Nonclassical diffusion with memory

We consider the nonclassical diffusion equation with hereditary memory on a bounded three‐dimensional domain. Setting the problem in the past history framework, and with very general assumptions on the memory kernel κ, we prove that the related solution semigroup possesses a global attractor of optimal regularity. Copyright © 2014 John Wiley & Sons, Ltd.     

Solution operators of three time variables for fractional linear problems

It is well known that the time fractional equation where is the fractional time derivative in the sense of Caputo of u does not generate a dynamical system in the standard sense. In this paper, we study the algebraic properties of the solution operator T(t,s,τ) for that equation with u(s) = v. We apply this theory to linear time fractional PDEs with constant coefficients. These equations are solved by the Fourier multiplier techniques. It appears that their solution exhibits some singularity, which leads us to introduce a new kind of solution for abstract time fractional problems. Copyright © 2016 John Wiley & Sons, Ltd.     

Positive solutions for a fourth order discrete p‐Laplacian boundary value problem

In this paper, we study the existence and multiplicity of positive solutions for the following fourth order nonlinear discrete p‐Laplacian boundary value problem where φ_p(s) = | s | ^{p ? 2}s, p > 1, is continuous, T is an integer with T ≥ 5 and . By virtue of Jensen's discrete inequalities, we use fixed point index theory to establish our main results. Copyright © 2013 John Wiley & Sons, Ltd.     

Global attractors for the coupled suspension bridge system with temperature

This paper deals with the long‐term properties of the thermoelastic nonlinear string‐beam system related to the well‐known Lazer–McKenna suspension bridge model (0.1) In particular, no mechanical dissipation occurs in the equations, because the loss of energy is entirely due to thermal effects. The existence of regular global attractors for the associated solution semigroup is proved (without resorting to a bootstrap argument) for time‐independent supplies f,g,h and any . Copyright © 2015 John Wiley & Sons, Ltd.     

Boundary value problem for p−Laplacian Caputo fractional difference equations with fractional sum boundary conditions

In this paper, we consider a discrete fractional boundary value problem of the form where 0 < α,β≤1, 1 < α + β≤2, 0 < γ≤1, , ρ is a constant, and denote the Caputo fractional differences of order α and β, respectively, is a continuous function, and ?_p is the p‐Laplacian operator. The existence of at least one solution is proved by using Banach fixed point theorem and Schaefer's fixed point theorem. Some illustrative examples are also presented. Copyright © 2015 John Wiley & Sons, Ltd.     

A general stability result in a Timoshenko system with infinite memory: A new approach

In this paper, we consider a Timoshenko system in the presence of an infinite memory, where the relaxation function satisfies a relation of the form Under the same hypothesis on g and ξ imposed for the finite memory case, we establish some general decay results for the equal and nonequal speed propagation cases. Our results improve in some situations some known decay rates. Also, some applications to other problems are discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

A new Beale–Kato–Majda criteria for the 3D magneto‐micropolar fluid equations in the Orlicz–Morrey space

In this work, we are interested in the study of regularity for the three‐dimensional magneto‐micropolar fluid equations in Orlicz–Morrey spaces. If the velocity field satisfies or the gradient field of velocity satisfies then we show that the solution remains smooth on [0,T]. In view of the embedding with 2 < p < 3 ∕ r and P > 1, we see that our result extends the result of Yuan and that of Gala. Copyright © 2012 John Wiley & Sons, Ltd.     

Global existence and optimal decay rates of solutions to conservation laws with diffusion‐type terms

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion‐type source term. Based on a low‐frequency and high‐frequency decomposition, Green's function method and the classical energy method, we not only obtain L² time‐decay estimates but also establish the global existence of solutions to Cauchy problem when the initial data u₀(x) satisfies the smallness condition on , but not on . Furthermore, by taking a time‐frequency decomposition, we obtain the optimal decay estimates of solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

A remark on the logarithmically improved regularity criterion for the micropolar fluid equations in terms of the pressure

In this paper, we study the regularity criterion of weak solutions of the three dimensional micropolar fluid flows. It is proved that if the pressure satisfies where denotes the critical Besov space, then the weak solution (u,w) becomes a regular solution on (0,T]. This regularity criterion can be regarded as log in time improvements of the standard Serrin's criteria established before. Copyright © 2011 John Wiley & Sons, Ltd.     

Existence and uniqueness of solutions of sequential nonlinear fractional difference equations with three‐point fractional sum boundary conditions

In this paper, we consider a discrete fractional boundary value problem of the form: where 0 < α,β≤1, 1 < α + β≤2, λ and ρ are constants, γ > 0, , is a continuous function, and E_βx(t) = x(t + β ? 1). The existence and uniqueness of solutions are proved by using Banach's fixed point theorem. An illustrative example is also presented. Copyright © 2014 John Wiley & Sons, Ltd.     

Existence of positive solutions for nonlinear Kirchhoff type problems in with critical Sobolev exponent

In this paper, we study the following nonlinear problem of Kirchhoff type with critical Sobolev exponent: where a, b > 0 are constants. Under certain assumptions on the sign‐changing function f(x,u), we prove the existence of positive solutions by variational methods. Our main results can be viewed as a partial extension of a recent result of He and Zou in [Journal of Differential Equations, 2012] concerning the existence of positive solutions to the nonlinear Kirchhoff problem where ϵ > 0 is a parameter, V (x) is a positive continuous potential, and with 4 < p < 6 and satisfies the Ambrosetti–Rabinowitz type condition. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite‐dimensional global attractor for a semi‐discrete fractional nonlinear Schrödinger equation

We consider a semi‐discrete in time Crank–Nicolson scheme to discretize a weakly damped forced nonlinear fractional Schrödinger equation u _t ?i (?Δ)^α u +i |u |²u +γ u =f for considered in the the whole space . We prove that such semi‐discrete equation provides a discrete infinite‐dimensional dynamical system in that possesses a global attractor in . We show also that if the external force is in a suitable weighted Lebesgue space, then this global attractor has a finite fractal dimension. Copyright © 2017 John Wiley & Sons, Ltd.     

Existence of positive solutions for singular impulsive differential equations with integral boundary conditions

In this paper, we study the existence of positive solutions for singular impulsive differential equations with integral boundary conditions where the nonlinearity f(t,u,v) may be singular at u = 0 and v = 0. The proof is based on the theory of Leray–Schauder degree, together with a truncation technique. Some recent results in the literature are generalized and improved. Copyright © 2014 John Wiley & Sons, Ltd.     

Asymptotic behavior to a chemotaxis consumption system with singular sensitivity

We consider a chemotaxis consumption system with singular sensitivity , v_t=εΔv−uv in a bounded domain with χ,α,ε>0. The global existence of classical solutions is obtained with n=1. Moreover, for any global classical solution (u,v) to the case of n,α≥1, it is shown that v converges to 0 in the L^∞‐norm as t→∞ with the decay rate established whenever ε∈(ε₀,1) with .     

Boundedness in a parabolic‐elliptic chemotaxis system with nonlinear diffusion and sensitivity and logistic source

In this paper, we study the zero‐flux chemotaxis‐system where Ω is a bounded and smooth domain of , n≥1, and where , k,μ>0 and α≤1. For any v≥0, the chemotactic sensitivity function is assumed to behave as the prototype χ(v)=χ₀/(1+av)², with a≥0 and χ₀>0. We prove that for any nonnegative and sufficiently regular initial data u(x,0), the corresponding initial‐boundary value problem admits a unique global bounded classical solution if α<1; indeed, for α=1, the same conclusion is obtained provided μ is large enough. Finally, we illustrate the range of dynamics present within the chemotaxis system in 1, 2, and 3 dimensions by means of numerical simulations.     

A Liouville theorem for weighted p−Laplace operator on smooth metric measure spaces

We prove that on a smooth metric measure space with m ?Bakry–Émery curvature bounded from below by ?(m ? 1)K for some constant K ≥0 (i.e., Ric_{f ,m} ≥?(m ? 1)K ), the following degenerate elliptic equation (0.1) has no nonconstant positive solution when p > 1 and constant λ _{f ,p} satisfies Our approach is based on the local Sobolev inequality and the Moser's iterative technique and is different from Cheng‐Yau's method, which was used by Wang‐Zhu in 2012 to derive a same Liouville theorem when 1 < p ≤2, Ric_{f ,m} ≥?(m ? 1)K and the sectional curvature is bounded from below. Copyright © 2016 John Wiley & Sons, Ltd.     

Galerkin finite element method for generalized Forchheimer equation of slightly compressible fluids in porous media

We consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for the density. We study Galerkin finite elements method for the initial boundary value problem. The existence and uniqueness of the approximation are proved. A prior estimates for the solutions in , time derivative in and gradient in , with a∈(0,1) are established. Error estimates for the density variable are derived in several norms for both continuous and discrete time procedures. Numerical experiments using backward Euler scheme confirm the theoretical analysis regarding convergence rates. Copyright © 2017 John Wiley & Sons, Ltd.     

Blow‐up analysis for a class of nonlinear reaction diffusion equations with Robin boundary conditions

This paper is devoted to the study of the blow‐up phenomena of following nonlinear reaction diffusion equations with Robin boundary conditions: Here, is a bounded convex domain with smooth boundary. With the aid of a differential inequality technique and maximum principles, we establish a blow‐up or non–blow‐up criterion under some appropriate assumptions on the functions f,g,ρ,k, and u₀. Moreover, we dedicate an upper bound and a lower bound for the blow‐up time when blowup occurs.     

Global existence and boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant and logistic source

This paper deals with the following chemotaxis system: under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. Here, δ and χ are some positive constants and f is a smooth function that satisfies with some constants a ?0,b  > 0, and γ  > 1. We prove that the classical solutions to the preceding system are global and bounded provided that Copyright © 2016 John Wiley & Sons, Ltd.     

Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source

We study a quasilinear parabolic–elliptic Keller–Segel system involving a source term of logistic type u_t = ? ? (?(u) ? u) ? χ ? ? (u ? v) + g(u), ? Δv = ? v + u in Ω × (0,T), subject to nonnegative initial data and the homogeneous Neumann boundary condition in a bounded domain with smooth boundary, n ≥ 1, χ > 0, ? ≥ c₁s^p for s ≥ s₀ > 1, and g(s) ≤ as ? μs² for s > 0 with a,g(0) ≥ 0, μ > 0. There are three nonlinear mechanisms included in the chemotaxis model: the nonlinear diffusion, aggregation and logistic absorption. The interaction among the triple nonlinearities shows that together with the nonlinear diffusion, the logistic absorption will dominate the aggregation such that the unique classical solution of the system has to be global in time and bounded, regardless of the initial data, whenever , or, equivalently, , which enlarge the parameter range , or , required by globally bounded solutions of the quasilinear K‐S system without the logistic source. Copyright © 2013 John Wiley & Sons, Ltd.