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.

A porous material that has been contaminated with a hazardous chemical agent is typically decontaminated by applying a cleanser solution to the surface and allowing the cleanser to react into the porous material, neutralising the agent. The agent and cleanser are often immiscible fluids and so, if the porous material is initially saturated with agent, a reaction front develops with the decontamination reaction occurring at this interface between the fluids. We investigate the effect of different initial agent configurations within the pore space on the decontamination process. Specifically, we compare the decontamination of a material initially saturated by the agent with the situation when, initially, the agent only coats the walls of the pores (referred to as the ‘agent-on-walls’ case). In previous work (Luckins et al., European Journal of Applied Mathematics, 31(5):782–805, 2020), we derived homogenised models for both of these decontamination scenarios, and in this paper we explore the solutions of these two models. We find that, for an identical initial volume of agent, the decontamination time is generally much faster for the agent-on-walls case compared with the initially saturated case, since the surface area on which the reaction can occur is greater. However for sufficiently deep spills of contaminant, or sufficiently slow reaction rates, decontamination in the agent-on-walls scenario can be slower. We also show that, in the limit of a dilute cleanser with a deep initial agent spill, the agent-on-walls model exhibits behaviour akin to a Stefan problem of the same form as that arising in the initially saturated model. The decontamination time is shown to decrease with both the applied cleanser concentration and the rate of the chemical reaction. However, increasing the cleanser concentration is also shown to result in lower decontamination efficiency, with an increase in the amount of cleanser chemical that is wasted.

In this article, we are concerned with the tempered fractional parabolic problem

$$ \begin{align*}\frac{\partial u}{\partial t}(x, t)-\left(\Delta+\lambda\right)^{\frac{\alpha}{2}} u(x, t)=f(u(x, t)), \end{align*} $$

where $-\left (\Delta +\lambda \right )^{\frac {\alpha }{2}}$ is a tempered fractional operator with $\alpha \in (0,2)$ and $\lambda $ is a sufficiently small positive constant. We first establish maximum principle principles for problems involving tempered fractional parabolic operators. And then, we develop the direct sliding methods for the tempered fractional parabolic problem, and discuss how they can be used to establish monotonicity results of solutions to the tempered fractional parabolic problem in various domains. We believe that our theory and methods can be conveniently applied to study parabolic problems involving other nonlocal operators.

Exact solutions are constructed for a class of nonlinear hyperbolic reaction-diffusion equations in two-space dimensions. Reduction of variables and subsequent solutions follow from a special nonclassical symmetry that uncovers a conditionally integrable system, equivalent to the linear Helmholtz equation. The hyperbolicity is commonly associated with a speed limit due to a delay, $\tau $, between gradients and fluxes. With lethal boundary conditions on a circular domain wherein a species population exhibits logistic growth of Fisher–KPP type with equal time lag, the critical domain size for avoidance of extinction does not depend on $\tau $. A diminishing exact solution within a circular domain is also constructed, when the reaction represents a weak Allee effect of Huxley type. For a combustion reaction of Arrhenius type, the only known exact solution that is finite but unbounded is extended to allow for a positive $\tau $.

We study curve-shortening flow for twisted curves in $\mathbb {R}^3$ (that is, curves with nowhere vanishing curvature $\kappa $ and torsion $\tau $) and define a notion of torsion-curvature entropy. Using this functional, we show that either the curve develops an inflection point or the eventual singularity is highly irregular (and likely impossible). In particular, it must be a Type-II singularity which admits sequences along which ${\tau }/{\kappa ^2} \to \infty $. This contrasts strongly with Altschuler’s planarity theorem, which shows that ${\tau }/{\kappa } \to 0$ along any essential blow-up sequence.

The dynamics of interfaces in slow diffusion equations with strong absorption are studied. Asymptotic methods are used to give descriptions of the behaviour local to a comprehensive range of possible singular events that can occur in any evolution. These events are: when an interface changes its direction of propagation (reversing and anti-reversing), when an interface detaches from an absorbing obstacle (detaching), when two interfaces are formed by film rupture (touchdown) and when the solution undergoes extinction. Our account of extinction and self-similar reversing and anti-reversing is built upon previous work; results on non-self-similar reversing and anti-reversing and on the various types of detachment and touchdown are developed from scratch. In all cases, verification of the asymptotic results against numerical solutions to the full PDE is provided. Self-similar solutions, both of the full equation and of its asymptotic limits, play a central role in the analysis.

In this paper, we consider the dynamical behaviour of a reaction–diffusion model for a population residing in a one-dimensional habit, with emphasis on the effects of boundary conditions and protection zone. We assume that the population is subjected to a strong Allee effect in its natural domain but obeys a monostable nonlinear growth in the protection zone $[L_1,\, L_2]$ with two constants satisfying $0\leq L_1< L_2$, and the general Robin condition is imposed on $x=0$ (i.e. $u(t,\,0)=bu_x(t,\,0)$ with $b\geq 0$). We show the existence of two critical values $0< L_*\leq L^*$, and prove that a vanishing–transition–spreading trichotomy result holds when the length of protection zone is smaller than $L_*$; a transition–spreading dichotomy result holds when the length of protection zone is between $L_*$ and $L^*$; only spreading happens when the length of protection zone is larger than $L^*$. Based on the properties of $L_*$, we obtain the precise strategies for an optimal protection zone: if $b$ is large (i.e. $b\geq 1/\sqrt {-g'(0)}$), the protection zone should start from somewhere near $0$; while if $b$ is small (i.e. $b< 1/\sqrt {-g'(0)}$), then the protection zone should start from somewhere away from $0$, and as far away from $0$ as possible.

We study a 3D ternary system which combines an interface energy with a long-range interaction term. Several such systems were derived as a sharp-interface limit of the Nakazawa–Ohta density functional theory of triblock copolymers. Both the binary case in 2D and 3D, and the ternary case in 2D, are quite well understood, whereas very little is known about the ternary case in 3D. In particular, it is even unclear whether minimizers are made of finitely many components. In this paper, we provide a positive answer to this, by proving that the number of components in a minimizer is bounded from above by a computable quantity depending only on the total masses and the interaction coefficients. There are two key difficulties, namely, the impossibility to decouple the long-range interaction from the perimeter term, and the absence of a quantitative isoperimetric inequality with two mass constraints in 3D. Therefore, the actual shape of minimizers is unknown, even for small masses, making the construction of suitable competing configurations significantly more delicate.

We first prove that the realization $A_{\mathrm {min}}$ of $A:={\operatorname {\mathrm {div}}}(Q\nabla )-V$ in $L^2({\mathbb {R}}^d)$ with unbounded coefficients generates a symmetric sub-Markovian and ultracontractive semigroup on $L^2({\mathbb {R}}^d)$ which coincides on $L^2({\mathbb {R}}^d)\cap C_b({\mathbb {R}}^d)$ with the minimal semigroup generated by a realization of $A$ on $C_b({\mathbb {R}}^d)$. Moreover, using time-dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernel of $A$ and deduce some spectral properties of $A_{\min }$ in the case of polynomially and exponentially growing diffusion and potential coefficients.

We carry out the extended symmetry analysis of an ultraparabolic Fokker–Planck equation with three independent variables, which is also called the Kolmogorov equation and is singled out within the class of such Fokker–Planck equations by its remarkable symmetry properties. In particular, its essential Lie invariance algebra is eight-dimensional, which is the maximum dimension within the above class. We compute the complete point symmetry pseudogroup of the Fokker–Planck equation using the direct method, analyse its structure and single out its essential subgroup. After listing inequivalent one- and two-dimensional subalgebras of the essential and maximal Lie invariance algebras of this equation, we exhaustively classify its Lie reductions, carry out its peculiar generalised reductions and relate the latter reductions to generating solutions with iterative action of Lie-symmetry operators. As a result, we construct wide families of exact solutions of the Fokker–Planck equation, in particular, those parameterised by an arbitrary finite number of arbitrary solutions of the (1+1)-dimensional linear heat equation. We also establish the point similarity of the Fokker–Planck equation to the (1+2)-dimensional Kramers equations whose essential Lie invariance algebras are eight-dimensional, which allows us to find wide families of exact solutions of these Kramers equations in an easy way.

The main purpose of this paper is to capture the asymptotic behaviour for solutions to a class of nonlinear elliptic and parabolic equations with the anisotropic weights consisting of two power-type weights of different dimensions near the degenerate or singular point, especially covering the weighted p-Laplace equations and weighted fast diffusion equations. As a consequence, we also establish the local Hölder estimates for their solutions in the presence of single power-type weights.

This paper concerns the monostable cooperative system with nonlocal diffusion and free boundaries, which has recently been discussed by Du and Ni [J. Differential equations 308(2021) 369-420 and arXiv:2010.01244]. We here aim at four aspects: the first is to give more accurate estimates for the longtime behaviours of the solution; the second is to discuss the limits of solution pair of a semi-wave problem; the third is to investigate the asymptotic behaviours of the corresponding Cauchy problem; the last is to study the limiting profiles of the solution as one of the expanding rates of free boundaries converges to $\infty$. Moreover, some epidemic models are given to illustrate their own rich longtime behaviours, which are quite different from those of the relevant existing works.

This work is devoted to the study of the sub-critical case of an anisotropic fully parabolic Keller–Segel chemotaxis system. We prove the existence of nonnegative weak solutions of (1.1) without restriction on the size of the initial data.

In this paper, we study the asymptotic profiles of positive solutions for diffusive logistic equations. The aim is to study the sharp effect of linear growth and nonlinear function. Both the classical reaction-diffusion equation and nonlocal dispersal equation are investigated. Our main results reveal that the linear and nonlinear parts of reaction term play quite different roles in the study of positive solutions.

New classes of conditionally integrable systems of nonlinear reaction–diffusion equations are introduced. They are obtained by extending a well-known nonclassical symmetry of a scalar partial differential equation to a vector equation. New exact solutions of nonlinear predator–prey systems with cross-diffusion are constructed. Infinite dimensional classes of exact solutions are made available for such nonlinear systems. Some of these solutions decay towards extinction and some oscillate or spiral around an interior fixed point. It is shown that the conditionally integrable systems are closely related to the standard diffusive Lotka–Volterra system, but they have additional features.

Loss functions with a large number of saddle points are one of the major obstacles for training modern machine learning (ML) models efficiently. First-order methods such as gradient descent (GD) are usually the methods of choice for training ML models. However, these methods converge to saddle points for certain choices of initial guesses. In this paper, we propose a modification of the recently proposed Laplacian smoothing gradient descent (LSGD) [Osher et al., arXiv:1806.06317], called modified LSGD (mLSGD), and demonstrate its potential to avoid saddle points without sacrificing the convergence rate. Our analysis is based on the attraction region, formed by all starting points for which the considered numerical scheme converges to a saddle point. We investigate the attraction region’s dimension both analytically and numerically. For a canonical class of quadratic functions, we show that the dimension of the attraction region for mLSGD is $\lfloor (n-1)/2\rfloor$, and hence it is significantly smaller than that of GD whose dimension is $n-1$.

We apply capacities to explore the space–time fractional dissipative equation: (0.1)

$$ \begin{align} \left\{\begin{aligned} &\partial^{\beta}_{t}u(t,x)=-\nu(-\Delta)^{\alpha/2}u(t,x)+f(t,x),\quad (t,x)\in\mathbb R^{1+n}_{+},\\ &u(0,x)=\varphi(x),\ x\in\mathbb R^{n}, \end{aligned}\right. \end{align} $$

where $\alpha>n$ and $\beta \in (0,1)$. In this paper, we focus on the regularity and the blow-up set of mild solutions to (0.1). First, we establish the Strichartz-type estimates for the homogeneous term $R_{\alpha ,\beta }(\varphi )$ and inhomogeneous term $G_{\alpha ,\beta }(g)$, respectively. Second, we obtain some space–time estimates for $G_{\alpha ,\beta }(g).$ Based on these estimates, we prove that the continuity of $R_{\alpha ,\beta }(\varphi )(t,x)$ and the Hölder continuity of $G_{\alpha ,\beta }(g)(t,x)$ on $\mathbb {R}^{1+n}_+,$ which implies a Moser–Trudinger-type estimate for $G_{\alpha ,\beta }.$ Then, for a newly introduced $L^{q}_{t}L^p_{x}$-capacity related to the space–time fractional dissipative operator $\partial ^{\beta }_{t}+(-\Delta )^{\alpha /2},$ we perform the geometric-measure-theoretic analysis and establish its basic properties. Especially, we estimate the capacity of fractional parabolic balls in $\mathbb {R}^{1+n}_+$ by using the Strichartz estimates and the Moser–Trudinger-type estimate for $G_{\alpha ,\beta }.$ A strong-type estimate of the $L^{q}_{t}L^p_{x}$-capacity and an embedding of Lorentz spaces are also derived. Based on these results, especially the Strichartz-type estimates and the $L^{q}_{t}L^p_{x}$-capacity of fractional parabolic balls, we deduce the size, i.e., the Hausdorff dimension, of the blow-up set of solutions to (0.1).

We investigate a reaction–diffusion problem in a two-component porous medium with a nonlinear interface condition between the different components. One component is connected and the other one is disconnected. The ratio between the microscopic pore scale and the size of the whole domain is described by the small parameter $\epsilon$. On the interface between the components, we consider a dynamic Wentzell-boundary condition, where the normal fluxes from the bulk domains are given by a reaction–diffusion equation for the traces of the bulk solutions, including nonlinear reaction kinetics depending on the solutions on both sides of the interface. Using two-scale techniques, we pass to the limit $\epsilon \to 0$ and derive macroscopic models, where we need homogenisation results for surface diffusion. To cope with the nonlinear terms, we derive strong two-scale convergence results.

In this paper, we consider an initial-boundary value problem of Hsieh's equation with conservative nonlinearity. The global unique solvability in the framework of Sobolev is established. In particular, one of our main motivations is to investigate the boundary layer (BL) effect and the convergence rates as the diffusion parameter $\beta$ goes zero. It is shown that the BL-thickness is of the order $O(\beta ^{\gamma })$ with $0<\gamma <\frac {1}{2}$. We need to point out that, different from the previous work on nonconservative form of Hsieh's equations, the conservative nonlinearity $(\psi ^{\beta }\theta ^{\beta })_x$ implies that new nonlinear term $\psi _x^{\beta }\theta ^{\beta }$ needs to be handled. It is important that more regularities on the solution to the limit problem are required to obtain the convergence rates and BL-thickness. It is more difficult for initial-boundary problem due to the lack of boundary conditions (especially, higher-order derivatives) prevents us from applying the integration by part to derive the energy estimates directly. Thus it is more complicated than the case of nonconservative form. Consequently more subtle mathematical analysis needs to be introduced to overcome the difficulties.

In a ball $\Omega \subset \mathbb {R}^{n}$ with $n\ge 2$, the chemotaxis system

\[ \left\{ \begin{array}{@{}l} u_t = \nabla \cdot \big( D(u)\nabla u\big) + \nabla\cdot \big(\dfrac{u}{v} \nabla v\big), \\ 0=\Delta v - uv \end{array} \right. \]

$u$

$v$

$D\in C^{3}([0,\infty ))$

$0< D(\xi ) \le {K_D} (\xi +1)^{-\alpha }$

$\xi >0$

${K_D}>0$

$\alpha >0$

$\Omega$