This paper is concerned with two frequency-dependent susceptible–infected–susceptible epidemic reaction–diffusion models in heterogeneous environment, with a cross-diffusion term modelling the effect that susceptible individuals tend to move away from higher concentration of infected individuals. It is first shown that the corresponding Neumann initial-boundary value problem in an n-dimensional bounded smooth domain possesses a unique global classical solution which is uniformly in-time bounded regardless of the strength of the cross-diffusion and the spatial dimension n. It is further shown that, even in the presence of cross-diffusion, the models still admit threshold-type dynamics in terms of the basic reproduction number $\mathcal {R}_0$ – i.e. the unique disease-free equilibrium is globally stable if $\mathcal {R}_0\lt1$, while if $\mathcal {R}_0\gt1$, the disease is uniformly persistent and there is an endemic equilibrium (EE), which is globally stable in some special cases with weak chemotactic sensitivity. Our results on the asymptotic profiles of EE illustrate that restricting the motility of susceptible population may eliminate the infectious disease entirely for the first model with constant total population but fails for the second model with varying total population. In particular, this implies that such cross-diffusion does not contribute to the elimination of the infectious disease modelled by the second one.

The present work is part of a series of studies conducted by the authors on analytical models of avascular tumour growth that exhibit both geometrical anisotropy and physical inhomogeneity. In particular, we consider a tumour structure formed in distinct ellipsoidal regions occupied by cell populations at a certain stage of their biological cycle. The cancer cells receive nutrient by diffusion from an inhomogeneous supply and they are subject to also an inhomogeneous pressure field imposed by the tumour microenvironment. It is proved that the lack of symmetry is strongly connected to a special condition that should hold between the data imposed by the tumour’s surrounding medium, in order for the ellipsoidal growth to be realizable, a feature already present in other non-symmetrical yet more degenerate models. The nutrient and the inhibitor concentration, as well as the pressure field, are provided in analytical fashion via closed-form series solutions in terms of ellipsoidal eigenfunctions, while their behaviour is demonstrated by indicative plots. The evolution equation of all the tumour’s ellipsoidal interfaces is postulated in ellipsoidal terms and a numerical implementation is provided in view of its solution. From the mathematical point of view, the ellipsoidal system is the most general coordinate system that the Laplace operator, which dominates the mathematical models of avascular growth, enjoys spectral decomposition. Therefore, we consider the ellipsoidal model presented in this work, as the most general analytic model describing the avascular growth in inhomogeneous environment. Additionally, due to the intrinsic degrees of freedom inherited to the model by the ellipsoidal geometry, the ellipsoidal model presented can be adapted to a very populous class of avascular tumours, varying in figure and in orientation.

We consider the pricing of European options under a modified Black–Scholes equation having fractional derivatives in the “spatial” (price) variable. To be specific, the underlying price is assumed to follow a geometric Koponen–Boyarchenko–Levendorski process. This pure jump Lévy process could better capture the real behaviour of market data. Despite many difficulties caused by the “globalness” of the fractional derivatives, we derive an explicit closed-form analytical solution by solving the fractional partial differential equation analytically, using the Fourier transform technique. Based on the newly derived formula, we also examine, in theory, many basic properties of the option price under the current model. On the other hand, for practical purposes, we impose a reliable implementation method for the current formula so that it can be easily used in the trading market. With the numerical results, the impact of different parameters on the option price are also investigated.

We propose a numerical method for the simulation of a quasi-linear parabolic biofilm model that exhibits three non-linear diffusion effects: (i) a power law degeneracy, (ii) a super diffusion singularity and (iii) non-linear cross-diffusion. The method is based on a spatial Finite Volume discretisation in which cross-diffusion terms are formally treated as convection terms. Time-integration of the resulting semi-discretised system is carried out using an error-controlled, time-adaptive, embedded Rosenbrock–Wanner method. We compare several variants of the method and two variants of the model to investigate how details such as the choice cross-diffusion coefficients, and specific variants of the time integrator affect simulation time.

We study a class of parabolic equations which can be viewed as a generalized mean curvature flow acting on cylindrically symmetric surfaces with a Dirichlet condition on the boundary. We prove the existence of a unique solution by means of an approximation scheme. We also develop the theory of asymptotic stability for solutions of general parabolic problems.

We study the effect of algebraically localized impurities on striped phases in one spatial dimension. We therefore develop a functional-analytic framework that allows us to cast the perturbation problem as a regular Fredholm problem despite the presence of the essential spectrum, caused by the soft translational mode. Our results establish the selection of jumps in wavenumber and phase, depending on the location of the impurity and the average wavenumber in the system. We also show that, for select locations, the jump in the wavenumber vanishes.

We consider a curvature flow $V=\unicode[STIX]{x1D705}+A$ in a two-dimensional undulating cylinder $\unicode[STIX]{x1D6FA}$ described by $\unicode[STIX]{x1D6FA}:=\{(x,y)\in \mathbb{R}^{2}\mid -g_{1}(y)<x<g_{2}(y),y\in \mathbb{R}\}$, where $V$ is the normal velocity of a moving curve contacting the boundaries of $\unicode[STIX]{x1D6FA}$ perpendicularly, $\unicode[STIX]{x1D705}$ is its curvature, $A>0$ is a constant and $g_{1}(y),g_{2}(y)$ are positive smooth functions. If $g_{1}$ and $g_{2}$ are periodic functions and there are no stationary curves, Matano et al. [‘Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit’, Netw. Heterog. Media1 (2006), 537–568] proved the existence of a periodic travelling wave. We consider the case where $g_{1},g_{2}$ are general nonperiodic positive functions and the problem has some stationary curves. For each stationary curve $\unicode[STIX]{x1D6E4}$ unstable from above/below, we construct an entire solution growing out of it, that is, a solution curve $\unicode[STIX]{x1D6E4}_{t}$ which increases/decreases monotonically, converging to $\unicode[STIX]{x1D6E4}$ as $t\rightarrow -\infty$ and converging to another stationary curve or to $+\infty /-\infty$ as $t\rightarrow \infty$.

We study the dynamics of a reaction–diffusion–advection equation $u_{t}=u_{xx}-au_{x}+f(u)$ on the right half-line with Robin boundary condition $u_{x}=au$ at $x=0$, where $f(u)$ is a combustion nonlinearity. We show that, when $0<a<c$ (where $c$ is the travelling wave speed of $u_{t}=u_{xx}+f(u)$), $u$ converges in the $L_{loc}^{\infty }([0,\infty ))$ topology either to $0$ or to a positive steady state; when $a\geq c$, a solution $u$ starting from a small initial datum tends to $0$ in the $L^{\infty }([0,\infty ))$ topology, but this is not true for a solution starting from a large initial datum; when $a>c$, such a solution converges to $0$ in $L_{loc}^{\infty }([0,\infty ))$ but not in $L^{\infty }([0,\infty ))$ topology.

This paper is concerned with the global dynamics and spreading speeds of a partially degenerate non-local dispersal system with monostable nonlinearity in periodic habitats. We first obtain the existence of the principal eigenvalue for a periodic eigenvalue problem with partially degenerate non-local dispersal. Then we study the coexistence and extinction dynamics. Finally, the existence and characterization of spreading speeds are considered. In particular, we show that the spreading speed is linearly determinate. Overall, we extend the existing results on global dynamics and spreading speeds for the degenerate reaction–diffusion system to the degenerate non-local dispersal case. The extension is non-trivial and meaningful.

The paper studies large time behaviour of solutions to the Keller–Segel system with quadratic degradation in a liquid environment, as given by

under Neumann boundary conditions in a bounded domain Ω ⊂ ℝn, where n ≥ 1 is arbitrary. It is shown that whenever U : Ω × (0,∞) → ℝn is a bounded and sufficiently regular solenoidal vector field any non-trivial global bounded solution of (⋆) approaches the trivial equilibrium at a rate that, with respect to the norm in either of the spaces L1(Ω) and L∞(Ω), can be controlled from above and below by appropriate multiples of 1/(t + 1). This underlines that, even up to this quantitative level of accuracy, the large time behaviour in (⋆) is essentially independent not only of the particular fluid flow, but also of any effect originating from chemotactic cross-diffusion. The latter is in contrast to the corresponding Cauchy problem, for which known results show that in the n = 2 case the presence of chemotaxis can significantly enhance biomixing by reducing the respective spatial L1 norms of solutions.

A classification of the behaviour of the solutions f(·, a) to the ordinary differential equation (|f′|p-2f′)′ + f - |f′|p-1 = 0 in (0,∞) with initial condition f(0, a) = a and f′(0, a) = 0 is provided, according to the value of the parameter a > 0 when the exponent p takes values in (1, 2). There is a threshold value a* that separates different behaviours of f(·, a): if a > a*, then f(·, a) vanishes at least once in (0,∞) and takes negative values, while f(·, a) is positive in (0,∞) and decays algebraically to zero as r→∞ if a ∊ (0, a*). At the threshold value, f(·, a*) is also positive in (0,∞) but decays exponentially fast to zero as r→∞. The proof of these results relies on a transformation to a first-order ordinary differential equation and a monotonicity property with respect to a > 0. This classification is one step in the description of the dynamics near the extinction time of a diffusive Hamilton–Jacobi equation with critical gradient absorption and fast diffusion.

For bounded domains Ω, we prove that the Lp-norm of a regular function with compact support is controlled by weighted Lp-norms of its gradient, where the weight belongs to a class of symmetric non-negative definite matrix-valued functions. The class of weights is defined by regularity assumptions and structural conditions on the degeneracy set, where the determinant vanishes. In particular, the weight A is assumed to have rank at least 1 when restricted to the normal bundle of the degeneracy set S. This generalization of the classical Poincaré inequality is then applied to develop a robust theory of first-order Lp-based Sobolev spaces with matrix-valued weight A. The Poincaré inequality and these Sobolev spaces are then applied to produce various results on existence, uniqueness and qualitative properties of weak solutions to boundary-value problems for degenerate elliptic, degenerate parabolic and degenerate hyperbolic partial differential equations (PDEs) of second order written in divergence form, where A is calibrated to the matrix of coefficients of the second-order spatial derivatives. The notion of weak solution is variational: the spatial states belong to the matrix-weighted Sobolev spaces with p = 2. For the degenerate elliptic PDEs, the Dirichlet problem is treated by the use of the Poincaré inequality and Lax–Milgram theorem, while the treatment of the Cauchy–Dirichlet problem for the degenerate evolution equations relies only on the Poincaré inequality and the parabolic and hyperbolic counterparts of the Lax–Milgram theorem.

In this paper the existence and uniqueness of weak and strong solutions for a non-autonomous non-local reaction–diffusion equation is proved. Furthermore, the existence of minimal pullback attractors in the L2-norm in the frameworks of universes of fixed bounded sets and those given by a tempered growth condition is established, along with some relationships between them. Finally, we prove the existence of minimal pullback attractors in the H1-norm and study relationships among these new families and those given previously in the L2 context. We also present new results in the autonomous framework that ensure the existence of global compact attractors as a particular case.

We give a weak-Lp Serrin-type regularity criterion for a weak solution to the three-dimensional magnetohydrodynamics equations in a bounded domain Ω ⊂ ℝ3.

The convergence and blow-up results are established for the evolution of non-simple closed curves in an area-preserving curvature flow. It is shown that the global solution starting from a locally convex curve converges to an m-fold circle if the enclosed algebraic area A0 is positive, and evolves into a point if A0 = 0.

In this work, we examine the mathematical analysis and numerical simulation of pattern formation in a subdiffusive multicomponents fractional-reaction-diffusion system that models the spatial interrelationship between two preys and predator species. The major result is centered on the analysis of the system for linear stability. Analysis of the main model reflects that the dynamical system is locally and globally asymptotically stable. We propose some useful theorems based on the existence and permanence of the species to validate our theoretical findings. Reliable and efficient methods in space and time are formulated to handle any space fractional reaction-diffusion system. We numerically present the complexity of the dynamics that are theoretically discussed. The simulation results in one, two and three dimensions show some amazing scenarios.

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space, which evolve by an arbitrary (nonhomogeneous) function of the radii of curvature (RoC). We determine conditions for parabolic flows that ensure the boundedness of various geometric quantities and investigate some examples. As a new tool, we introduce the RoC diagram of a surface and its hyperbolic or anti-de Sitter metric. The relationship between the RoC diagram and the properties of Weingarten surfaces is also discussed.

Semi-analytical solutions are derived for the Brusselator system in one- and two-dimensional domains. The Galerkin method is processed to approximate the governing partial differential equations via a system of ordinary differential equations. Both steady-state concentrations and transient solutions are obtained. Semi-analytical results for the stability of the model are presented for the identified critical parameter value at which a Hopf bifurcation occurs. The impact of the diffusion coefficients on the system is also considered. The results show that diffusion acts to stabilize the systems better than the equivalent nondiffusive systems with the increasing critical value of the Hopf bifurcation. Comparison between the semi-analytical and numerical solutions shows an excellent agreement with the steady-state transient solutions and the parameter values at which the Hopf bifurcations occur. Examples of stable and unstable limit cycles are given, and Hopf bifurcation points are shown to confirm the results previously calculated in the Hopf bifurcation map. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with the numerical solutions of partial differential equations.

We introduce a multi-species chemotaxis type system admitting an arbitrarily large number of population species, all of which are attracted versus repelled by a single chemical substance. The production versus destruction rates of the chemotactic substance by the species is described by a probability measure. For such a model, we investigate the variational structures, in particular, we prove the existence of Lyapunov functionals, we establish duality properties as well as a logarithmic Hardy–Littlewood–Sobolev type inequality for the associated free energy. The latter inequality provides the optimal critical value for the conserved total population mass.