This paper investigates the separation property in binary phase-segregation processes modelled by Cahn-Hilliard type equations with constant mobility, singular entropy densities and different particle interactions. Under general assumptions on the entropy potential, we prove the strict separation property in both two and three-space dimensions. Namely, in 2D, we notably extend the minimal assumptions on the potential adopted so far in the literature, by only requiring a mild growth condition of its first derivative near the singular points $\pm 1$, without any pointwise additional assumption on its second derivative. For all cases, we provide a compact proof using De Giorgi’s iterations. In 3D, we also extend the validity of the asymptotic strict separation property to the case of fractional Cahn-Hilliard equation, as well as show the validity of the separation when the initial datum is close to an ‘energy minimizer’. Our framework offers insights into statistical factors like particle interactions, entropy choices and correlations governing separation, with broad applicability.

This paper deals with the following quasilinear chemotaxis system with consumption of chemoattractant

\[ \left\{\begin{array}{@{}ll} u_t=\Delta u^{m}-\nabla\cdot(u\nabla v),\quad & x\in \Omega,\quad t>0,\\ v_t=\Delta v-uv,\quad & x\in \Omega,\quad t>0\\ \end{array}\right. \]

$\Omega \subset \mathbb {R}^N(N=3,\,4,\,5)$

$\partial \Omega$

$m>\max \{1,\,\frac {3N-2}{2N+2}\}$

$(\bar {u}_0,\,0)$

$t\rightarrow \infty$

$\bar {u}_0=\frac {1}{|\Omega |}\int _\Omega u_0$

$m>\frac {3N-2}{2N}$

$N\geq 3$

$m>\frac {3N}{2N+2}$

$N\geq 2$

In Communications in Contemporary Mathematics 24 3, (2022),the authors have developed a method for constructing G-invariant partial differential equations (PDEs) imposed on hypersurfaces of an $(n+1)$-dimensional homogeneous space $G/H$, under mild assumptions on the Lie group G. In the present paper, the method is applied to the case when $G=\mathsf{PGL}(n+1)$ (respectively, $G=\mathsf{Aff}(n+1)$) and the homogeneous space $G/H$ is the $(n+1)$-dimensional projective $\mathbb{P}^{n+1}$ (respectively, affine $\mathbb{A}^{n+1}$) space, respectively. The main result of the paper is that projectively or affinely invariant PDEs with n independent and one unknown variables are in one-to-one correspondence with invariant hypersurfaces of the space of trace-free cubic forms in n variables with respect to the group $\mathsf{CO}(d,n-d)$ of conformal transformations of $\mathbb{R}^{d,n-d}$.

In this paper, we derive the effective model describing a thin-domain flow with permeable boundary through which the fluid is injected into the domain. We start with incompressible Stokes system and perform the rigorous asymptotic analysis. Choosing the appropriate scaling for the injection leads to a compressible effective model. In this paper, we derive the effective model describing a thin-domain flow with permeable boundary through which the fluid is injected into the domain. We start with incompressible Stokes system and perform the rigorous asymptotic analysis. Choosing the appropriate scaling for the injection leads to a compressible effective model.

In this paper, we are interested in positive solutions of

$$ \begin{align*}\left\{ \begin{array}{@{}ll} -\Delta u = a(x)v^{p-1}, \quad &\text{ in } \Omega,\\ -\Delta v = b(x)u^{q-1}, \quad &\text{ in } \Omega,\\ u,v>0, \quad &\text{ in } \Omega,\\ u=v=0, \quad &\text{ on } \partial\Omega, \end{array} \right. \end{align*} $$

$\Omega $

${\mathbb {R}}^N (N \ge 3)$

$ a(x), b(x)$

$\Bigl \lfloor \frac {N}{2} \Bigr \rfloor $

$\frac {N}{2}$

$ a(x)=b(x)=1$

$\Omega $

$$ \begin{align*}(p-1)(q-1)> \Big(1+\frac{2N}{\lambda_H}\Big)^2\left(\frac{q}{p}\right),\end{align*} $$

$\lambda _H$

$\Omega .$

$\lambda _H$

$q.$

We show a result on propagation of the anisotropic Gabor wave front set for linear operators with a tempered distribution Schwartz kernel. The anisotropic Gabor wave front set is parametrized by a positive parameter relating the space and frequency variables. The anisotropic Gabor wave front set of the Schwartz kernel is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by any partial differential operator with constant coefficients, real symbol and order at least two.

Motivated by the impact of worsening climate conditions on vegetation patches, we study dynamic instabilities in an idealised Ginzburg–Landau model. Our main results predict time instances of sudden drops in wavenumber and the resulting target states. The changes in wavenumber correspond to the annihilation of individual vegetation patches when resources are scarce and cannot support the original number of patches. Drops happen well after the primary pattern has destabilised at the Eckhaus boundary and key to distinguishing between the disappearance of 1,2 or more patches during the drop are complex spatio-temporal resonances in the linearisation at the unstable pattern. We support our results with numerical simulations and expect our results to be conceptually applicable universally near the Eckhaus boundary, in particular in more realistic models.

Introducing a pair-parameter matrix Mittag–Leffler function, we study the uniqueness and Hyers–Ulam stability to a new fractional nonlinear partial integro-differential equation with variable coefficients and a mixed boundary condition using Banach’s contractive principle as well as Babenko’s approach in a Banach space. These investigations have serious applications since uniqueness and stability analysis are essential topics in various research fields. The techniques used also work for different types of differential equations with initial or boundary conditions, as well as integral equations. Moreover, we present a Python code to compute approximate values of our newly established pair-parameter matrix Mittag–Leffler functions, which extend the multivariate Mittag–Leffler function. A few examples are given to show applications of the key results obtained.

Strong unique continuation properties and a classification of the asymptotic profiles are established for the fractional powers of a Schrödinger operator with a Hardy-type potential, by means of an Almgren monotonicity formula combined with a blow-up analysis.

In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential,

\[ (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\ \mathbb{R}^n, \]

$n \geq 1$

$0< s<1$

$\omega >-\lambda _{1,s}$

$2< p< {2n}/{(n-2s)^+}$

$\lambda _{1,s}>0$

$(-\Delta )^s + |x|^2$

$(-\Delta )^s$

$\mathcal {F}((-\Delta )^{s}u)(\xi )=|\xi |^{2s} \mathcal {F}(u)(\xi )$

$\xi \in \mathbb {R}^n$

$\mathcal {F}$

This article is concerned with the problem of determining an unknown source of non-potential, external time-dependent perturbations of an incompressible fluid from large-scale observations on the flow field. A relaxation-based approach is proposed for accomplishing this, which makes use of a nonlinear property of the equations of motions to asymptotically enslave small scales to large scales. In particular, an algorithm is introduced that systematically produces approximations of the flow field on the unobserved scales in order to generate an approximation to the unknown force; the process is then repeated to generate an improved approximation of the unobserved scales, and so on. A mathematical proof of convergence of this algorithm is established in the context of the two-dimensional Navier–Stokes equations with periodic boundary conditions under the assumption that the force belongs to the observational subspace of phase space; at each stage in the algorithm, it is shown that the model error, represented as the difference between the approximating and true force, asymptotically decreases to zero in a geometric fashion provided that sufficiently many scales are observed and certain parameters of the algorithm are appropriately tuned.

In this paper, we study the singular boundary value problem

$$ \begin{align*} \begin{cases} \Delta_\infty^h u=\lambda f(x,u,Du) \quad &\mathrm{in}\; \Omega, \\ u>0\quad &\mathrm{in}\; \Omega,\\ u=0 \quad &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*} $$

where $\lambda>0$ is a parameter, $h>1$ and $\Delta _\infty ^h u=|Du|^{h-3} \langle D^2uDu,Du \rangle $ is the highly degenerate and h-homogeneous operator related to the infinity Laplacian. The nonlinear term $f(x,t,p):\Omega \times (0,\infty )\times \mathbb {R}^{n}\rightarrow \mathbb {R}$ is a continuous function and may exhibit singularity at $t\rightarrow 0^{+}$. We establish the comparison principle by the double variables method for the general equation $\Delta _\infty ^h u=F(x,u,Du)$ under some conditions on the term $F(x,t,p)$. Then, we establish the existence of viscosity solutions to the singular boundary value problem in a bounded domain based on Perron’s method and the comparison principle. Finally, we obtain the existence result in the entire Euclidean space by the approximation procedure. In this procedure, we also establish the local Lipschitz continuity of the viscosity solution.

Existence of specific eternal solutions in exponential self-similar form to the following quasilinear diffusion equation with strong absorption

\[ \partial_t u=\Delta u^m-|x|^{\sigma}u^q, \]

$(t,\,x)\in (0,\,\infty )\times \mathbb {R}^N$

$m>1$

$q\in (0,\,1)$

$\sigma =\sigma _c:=2(1-q)/ (m-1)$

\[ u(t,x)={\rm e}^{-\alpha t}f(|x|\,{\rm e}^{\beta t}), \quad \alpha=\frac{2}{m-1}\beta, \]

$\beta ^*\in (0,\,\infty )$

$(u_A)_{A>0}$

$(f_A)_{A>0}$

$f_A(0)=A$

$f_A'(0)=0$

$\sigma \in (0,\,\sigma _c)$

For $s\in [\tfrac {1}{2},\, 1)$, let $u$ solve $(\partial _t - \Delta )^s u = Vu$ in $\mathbb {R}^{n} \times [-T,\, 0]$ for some $T>0$ where $||V||_{ C^2(\mathbb {R}^n \times [-T, 0])} < \infty$. We show that if for some $0<\mathfrak {K} < T$ and $\epsilon >0$

\[ {\unicode{x2A0D}}-_{[-\mathfrak{K},\, 0]} u^2(x, t) {\rm d}t \leq Ce^{-|x|^{2+\epsilon}}\ \forall x \in \mathbb{R}^n, \]

$u \equiv 0$

$\mathbb {R}^{n} \times [-T,\, 0]$

In this paper, we are concerned with the ground states of the following planar Kirchhoff-type problem:

\[ -\left(1+b\displaystyle\int_{\mathbb{R}^2}|\nabla u|^2\,{\rm d}x\right)\Delta u+\omega u=|u|^{p-2}u, \quad x\in\mathbb{R}^2. \]

$b,\, \omega >0$

$p>2$

We analyse the asymptotic dynamics of quasilinear parabolic equations when solutions may grow up (i.e. blow up in infinite time). For such models, there is a global attractor which is unbounded and the semiflow induces a nonlinear dynamics at infinity by means of a Poincaré projection. In case the dynamics at infinity is given by a semilinear equation, then it is gradient, consisting of the so-called equilibria at infinity and their corresponding heteroclinics. Moreover, the diffusion and reaction compete for the dimensionality of the induced dynamics at infinity. If the equilibria are hyperbolic, we explicitly prove the occurrence of heteroclinics between bounded equilibria and/or equilibria at infinity. These unbounded global attractors describe the space of admissible initial data at event horizons of certain black holes.

In this paper, we consider the following non-linear system involving the fractional Laplacian0.1

\begin{equation} \left\{\begin{array}{@{}ll} (-\Delta)^{s} u (x)= f(u,\,v), \\ (-\Delta)^{s} v (x)= g(u,\,v), \end{array} \right. \end{equation}

$0< s<1$

$x_n$

Conventional preytaxis systems assume that prey-tactic velocity is proportional to the prey density gradient. However, many experiments exploring the predator–prey interactions show that it is the predator’s acceleration instead of velocity that is proportional to the prey density gradient in the prey-tactic movement, which we call preytaxis with prey-induced acceleration. Mathematical models of preytaxis with prey-induced acceleration were proposed by Arditi et al. ((2001) Theor. Popul. Biol. 59(3), 207–221) and Sapoukhina et al. ((2003) Am. Nat. 162(1), 61–76) to interpret the spatial heterogeneity of predators and prey observed in experiments. This paper is devoted to exploring the qualitative behaviour of such preytaxis systems with prey-induced acceleration and establishing the global existence of classical solutions with uniform-in-time bounds in all spatial dimensions. Moreover, we prove the global stability of spatially homogeneous prey-only and coexistence steady states with decay rates under certain conditions on system parameters. For the parameters outside the stability regime, we perform linear stability analysis to find the possible patterning regimes and use numerical simulations to demonstrate that spatially inhomogeneous time-periodic patterns will typically arise from the preytaxis system with prey-induced acceleration. Noticing that conventional preytaxis systems are unable to produce spatial patterns, our results imply that the preytaxis with prey-induced acceleration is indeed more appropriate than conventional preytaxis to interpret the spatial heterogeneity resulting from predator–prey interactions.

For $s_1,\,s_2\in (0,\,1)$ and $p,\,q \in (1,\, \infty )$, we study the following nonlinear Dirichlet eigenvalue problem with parameters $\alpha,\, \beta \in \mathbb {R}$ driven by the sum of two nonlocal operators:

\[ (-\Delta)^{s_1}_p u+(-\Delta)^{s_2}_q u=\alpha|u|^{p-2}u+\beta|u|^{q-2}u\ \text{in }\Omega, \quad u=0\ \text{in } \mathbb{R}^d \setminus \Omega, \quad \mathrm{(P)} \]

$\Omega \subset \mathbb {R}^d$

$\alpha,\,\beta$

$(\alpha,\, \beta )$

$p$

$q$

This article is devoted to the analysis of the parabolic–parabolic chemotaxis system with multi-components over $\mathbb{R}^2$. The optimal small initial condition on the global existence of solutions for multi-species chemotaxis model in the fully parabolic situation had not been attained as far as the author knows. In this paper, we prove that under the sub-critical mass condition, any solutions to conflict-free system exist globally. Moreover, the global existence of solutions to system with strong self-repelling effect has been discussed even for large initial data. The proof is based on the modified free energy functional and the Moser–Trudinger inequality for system.