We establish some existence results for a class of critical $N$-Laplacian problems in a bounded domain in $\mathbb {R}^{N}$. In the absence of a suitable direct sum decomposition of the underlying Sobolev space to which the classical linking theorem can be applied, we use an abstract linking theorem based on the $\mathbb {Z}_2$-cohomological index to obtain a non-trivial critical point.

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

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

$u$

$v$

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

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

$\xi >0$

${K_D}>0$

$\alpha >0$

$\Omega$

We prove the existence of nontrivial ground state solutions of the critical quasilinear Hénon equation $\displaystyle -\Delta _p u=|x|^{\alpha _1}|u|^{p^{*}(\alpha _1)-2}u-|x|^{\alpha _2}|u|^{p^{*}(\alpha _2)-2}u\ \ {\rm in}\ \mathbb {R}^{N}.$ It is a new problem in the sense that the signs of the coefficients of critical terms are opposite.

Savin [‘ $\mathcal {C}^{1}$ regularity for infinity harmonic functions in two dimensions’, Arch. Ration. Mech. Anal. 3(176) (2005), 351–361] proved that every planar absolutely minimizing Lipschitz (AML) function is continuously differentiable whenever the ambient space is Euclidean. More recently, Peng et al. [‘Regularity of absolute minimizers for continuous convex Hamiltonians’, J. Differential Equations 274 (2021), 1115–1164] proved that this property remains true for planar AML functions for certain convex Hamiltonians, using some Euclidean techniques. Their result can be applied to AML functions defined in two-dimensional normed spaces with differentiable norm. In this work we develop a purely non-Euclidean technique to obtain the regularity of planar AML functions in two-dimensional normed spaces with differentiable norm.

This article deals with kinetic Fokker–Planck equations with essentially bounded coefficients. A weak Harnack inequality for nonnegative super-solutions is derived by considering their log-transform and adapting an argument due to S. N. Kružkov (1963). Such a result rests on a new weak Poincaré inequality sharing similarities with the one introduced by W. Wang and L. Zhang in a series of works about ultraparabolic equations (2009, 2011, 2017). This functional inequality is combined with a classical covering argument recently adapted by L. Silvestre and the second author (2020) to kinetic equations.

In this paper, we establish a new fractional interpolation inequality for radially symmetric measurable functions on the whole space $R^{N}$ and a new compact imbedding result about radially symmetric measurable functions. We show that the best constant in the new interpolation inequality can be achieved by a radially symmetric function. As applications of this compactness result, we study the existence of ground states of the nonlinear fractional Schrödinger equation on the whole space $R^{N}$. We also prove an existence result of standing waves and prove their orbital stability.

This paper focuses on a 2D magnetohydrodynamic system with only horizontal dissipation in the domain $\Omega = \mathbb {T}\times \mathbb {R}$ with $\mathbb {T}=[0,\,1]$ being a periodic box. The goal here is to understand the stability problem on perturbations near the background magnetic field $(1,\,0)$. Due to the lack of vertical dissipation, this stability problem is difficult. This paper solves the desired stability problem by simultaneously exploiting two smoothing and stabilizing mechanisms: the enhanced dissipation due to the coupling between the velocity and the magnetic fields, and the strong Poincaré type inequalities for the oscillation part of the solution, namely the difference between the solution and its horizontal average. In addition, the oscillation part of the solution is shown to converge exponentially to zero in $H^{1}$ as $t\to \infty$. As a consequence, the solution converges to its horizontal average asymptotically.

In this paper, we consider the following two-component Schrödiner system

\[ \begin{cases} -\Delta u_1 +V_1(x) u_1= \mu_1 u_1^{3} +\beta u_1u_2^{2} & \text{in}\;\mathbb{R}^{3},\\ -\Delta u_2+V_2(x) u_2= \mu_2 u_2^{3} +\beta u_1^{2}u_2 & \text{in}\;\mathbb{R}^{3}. \end{cases} \]

This paper considers a model for oncolytic virotherapy given by the doubly haptotactic cross-diffusion system

\[ \left\{\begin{array}{@{}ll} u_t=D_u\Delta u-\xi_u\nabla\cdot(u\nabla v)+\mu_u u(1-u)-\rho uz,\\ v_t={-} (\alpha_u u+\alpha_w w)v,\\ w_t=D_w\Delta w-\xi_w\nabla\cdot(w\nabla v)- w+\rho uz,\\ z_t=D_z\Delta z-\delta_z z- \rho uz+\beta w, \end{array}\right. \]

$D_u,D_w,D_z,\xi _u,\xi _w,\delta _z,\rho$

$\alpha _u,\alpha _w,\mu _u,\beta$

$\Omega \subset {\mathbb {R}}^{2}$

$\beta <1$

$(u,v,w,z)$

$(1, 0, 0, 0)$

$(L^{\infty }(\Omega ))^{4}$

$t\rightarrow \infty$

We study a variant of the mean curvature flow for closed, convex hypersurfaces where the normal velocity is a non-homogeneous function of the principal curvatures. We show that if the initial hypersurface satisfies a certain pinching condition, then this is preserved and the flow converges to a sphere under rescaling.

We prove the existence of a solution for a class of activator–inhibitor system of type $- \Delta u +u = f(u) -v$ , $-\Delta v+ v=u$ in $\mathbb{R}^{N}$ . The function f is a general nonlinearity which can grow polynomially in dimension $N\geq 3$ or exponentiallly if $N=2$ . We are able to treat f when it has critical growth corresponding to the Sobolev space we work with. We transform the system into an equation with a nonlocal term. We find a critical point of the corresponding energy functional defined in the space of functions with norm endowed by a scalar product that takes into account such nonlocal term. For that matter, and due to the lack of compactness, we deal with weak convergent minimizing sequences and sequences of Lagrange multipliers of an action minima problem.

In this paper, we prove several results on the exponential decay in $L^{2}$ norm of the KdV equation on the real line with localized dampings. First, for the linear KdV equation, the exponential decay holds if and only if the averages of the damping coefficient on all intervals of a fixed length have a positive lower bound. Moreover, under the same damping condition, the exponential decay holds for the (nonlinear) KdV equation with small initial data. Finally, with the aid of certain properties of propagation of regularity in Bourgain spaces for solutions of the associated linear system and the unique continuation property, the exponential decay for the KdV equation with large data holds if the damping coefficient has a positive lower bound on $E$, where $E$ is equidistributed over the real line and the complement $E^{c}$ has a finite Lebesgue measure.

This corrigendum corrects a result in [1].

We consider the following inhomogeneous problems

\[ \begin{cases} \epsilon^{2}\mbox{div}(a(x)\nabla u(x))+f(x,u)=0 & \text{ in }\Omega,\\ \frac{\partial u}{\partial \nu}=0 & \text{ on }\partial \Omega,\\ \end{cases} \]

$\Omega$

$\mathbb {R}^{N}$

$\epsilon >0$

$a$

$L^{1}$

In this study, we consider the nonclassical diffusion equations with time-dependent memory kernels

\begin{equation*} u_{t} - \Delta u_t - \Delta u - \int_0^\infty k^{\prime}_{t}(s) \Delta u(t-s) ds + f( u) = g \end{equation*}

$\Omega \subset \mathbb{R}^N,\, N\geq 3$

$\mathcal{A}=\{A_t\}_{t\in\mathbb{R}}$

$H_0^1(\Omega)\times L^2_{\mu_t}(\mathbb{R}^+,H_0^1(\Omega))$

$k_t$

$m\delta_0$

$t\to \infty$

\begin{equation*}u_{t} - \Delta u_{t} - (1+m)\Delta u + f( u) = g. \end{equation*}

$k_t(\!\cdot\!)$

We consider the following class of quasilinear Schrödinger equations proposed in plasma physics and nonlinear optics $-\Delta u+V(x)u+\frac {\kappa }{2}[\Delta (u^{2})]u=h(u)$ in the whole two-dimensional Euclidean space. We establish the existence and qualitative properties of standing wave solutions for a broader class of nonlinear terms $h(s)$ with the critical exponential growth. We apply the dual approach to obtain solutions in the usual Sobolev space $H^{1}(\mathbb {R}^{2})$ when the parameter $\kappa >0$ is sufficiently small. Minimax techniques, Trudinger–Moser inequality and the Nash–Moser iteration method play an essential role in establishing our results.

We study the optimal investment strategy to minimize the probability of lifetime ruin under a general mortality hazard rate. We explore the error between the minimum probability of lifetime ruin and the achieved probability of lifetime ruin if one follows a simple investment strategy inspired by earlier work in this area. We also include numerical examples to illustrate the estimation. We show that the nearly optimal probability of lifetime ruin under the simplified investment strategy is quite close to the original minimum probability of lifetime ruin under reasonable parameter values.