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In this paper we study a class of nonlinear quasi-linear diffusion equations involving the fractional p (·) -Laplacian with variable exponents, which is a fractional version of the nonhomogeneous p (·) -Laplace operator. The paper is divided into two parts. In the first part, under suitable conditions on the nonlinearity f, we analyze the problem (P 1) in a bounded domain Ω of R N and we establish the well-posedness of solutions by using techniques of monotone operators. We also study the large-time behaviour and extinction of solutions and we prove that the fractional p (·) -Laplacian operator generates a (nonlinear) submarkovian semigroup on L 2 (Ω). In the second part of the paper we establish the existence of global attractors for problem (P 2) under certain conditions in the potential V. Our results are new in the literature, both for the case of variable exponents and for the fractional p-laplacian case with constant exponent. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the initial value problem(IVP henceforth) associated with the higher order nonlinear dispersive equation given in Jones et al. (Int J Math Math Sci 24:371–377, 2000): ∂ t u + α ∂ x 7 u + β ∂ x 5 u + γ ∂ x 3 u + μ ∂ x u + λ u ∂ x u = 0 , x ∈ R , t ∈ R , u (x , 0) = u 0 (x) , x ∈ R with the initial data in the Sobolev space H s (R). Benefited from the ideas of Huo and Jia (Z Angew Math Phys 59:634–646, 2008), Zhang et al. (Acta Math Sci 37B(2):385–394, 2017) and Zhang and Huang (Math Methods Appl Sci 39(10):2488–2513, 2016) that is, using Fourier restriction norm method, Tao's [k, Z]-multiplier method and the contraction mapping principle, we prove that IVP is locally well-posed for the initial data u 0 ∈ H s (R) with s ≥ - 5 8 . Moreover, based on the local well-posedness and conservation law, we establish the global well-posedness for the initial data u 0 ∈ H s (R) with s = 0 . [ABSTRACT FROM AUTHOR]

In this paper, we study the time periodic traveling wave solutions for a periodic SIR epidemic model with diffusion and standard incidence. We establish the existence of periodic traveling waves by investigating the fixed points of a nonlinear operator defined on an appropriate set of periodic functions. Then we prove the nonexistence of periodic traveling via the comparison arguments combined with the properties of the spreading speed of an associated subsystem. [ABSTRACT FROM AUTHOR]

The first part of this paper is a general approach towards chaotic dynamics for a continuous map $$f:X\supset M\rightarrow X$$ which employs the fixed point index and continuation. The second part deals with the differential equation with state-dependent delay. For a suitable parameter $$\alpha $$ close to $$5\pi /2$$ we construct a delay functional $$d_{{\varDelta }}$$ , constant near the origin, so that the previous equation has a homoclinic solution, $$h(t)\rightarrow 0$$ as $$t\rightarrow \pm \infty $$ , with certain regularity properties of the linearization of the semiflow along the flowline $$t\mapsto h_t$$ . The third part applies the method from the beginning to a return map which describes solution behaviour close to the homoclinic loop, and yields the existence of chaotic motion. [ABSTRACT FROM AUTHOR]

We study the existence and nonexistence of traveling waves of a general diffusive Kermack-McKendrick SIR model with standard incidence where the total population is not constant. The three classes, susceptible S, infected I and removed R, are all involved in the traveling wave solutions. We show that the minimum wave speed of traveling waves for the three-dimensional non-monotonic system can be derived from its linearizaion at the initial disease-free equilibrium. The proof in this paper is based on Schauder fixed point theorem and Laplace transform. Our study provides a promising method to deal with high dimensional epidemic models. [ABSTRACT FROM AUTHOR]

We give sufficient conditions for the boundedness of the blow-up set and no boundary blow-up for type I blowing up solutions to a nonlinear parabolic system with unequal elliptic operators. We introduce a new method to simplify the nonlinear parabolic system into a scalar nonlinear parabolic inequality and investigate qualitative properties of the blow-up set. [ABSTRACT FROM AUTHOR]

In order to prove numerically the global existence and uniqueness of smooth solutions of a fourth order, nonlinear PDE, we derive rigorous a-posteriori upper bounds on the supremum of the numerical range of the linearized operator. These bounds also have to be easily computable in order to be applicable to our rigorous a-posteriori methods, as we use them in each time-step of the numerical discretization. The final goal is to establish global bounds on smooth local solutions, which then establish global uniqueness. [ABSTRACT FROM AUTHOR]