs that do not contain a rainbow face, i.e., a face with vertices of mutually distinct colors. If G is a 3 ‐connected plane graph with n vertices, then the number of colors in such a coloring does not exceed . If G is 4 ‐connected, then the number of colors is at most , and for n≡3(mod8), it is at most . Finally, if G is 5 ‐connected, then the number of colors is at most . The bounds for 3 ‐connected and 4 ‐connected plane graphs are the best possible as we exhibit constructions of graphs with colorings matching the bounds. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 129–145, 2010     

Visco‐elasto‐plastic model for martensitic phase transformation in shape‐memory alloys

Evolution of fine structure in martensite undergoing an isothermal process is modelled on a microscopic level by using a positive homogeneous dissipation potential which can reflect a specific energy needed for a phase transformation between different variants of martensite. The model thus naturally incorporates an activation phenomenon. Existence of a weak solution is proved together with convergence of finite‐element approximations. Numerical experiments showing the expected rate‐independent hysteresis response are also presented. Copyright © 2002 John Wiley & Sons, Ltd.