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Several types of differential equations, such as delay differential equations, age-structure models in population dynamics, evolution equations with boundary conditions, can be written as semilinear Cauchy problems with an operator which is not densely defined in its domain. The goal of this paper is to develop a center manifold theory for semilinear Cauchy problems with non-dense domain. Using Liapunov-Perron method and following the techniques of Vanderbauwhede et al. in treating infinite dimensional systems, we study the existence and smoothness of center manifolds for semilinear Cauchy problems with non-dense domain. As an application, we use the center manifold theorem to establish a Hopf bifurcation theorem for age structured models.

In this paper, we extend to the non-Hille–Yosida case a variation of constants formula for a nonautonomous and nonhomogeneous Cauchy problems first obtained by Gühring and Räbiger. By using this variation of constants formula, we derive a necessary and sufficient condition for the existence of an exponential dichotomy for the evolution family generated by the associated nonautonomous homogeneous problem. We also prove a persistence result of the exponential dichotomy for small perturbations. Finally, we illustrate our results by considering two examples. The first example is a parabolic equation with nonlocal and nonautonomous boundary conditions, and the second example is an age-structured model that is a hyperbolic equation.

The aim of this paper is to show how a weakly dispersive perturbation of the inviscid Burgers equation improves (enlarges) the space of resolution of the local Cauchy problem. More generally we will review several problems arising from weak dispersive perturbations of nonlinear hyperbolic equations or systems. [PUBLICATION ABSTRACT]

In this survey paper, we start with a discussion of the general fractional derivative (GFD) introduced by A. Kochubei in his recent publications. In particular, a connection of this derivative to the corresponding fractional integral and the Sonine relation for their kernels are presented. Then we consider some fractional ordinary differential equations (ODEs) with the GFD including the relaxation equation and the growth equation. The main part of the paper is devoted to the fractional partial differential equations (PDEs) with the GFD. We discuss both the Cauchy problems and the initial-boundary-value problems for the time-fractional diffusion equations with the GFD. In the final part of the paper, some results regarding the inverse problems for the differential equations with the GFD are presented.

In this article we consider an abstract Cauchy problem with the Hilfer fractional derivative and an almost sectorial operator. We introduce a suitable definition of a mild solution for this evolution equation and establish the existence result for a mild solution. We also give an example to highlight the applicability of theoretical results established.

We consider the Cauchy problem of three-dimensional isentropic compressible magnetohydrodynamic equations in the present paper. For regular initial data with small energy but possibly large oscillations, we prove the global well-posedness of classical solution, where the flow density is allowed to contain vacuum states, and the large-time behavior of the solution is also shown. [PUBLICATION ABSTRACT]

We study the existence and uniqueness of solution of a nonlinear Cauchy problem involving the ψ-Hilfer fractional derivative. In addition, we discuss the Ulam–Hyers and Ulam–Hyers–Rassias stabilities of its solutions. A few examples are presented to illustrate the possible applications of our main results.

This paper is devoted to the Cauchy problem for the generalized damped Boussinesq equation with a nonlinear source term in the natural energy space. With the help of linear time-space estimates, we establish the local existence and uniqueness of solutions by means of the contraction mapping principle. The global existence and blow-up of the solutions at both subcritical and critical initial energy levels are obtained. Moreover, we construct the sufficient conditions of finite time blow-up of the solutions with arbitrary positive initial energy.

Both propagation and collision behavior of δ ′ waves in a model of three partial differential equations are investigated. These are Cauchy problems for this model with initial values involving the first-order derivative of Dirac measure. Based on an α -solution concept defined in the framework of multiplication of distributions, we obtain a unique α -solution for the propagation process of an δ ′ wave. This α -solution shows rigorously the location, velocity and strengths of the δ ′ wave. Besides, with the help of this result, we also derive an α -solution for the collision behavior of two δ ′ waves. Such α -solution not only reveals, respectively, the location, speed and strengths of each one of the two δ ′ waves before their interaction, but also describes the location, velocity and strengths of a new emerged δ ′ wave after their collision. Moreover, this work can be applied to investigate the Cauchy problems for a system of partial differential equations with general initial values involving the derivative of the Dirac measure.

In this paper we propose a new method to stabilize nonsymmetric indefinite problems. The idea is to solve a forward and an adjoint problem simultaneously using a suitable stabilized finite element method. Both stabilization of the element residual and of the jumps of certain derivatives of the discrete solution over element faces may be used. Under the assumption of well-posedness of the partial differential equation and its associated adjoint problem we prove optimal error estimates in $H^1$ and $L^2$ norms in an abstract framework. Some examples of problems that are neither symmetric nor coercive but that enter the abstract framework are given. First we treat indefinite convection-diffusion equations with nonsolenoidal transport velocity and either pure Dirichlet conditions or pure Neumann conditions and then a Cauchy problem for the Helmholtz operator. Some numerical illustrations are given. [PUBLICATION ABSTRACT]