This paper is focused on the existence and uniqueness of nonconstant steady states in a reaction–diffusion–ODE system, which models the predator–prey interaction with Holling-II functional response. Firstly, we aim to study the occurrence of regular stationary solutions through the application of bifurcation theory. Subsequently, by a generalized mountain pass lemma, we successfully demonstrate the existence of steady states with jump discontinuity. Furthermore, the structure of stationary solutions within a one-dimensional domain is investigated and a variety of steady-state solutions are built, which may exhibit monotonicity or symmetry. In the end, we create heterogeneous equilibrium states close to a constant equilibrium state using bifurcation theory and examine their stability.

This article is concerned with the spreading speed and traveling waves of a lattice prey–predator system with non-local diffusion in a periodic habitat. With the help of an associated scalar lattice equation, we derive the invasion speed for the predator. More specifically, when the dispersal kernel of the predator is exponentially bounded, the invasion speed is finite and can be characterized in terms of principal eigenvalues; while the dispersal kernel is algebraically decaying, the invasion speed is infinite and the accelerated spreading rate is obtained. Furthermore, the existence and non-existence of traveling waves connecting the semi-equilibrium point to a uniformly persistent state are established.

In this article, motivated by the regularity theory of the solutions of doubly nonlinear parabolic partial differential equations, the authors introduce the off-diagonal two-weight version of the parabolic Muckenhoupt class with time lag. Then the authors introduce the uncentered parabolic fractional maximal operator with time lag and characterize its two-weighted boundedness (including the endpoint case) in terms of these weights under an additional mild assumption (which is not necessary for one-weight case). The most novelty of this article exists in that the authors further introduce a new parabolic shaped domain and its corresponding parabolic fractional integral with time lag and, moreover, applying the aforementioned (two-)weighted boundedness of the parabolic fractional maximal operator with time lag, the authors characterize the (two-)weighted boundedness (including the endpoint case) of these parabolic fractional integrals in terms of the off-diagonal (two-weight) parabolic Muckenhoupt class with time lag; as applications, the authors further establish a parabolic weighted Sobolev embedding and a priori estimate for the solution of the heat equation. The key tools to achieve these include the parabolic Calderón–Zygmund-type decomposition, the chaining argument, and the parabolic Welland inequality, which is obtained by making the utmost of the geometrical relation between the parabolic shaped domain and the parabolic rectangle.

In this paper, we report the spatiotemporal dynamics of an intraguild predation (IGP)-type predator–prey model incorporating harvesting and prey-taxis. We first discuss the local and global existence of the classical solutions in N-dimensional space. It is found that the model has a global classical solution when controlling the prey-taxis coefficient in a certain range. Thereafter, we focus on the existence of the steady-state bifurcation. Moreover, we theoretically investigate the properties of the bifurcating solution near the steady-state bifurcation critical threshold. As a consequence, the spatial pattern formation of this model can be theoretically confirmed. Importantly, by means of rigorous theoretical derivation, we provide discriminant criteria on the stability of the bifurcating solution. Finally, the complicated patterns are numerically displayed. It is demonstrated that the harvesting and prey-taxis significantly affect the pattern formation of this IGP-type predator–prey model. Our main results of this paper reveal that (i) The repulsive prey-taxis could destabilize the spatial homogeneity, while the attractive prey-taxis effect and self-diffusion will stabilize the spatial homogeneity of this model. (ii) Numerical results suggest that over-harvesting for prey or predators is not advisable, it can lead to an ecological imbalance due to a significant reduction in population numbers. However, harvesting within a certain range is a feasible approach.

Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source

\begin{align*} |x|^{-2}\partial _tu=\Delta u+|x|^{\sigma }u^p, \quad (x,t)\in {\mathbb {R}}^N\times (0,T), \end{align*}

$p\gt 1$

$\sigma \ge -2$

$t\to \infty$

$\sigma =-2$

This paper is concerned with a predator–prey system with hunting cooperation and prey-taxis under homogeneous Neumann boundary conditions. We establish the existence of globally bounded solutions in two dimensions. In three or higher dimensions, the global boundedness of solutions is obtained for the small prey-tactic coefficient. By using hunting cooperation and prey species diffusion as bifurcation parameters, we conduct linear stability analysis and find that both hunting cooperation and prey species diffusion can drive the instability to induce Hopf, Turing and Turing–Hopf bifurcations in appropriate parameter regimes. It is also found that prey-taxis is a factor stabilizing the positive constant steady state. We use numerical simulations to illustrate various spatiotemporal patterns arising from the abovementioned bifurcations including spatially homogeneous and inhomogeneous time-periodic patterns, stationary spatial patterns and chaotic fluctuations.

The aim of this article is to extend the scope of the theory of regularity structures in order to deal with a large class of singular stochastic partial differential equations of the form

\begin{equation*}\partial_t u = \mathfrak{L} u+ F(u, \xi),\end{equation*}

where the differential operator $\mathfrak{L}$ fails to be elliptic. This is achieved by interpreting the base space $\mathbb{R}^{d}$ as a non-trivial homogeneous Lie group $\mathbb{G}$ such that the differential operator $\partial_t -\mathfrak{L}$ becomes a translation invariant hypoelliptic operator on $\mathbb{G}$. Prime examples are the kinetic Fokker-Planck operator $\partial_t -\Delta_v - v\cdot \nabla_x$ and heat-type operators associated with sub-Laplacians. As an application of the developed framework, we solve a class of parabolic Anderson type equations

\begin{equation*}\partial_t u = \sum_{i} X^2_i u + u (\xi-c)\end{equation*}

on the compact quotient of an arbitrary Carnot group.

We consider the harmonic map heat flow for maps $\mathbb {R}^{2} \to \mathbb {S}^2$. It is known that solutions to the initial value problem exhibit bubbling along a well-chosen sequence of times. We prove that every sequence of times admits a subsequence along which bubbling occurs. This is deduced as a corollary of our main theorem, which shows that the solution approaches the family of multi-bubble configurations in continuous time.

The present article is concerned with the Lyapunov stability of stationary solutions to the Allen–Cahn equation with a strong irreversibility constraint, which was first intensively studied in [2] and can be reduced to an evolutionary variational inequality of obstacle type. As a feature of the obstacle problem, the set of stationary solutions always includes accumulation points, and hence, it is rather delicate to determine the stability of such non-isolated equilibria. Furthermore, the strongly irreversible Allen–Cahn equation can also be regarded as a (generalized) gradient flow; however, standard techniques for gradient flows such as linearization and Łojasiewicz–Simon gradient inequalities are not available for determining the stability of stationary solutions to the strongly irreversible Allen–Cahn equation due to the non-smooth nature of the obstacle problem.

This paper is concerned with a singular limit of the Kobayashi–Warren–Carter system, a phase field system modelling the evolutions of structures of grains. Under a suitable scaling, the limit system is formally derived when the interface thickness parameter tends to zero. Different from many other problems, it turns out that the limit system is a system involving fractional time derivatives, although the original system is a simple gradient flow. A rigorous derivation is given when the problem is reduced to a gradient flow of a single-well Modica–Mortola functional in a one-dimensional setting.

We investigate a recent model proposed in the literature elucidating patterns driven by chemotaxis, similar to viscous fingering phenomena. Notably, this model incorporates a singular advection term arising from a modified formulation of Darcy’s law. It is noteworthy that this type of advection can also be well interpreted as a description of a radial fluid flow source surrounding an aggregation of cells. For the two-dimensional scenario, we establish a precise threshold delineating between blow-up and global solution existence. This threshold is contingent upon the pressure magnitude and the initial total mass of the aggregating cells.

Well-posedness in time-weighted spaces of certain quasilinear (and semilinear) parabolic evolution equations $u'=A(u)u+f(u)$ is established. The focus lies on the case of strict inclusions $\mathrm{dom}(f)\subsetneq \mathrm{dom}(A)$ of the domains of the nonlinearities $u\mapsto f(u)$ and $u\mapsto A(u)$. Based on regularizing effects of parabolic equations it is shown that a semiflow is generated in intermediate spaces. In applications this allows one to derive global existence from weaker a priori estimates. The result is illustrated by examples of chemotaxis systems.

This paper deals with a 4th-order parabolic equation involving the Frobenius norm of a Hessian matrix, subject to the Neumann boundary conditions. Some threshold results for blow-up or global or extinction solutions are obtained through classifying the initial energy and the Nehari energy. The bounds of blow-up time, decay estimates, and extinction rates are studied, respectively.

We study the global well-posedness and uniform boundedness of a two-dimensional reaction–advection–diffusion system with nonlinear advection. This strongly coupled system of nonlinear partial differential equations represents the continuum of a 2D lattice model designed to describe residential burglary, where each location is characterised by a tractability value that varies in both space and time. We show that the model with sublinear advection enhancement is globally well-posed, with a unique solution that is classical and uniformly bounded in time. Our results provide valuable insights into the development of urban crime models with nonlinear advection enhancements, making them suitable for broader applications, including nonlocal or heterogeneous near-repeat victimisation effects.

We prove the existence of solutions to the Kuramoto–Sivashinsky equation with low regularity data in function spaces based on the Wiener algebra and in pseudomeasure spaces. In any spatial dimension, we allow the data to have its antiderivative in the Wiener algebra. In one spatial dimension, we also allow data that are in a pseudomeasure space of negative order. In two spatial dimensions, we also allow data that are in a pseudomeasure space one derivative more regular than in the one-dimensional case. In the course of carrying out the existence arguments, we show a parabolic gain of regularity of the solutions as compared to the data. Subsequently, we show that the solutions are in fact analytic at any positive time in the interval of existence.

In this paper, we prove the global exstence of weak solutions for a porous medium dynamics of m species moving between two domains separated by a zero-thickness membrane. On this membrane, Kedem–Katchalsky conditions are considered, and the study is characterized by natural structural conditions applied to the nonlinear reactive terms. The global existence is established under the assumption that these reactive terms are bounded in $L^1$. This problem has already been analyzed in the linear diffusion case by Ciavolella and Perthame in Ciavolella and Perthame (2021, Journal of Evolution Equations 21, 1513–1540). The present work constitutes an extension for nonlinear diffusion, particularly of the porous medium type, in the form $\partial _t v_i - \Delta v_i^{r_i} = R_i$, for an exponent $r_i < 2$. The case $r_i \geq 2$ remains an open problem. This paper is an adaptation of the ideas from Ciavolella and Perthame (2021, Journal of Evolution Equations 21, 1513–1540), with new strategies to overcome the appearance of nonlinearity and degeneracy in the diffusion term.

Coffee berry diseases (CBD) pose significant threats to coffee production worldwide, affecting the livelihoods of millions of farmers and the global coffee market. Fractional calculus provides a powerful framework for describing non-local and memory-dependent phenomena, making it suitable for modelling the long-range interactions inherent in CBD spread. This study aims to formulate and analyse fractional order model for CBD transmission dynamics in the sense of Atangana–Baleanu–Caputo. Fixed point theorems were utilised to test the existence and uniqueness of the model’s solutions using fractional order. The basic reproduction number was calculated utilising the next-generation matrix. The model has locally asymptotically stable equilibrium positions (disease-free and endemic). Furthermore, the Lyapunov function was used to conduct a global stability analysis of the equilibrium locations. A numerical simulation of the CBD model was created using the fractional Adam–Bashforth–Moulton approach to validate the analytical findings. Our findings contribute to the development of more accurate predictive models and inform the design of targeted interventions to mitigate the impact of CBD on coffee production systems.

In this paper, we study the existence of travelling wave solutions and the spreading speed for the solutions of an age-structured epidemic model with nonlocal diffusion. Our proofs make use of the comparison principles both to construct suitable sub/super-solutions and to prove the regularity of travelling wave solutions.

We introduce a free boundary model to study the effect of vesicle transport onto neurite growth. It consists of systems of drift-diffusion equations describing the evolution of the density of antero- and retrograde vesicles in each neurite coupled to reservoirs located at the soma and the growth cones of the neurites, respectively. The model allows for a change of neurite length as a function of the vesicle concentration in the growth cones. After establishing existence and uniqueness for the time-dependent problem, we briefly comment on possible types of stationary solutions. Finally, we provide numerical studies on biologically relevant scales using a finite volume scheme. We illustrate the capability of the model to reproduce cycles of extension and retraction.

We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension,

\begin{equation*} u_t = D u_{xx} + u(1-\phi *u), \end{equation*}

$\phi *u$

$\phi (y) \equiv H\left (\frac{1}{4}-y^2\right )$

$u=1$

$D$

$\Delta _1 \approx 0.00297$

$D \ll 1$

$O(1)$

$u=O(1)$

$u$

$D \to 0^+$

$D \geq \Delta _1$

$2 \sqrt{D}$

$0 \lt D \lt \Delta _1$

$u=0$

$D$