It is known that a polynomial local diffeomorphism $(f,\, g): {\mathbb {R}}^{2} \to {\mathbb {R}}^{2}$ is a global diffeomorphism provided the higher homogeneous terms of $f f_x+g g_x$ and $f f_y+g g_y$ do not have real linear factors in common. Here, we give a weight-homogeneous framework of this result. Our approach uses qualitative theory of differential equations. In our reasoning, we obtain a result on polynomial Hamiltonian vector fields in the plane, generalization of a known fact.

In this paper, we prove the existence and regularity of pullback attractors for non-autonomous nonclassical diffusion equations with nonlocal diffusion when the nonlinear term satisfies critical exponential growth and the external force term $h \in L_{l o c}^{2}(\mathbb {R} ; H^{-1}(\Omega )).$ Under some appropriate assumptions, we establish the existence and uniqueness of the weak solution in the time-dependent space $\mathcal {H}_{t}(\Omega )$ and the existence and regularity of the pullback attractors.

We consider the no-flux initial-boundary value problem for the cross-diffusive evolution system:

\begin{eqnarray*}\left\{ \begin{array}{ll}u_t = u_{xx} - \chi \big(\frac{u}{v} \partial_x v \big)_x - uv +B_1(x,t),\qquad & x\in \Omega, \ t>0, \\[1mm]v_t = v_{xx} +uv - v + B_2(x,t),\qquad & x\in \Omega, \ t>0,\end{array} \right.\end{eqnarray*}

$\chi=2$

In bounded intervals $\Omega\subset\mathbb{R}$ and with prescribed suitably regular non-negative functions $B_1$ and $B_2$, we first prove the existence of global classical solutions for any choice of $\chi>0$ and all reasonably regular non-negative initial data.

We next address the issue of determining the qualitative behaviour of solutions under appropriate assumptions on the asymptotic properties of $B_1$ and $B_2$. Indeed, for arbitrary $\chi>0$, we obtain boundedness of the solutions given strict positivity of the average of $B_2$ over the domain; moreover, it is seen that imposing a mild decay assumption on $B_1$ implies that u must decay to zero in the long-term limit. Our final result, valid for all $\chi\in\left(0,\frac{\sqrt{6\sqrt{3}+9}}{2}\right),$ which contains the relevant value $\chi=2$, states that under the above decay assumption on $B_1$, if furthermore $B_2$ appropriately stabilises to a non-trivial function $B_{2,\infty}$, then (u,v) approaches the limit $(0,v_\infty)$, where $v_\infty$ denotes the solution of

\begin{eqnarray*}\left\{ \begin{array}{l}-\partial_{xx}v_\infty + v_\infty = B_{2,\infty},\qquad x\in \Omega, \\[1mm]\partial_x v_{\infty}=0,\qquad x\in\partial\Omega.\end{array} \right.\end{eqnarray*}

$\chi$

$\chi$

The Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation is one of the prototypical reaction–diffusion equations and is encountered in many areas, primarily in population dynamics. An important consideration for the phenomena modelled by diffusion equations is the length of the diffusive process. In this paper, three definitions of the critical time are given, and bounds are obtained by a careful construction of the upper and lower solutions. The comparison functions satisfy the nonlinear, but linearizable, partial differential equations of Fisher–KPP type. Results of the numerical simulations are displayed. Extensions to some classes of reaction–diffusion systems and an application to a spatially heterogeneous harvesting model are also presented.

This paper presents the current state of mathematical modelling of the electrochemical behaviour of lithium-ion batteries (LIBs) as they are charged and discharged. It reviews the models developed by Newman and co-workers, both in the cases of dilute and moderately concentrated electrolytes and indicates the modelling assumptions required for their development. Particular attention is paid to the interface conditions imposed between the electrolyte and the active electrode material; necessary conditions are derived for one of these, the Butler–Volmer relation, in order to ensure physically realistic solutions. Insight into the origin of the differences between various models found in the literature is revealed by considering formulations obtained by using different measures of the electric potential. Materials commonly used for electrodes in LIBs are considered and the various mathematical models used to describe lithium transport in them discussed. The problem of upscaling from models of behaviour at the single electrode particle scale to the cell scale is addressed using homogenisation techniques resulting in the pseudo-2D model commonly used to describe charge transport and discharge behaviour in lithium-ion cells. Numerical solution to this model is discussed and illustrative results for a common device are computed.

In this paper we study the existence and the analytic dependence upon domain perturbation of the solutions of a nonlinear nonautonomous transmission problem for the Laplace equation. The problem is defined in a pair of sets consisting of a perforated domain and an inclusion whose shape is determined by a suitable diffeomorphism $\phi$. First we analyse the case in which the inclusion is a fixed domain. Then we will perturb the inclusion and study the arising boundary value problem and the dependence of a specific family of solutions upon the perturbation parameter $\phi$.

For $1< p<\infty$ we prove an $L^{p}$-version of the generalized trace-free Korn inequality for incompatible tensor fields $P$ in $W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$. More precisely, let $\Omega \subset \mathbb {R}^{3}$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that

\[ \lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\leq c\,\left(\lVert{\operatorname{dev} \operatorname{sym} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \]

$P\in W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$

$P\in W^{1,p} (\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$

$P\times \nu =0$

$\partial \Omega$

$\nu$

$\partial \Omega$

$\operatorname {dev} P : = P -\frac 13 \operatorname {tr}(P) {\cdot } {\mathbb {1}}$

$P$

\begin{align*} &\lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}+\lVert{ \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\\ &\quad\leq c\,\left(\lVert{P}\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \end{align*}

$P\in W^{1,p}(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$

$\Gamma \subseteq \partial \Omega$

In this article, we give a comprehensive characterization of $L^1$-summability for the Navier-Stokes flows in the half space, which is a long-standing problem. The main difficulties are that $L^q-L^r$ estimates for the Stokes flow don't work in this end-point case: $q=r=1$; the projection operator $P: L^1\longrightarrow L^1_\sigma$ is not bounded any more; useful information on the pressure function is missing, which arises in the net force exerted by the fluid on the noncompact boundary. In order to achieve our aims, by making full use of the special structure of the half space, we decompose the pressure function into two parts. Then the knotty problem of handling the pressure term can be transformed into establishing a crucial and new weighted $L^1$-estimate, which plays a fundamental role. In addition, we overcome the unboundedness of the projection $P$ by solving an elliptic problem with homogeneous Neumann boundary condition.

We prove that immersions of planar domains are uniquely specified by their Jacobian determinant, curl function and boundary values. This settles the two-dimensional version of an outstanding conjecture related to a particular grid generation method in computer graphics.

In this paper, we study the existence and concentration behaviour of multi-peak standing waves for a singularly perturbed Davey–Stewartson system, which arises in the theory of shallow water waves. For this purpose, we first give a sharp threshold of the existence of ground-state solutions to the related limiting problem. Next, combining the penalization method and the regularity theory of elliptic equations, we construct a family of positive solutions concentrating around any prescribed finite set of local minima, possibly degenerate, of the potential. A feature of this analysis is that we do not need any uniqueness or non-degeneracy conditions for the limiting equation. To the best of our knowledge, this paper is the first study dealing with the study of concentrating solutions for Davey–Stewartson systems. We emphasize that with respect to the classical Schrödinger equation, the presence of a singular integral operator in the Davey–Stewartson system forces the implementation of new ideas to obtain the existence of multi-peak solutions.

We prove existence results of two solutions of the problem

\[ \begin{cases} L(u)+u^{m-1}=\lambda u^{p-1} & \text{in}\ \Omega,\\ u>0 & \text{in}\ \Omega,\\ u=0 & \text{on}\ \partial \Omega, \end{cases} \]

$L(v)=-\textrm {div}(M(x)\nabla v)$

$p\in (2,2^{*}]$

$\lambda$

$m$

$m\to +\infty$

In this paper, we investigate the existence and nonexistence of results for a class of Hamiltonian-Choquard-type elliptic systems. We show the nonexistence of classical nontrivial solutions for the problem

\[ \begin{cases} -\Delta u + u= ( I_{\alpha} \ast |v|^{p} )v^{p-1} \text{ in } \mathbb{R}^{N},\\ -\Delta v + v= ( I_{\beta} \ast |u|^{q} )u^{q-1} \text{ in } \mathbb{R}^{N}, \\ u(x),v(x) \rightarrow 0 \text{ when } |x|\rightarrow \infty, \end{cases} \]

$(N+\alpha )/p + (N+\beta )/q \leq 2(N-2)$

$N\geq 3$

$(N+\alpha )/p + (N+\beta )/q \geq 2N$

$N=2$

$I_{\alpha }$

$I_{\beta }$

\[ \begin{cases} -\Delta u + u= (I_{\alpha} \ast |v|^{\frac{\alpha}{2}+1})|v|^{\frac{\alpha}{2}-1}v + g(v) \hbox{ in } \mathbb{R}^{2},\\ - \Delta v + v= (I_{\beta} \ast |u|^{\frac{\beta}{2}+1})|u|^{\frac{\beta}{2}-1}u + f(u), \hbox{ in } \mathbb{R}^{2},\\ u,v \in H^{1}(\mathbb{R}^{2}), \end{cases} \]

$\alpha ,\,\beta \in (0,\,2)$

$f,\,g$

We compare the solutions of two Poisson problems in a spherical shell with Robin boundary conditions, one with given data, and one where the data have been cap symmetrized. When the Robin parameters are nonnegative, we show that the solution to the symmetrized problem has larger convex means. Sending one of the Robin parameters to $+\infty $, we obtain mixed Robin/Dirichlet comparison results in shells. We prove similar results on balls and prove a comparison principle on generalized cylinders with mixed Robin/Neumann boundary conditions.

In this paper, we study some properties of the generalized Fokker–Planck equation induced by the time-changed fractional Ornstein–Uhlenbeck process. First of all, we exploit some sufficient conditions to show that a mild solution of such equation is actually a classical solution. Then, we discuss an isolation result for mild solutions. Finally, we prove the weak maximum principle for strong solutions of the aforementioned equation and then a uniqueness result.

In this paper, we study a dissipative systems modelling electrohydrodynamics in incompressible viscous fluids. The system consists of the Navier–Stokes equations coupled with a classical Poisson–Nernst–Planck equations. In the three-dimensional case, we establish a global regularity criteria in terms of the middle eigenvalue of the strain tensor in the framework of the anisotropic Lorentz spaces for local smooth solution. The proof relies on the identity for entropy growth introduced by Miller in the Arch. Ration. Mech. Anal. [16].

In this study, we continue our study of the Cauchy problem associated with the Brinkman equations [see (1.1) and (1.2) below] which model fluid flow in certain types of porous media. Here, we will consider the flow in the upper half-space

\[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R}^{3}\left\vert z\geqslant 0\right.\right\}, \]

$z=0$

This paper is concerned with the global regularity problem on the micropolar Rayleigh-Bénard problem with only velocity dissipation in $\mathbb {R}^{d}$ with $d=2\ or\ 3$. By fully exploiting the special structure of the system, introducing two combined quantities and using the technique of Littlewood-Paley decomposition, we establish the global regularity of solutions to this system in $\mathbb {R}^{2}$. Moreover, we obtain the global regularity for fractional hyperviscosity case in $\mathbb {R}^{3}$ by employing various techniques including energy methods, the regularization of generalized heat operators on the Fourier frequency localized functions and logarithmic Sobolev interpolation inequalities.

In this study, we investigate the intial value problem (IVP) for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn–Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild solution as well as the smoothing effects of resolvent operators are proved. For the IVP associated with the first one, by using the Orlicz space with the function $\Xi (z)={\textrm {e}}^{|z|^{p}}-1$ and some embeddings between it and the usual Lebesgue spaces, we prove that the solution is a global-in-time solution or it shall blow up in a finite time if the initial value is regular. In the case of singular initial data, the local-in-time/global-in-time existence and uniqueness are derived. Also, the regularity of the mild solution is investigated. For the IVP associated with the second one, some modifications to the generalized formula are made to deal with the nonlinear term. We also establish some important estimates for the derivatives of resolvent operators, they are the basis for using the Picard sequence to prove the local-in-time existence of the solution.

We consider patterns formed in a two-dimensional thin film on a planar substrate with a Derjaguin disjoining pressure and periodic wettability stripes. We rigorously clarify some of the results obtained numerically by Honisch et al. [Langmuir 31: 10618–10631, 2015] and embed them in the general theory of thin-film equations. For the case of constant wettability, we elucidate the change in the global structure of branches of steady-state solutions as the average film thickness and the surface tension are varied. Specifically we find, by using methods of local bifurcation theory and the continuation software package AUTO, both nucleation and metastable regimes. We discuss admissible forms of spatially non-homogeneous disjoining pressure, arguing for a form that differs from the one used by Honisch et al., and study the dependence of the steady-state solutions on the wettability contrast in that case.

This paper is concerned with a prey–predator model with population flux by attractive transition. Our previous paper (Oeda and Kuto, 2018, Nonlinear Anal. RWA, 44, 589–615) obtained a bifurcation branch (connected set) of coexistence steady states which connects two semitrivial solutions. In Oeda and Kuto (2018, Nonlinear Anal. RWA, 44, 589–615), we also showed that any positive steady-state approaches a positive solution of either of two limiting systems, and moreover, one of the limiting systems is an equal diffusive competition model. This paper obtains the bifurcation structure of positive solutions to the other limiting system. Moreover, this paper implies that the global bifurcation branch of coexistence states consists of two parts, one of which is a simple curve running in a tubular domain near the set of positive solutions to the equal diffusive competition model, the other of which is a connected set characterized by positive solutions to the other limiting system.