This paper is concerned with a nonlocal reaction–diffusion system with double free boundaries and two time delays. The free boundary problem describes the evolution of faecally–orally transmitted diseases. We first show the well-posedness of global solution, and then establish the monotonicity and asymptotic property of basic reproduction number for the epidemic model without delays, which is defined by spectral radius of the next infection operator. By introducing the generalized principal eigenvalue defined in general domain, we obtain an upper bound of the limit value of basic reproduction number. We discuss the spreading and vanishing phenomena in terms of the basic production number. By employing the perturbed approximation method and monotone iteration method, we establish the existence, uniqueness and monotonicity of solution to semi-wave problem. When spreading occurs, we determine the asymptotic spreading speeds of free boundaries by constructing suitable upper and lower solutions from the semi-wave solutions. Moreover, spreading speeds for partially degenerate diffusion case are provided in a similar way.

A delayed reaction-diffusion system with free boundaries is investigated in this paper to understand how the bacteria spread spatially to larger area from the initial infected habitat. Under the assumptions that the nonlinearities are of monostable type and the initial values satisfy some compatible condition, we show that the free boundary problem is well-posed and discuss the long-time behaviour of solution (including spreading and vanishing) in terms of the spatial-temporal risk index. Furthermore, to determine the spreading speed of free boundaries when spreading occurs, we first study the distribution of roots of a transcendental equation containing a polynomial of degree four and then establish the existence and uniqueness of monotone solution to a delay-induced nonlocal semi-wave problem by employing the approximation method, lower-upper solutions technique and Schauder fixed point theorem. It is shown that time delays slow down the spreading of bacteria.

We study the local and global existence and uniqueness of mild solution for a general class of abstract differential equations with state-dependent argument. In the last section, some examples on partial differential equations with state-dependent argument are presented.

In this paper, we consider the monotone travelling wave solutions of a reaction–diffusion epidemic system with nonlocal delays. We obtain the existence of monotone travelling wave solutions by applying abstract existence results. By transforming the nonlocal delayed system to a non-delayed system and choosing suitable small positive constants to define a pair of new upper and lower solutions, we use the contraction technique to prove the asymptotic stability (up to translation) of monotone travelling waves. Furthermore, the uniqueness and Lyapunov stability of monotone travelling wave solutions will be established with the help of the upper and lower solution method and the exponential asymptotic stability.

We give necessary and sufficient conditions of the ${{L}^{p}}$-well-posedness (resp. $B_{p,\,q}^{s}$-wellposedness) for the second order degenerate differential equation with finite delays

$${{\left( Mu \right)}^{\prime \prime }}\left( t \right)+B{u}'\left( t \right)+Au\left( t \right)=G{{{u}'}_{t}}+F{{u}_{t}}+f\left( t \right),\left( t\in \left[ 0,2\pi \right] \right)$$

with periodic boundary conditions $\left( Mu \right)\,\left( 0 \right)\,=\,\left( Mu \right)\,\left( 2\pi \right),\,{{\left( Mu \right)}^{\prime }}\left( 0 \right)\,=\,{{\left( Mu \right)}^{\prime }}\left( 2\pi \right)$, where $A,\,B,\,\text{and}\,M$ are closed linear operators on a complex Banach space $X$ satisfying $D\left( A \right)\,\cap \,D\left( B \right)\,\subset \,D\left( M \right)$, $F\,\text{and}\,G$ are bounded linear operators from ${{L}^{p}}\left( \left[ -2\pi ,\,0 \right];\,X \right)\,\left( \text{resp}\text{.}\,\text{B}_{p,q}^{s}\left( \left[ -2\pi ,\,0 \right];\,X \right) \right)$ into $X$.

In this paper, by using operator-valued ${\dot{C}}^{\unicode[STIX]{x1D6FC}}$-Fourier multiplier results on vector-valued Hölder continuous function spaces, we give a characterization of the $C^{\unicode[STIX]{x1D6FC}}$-well-posedness for the third order differential equations $au^{\prime \prime \prime }(t)+u^{\prime \prime }(t)=Au(t)+Bu^{\prime }(t)+f(t)$, ($t\in \mathbb{R}$), where $A,B$ are closed linear operators on a Banach space $X$ such that $D(A)\subset D(B)$, $a\in \mathbb{C}$ and $0<\unicode[STIX]{x1D6FC}<1$.

Using known operator-valued Fourier multiplier results on vector-valued Hölder continuous function spaces ${{C}^{\alpha }}(\mathbb{R};\,X)$, we completely characterize the ${{C}^{\alpha }}$-well-posedness of the first order degenerate differential equations with finite delay $(Mu{)}'(t)\,=\,Au(t)\,+\,F{{u}_{t}}\,+\,f(t)$ for $t\,\in \,\mathbb{R}$ by the boundedness of the $(M,\,F)$-resolvent of A under suitable assumption on the delay operator $F$, where $A,M$ are closed linear operators on a Banach space $X$ satisfying $D(A)\,\cap \,D(M)\,\ne \,\{0\}$, the delay operator $F$ is a bounded linear operator from $C([-r,0];X)$ to $X$, and $r\,>\,0$ is fixed.

Uniformization transforms a pseudo-Poisson process with unequal intensities (leaving rates) into one with uniform intensity. Self-transitions is the price to pay. Two intensities arise when one considers an absorbing barrier of a Markov process as a body in its own right: a pair of Markov processes intertwined by an extended Chapman–Kolmogorov equation naturally arises. We show that Sauer's two-state space empathy theory handles such intertwined processes. The price of self-transitions is also avoided.

We study the existence of solutions for a class of abstract impulsive differential equations. Our technical framework allows us to study partial differential equations with impulsive conditions involving partial derivatives and nonlinear expressions of the solution. Some applications to impulsive partial differential equations are presented.

In this paper we discuss the existence of mild and classical solutions for a class of abstract non-autonomous neutral functional differential equations. An application to partial neutral differential equations is considered.