American Institute of Mathematical Sciences

We propose a structure-preserving finite difference scheme for the Cahn–Hilliard equation with a dynamic boundary condition using the discrete variational derivative method (DVDM) proposed by Furihata and Matsuo [14]. In this approach, it is important and essential how to discretize the energy which characterizes the equation. By modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a standard central difference operator as an approximation of an outward normal derivative on the discrete boundary condition of the scheme. We show that our proposed scheme is second-order accurate in space, although the previous structure-preserving scheme proposed by Fukao–Yoshikawa–Wada [13] is first-order accurate in space. Also, we show the stability, the existence, and the uniqueness of the solution for our proposed scheme. Computation examples demonstrate the effectiveness of our proposed scheme. Especially through computation examples, we confirm that numerical solutions can be stably obtained by our proposed scheme.

We consider the numerical construction of minimal Lagrangian graphs, which is related to recent applications in materials science, molecular engineering, and theoretical physics. It is known that this problem can be formulated as an additive eigenvalue problem for a fully nonlinear elliptic partial differential equation. We introduce and implement a two-step generalized finite difference method, which we prove converges to the solution of the eigenvalue problem. Numerical experiments validate this approach in a range of challenging settings. We further discuss the generalization of this new framework to Monge-Ampère type equations arising in optimal transport. This approach holds great promise for applications where the data does not naturally satisfy the mass balance condition, and for the design of numerical methods with improved stability properties.

We prove variation and oscillation \begin{document}$ L^p $\end{document}-inequalities associated with fractional derivatives of certain semigroups of operators and with the family of truncations of Riesz transforms in the inverse Gaussian setting. We also study these variational \begin{document}$ L^p $\end{document}-inequalities in a Banach-valued context by considering Banach spaces with the UMD-property and whose martingale cotype is fewer than the variational exponent. We establish \begin{document}$ L^p $\end{document}-boundedness properties for weighted difference involving the semigroups under consideration.

Motivated by the studies of the hydrodynamics of the tethered bacteria Thiovulum majus in a liquid environment, we consider the following chemotaxis system

\begin{document}$ \begin{equation*} \left\{ \begin{split} & n_t = \Delta n-\nabla\cdot\left(n\chi(c)\nabla{c}\right)+nc, &x\in \Omega, t>0, \ & c_t = \Delta c-{\bf u}\cdot\nabla c-nc, &x\in \Omega, t>0\ \end{split} \right. \end{equation*} $\end{document}

under homogeneous Neumann boundary conditions in a bounded convex domain \begin{document}$ \Omega\subset \mathbb{R}^d(d\in\{2, 3\}) $\end{document} with smooth boundary. For any given fluid \begin{document}$ {\bf u} $\end{document}, it is proved that if \begin{document}$ d = 2 $\end{document}, the corresponding initial-boundary value problem admits a unique global classical solution which is uniformly bounded, while if \begin{document}$ d = 3 $\end{document}, such solution still exists under the additional condition that \begin{document}$ 0<\chi\leq \frac{1}{16\|c(\cdot, 0)\|_{L^\infty(\Omega)}} $\end{document}.

We consider an initial boundary value problem of three-dimensional (3D) nonhomogeneous magneto-micropolar fluid equations in a bounded simply connected smooth domain with homogeneous Dirichlet boundary conditions for the velocity and micro-rotational velocity and Navier-slip boundary condition for the magnetic field. We prove the global existence and exponential decay of strong solutions provided that some smallness condition holds true. Note that although the system degenerates near vacuum, there is no need to require compatibility conditions for the initial data via time weighted techniques.

We consider in this paper the nonlinear elliptic equation with Neumann boundary condition

\begin{document}$ \begin{align*} \begin{cases} \Delta u = a|u|^{m-1}u\, \, \mbox{ in }\, \, \mathbb{R}^{n+1}_{+}\\ \dfrac{\partial u}{\partial t} = b|u|^{\eta-1}u+f\, \, \mbox{ on }\, \, \partial \mathbb{R}^{n+1}_{+}. \end{cases} \end{align*} $\end{document}

For \begin{document}$ a, b\neq 0 $\end{document}, \begin{document}$ m>\frac{n+1}{n-1} $\end{document}, \begin{document}$ (n>1) $\end{document}, \begin{document}$ \eta = \frac{m+1}{2} $\end{document} and small data \begin{document}$ f\in L^{\frac{nq}{n+1}, \infty}(\partial \mathbb{R}^{n+1}_{+}) $\end{document}, \begin{document}$ q = \frac{(n+1)(m-1)}{m+1} $\end{document} we prove that the problem is solvable. More precisely, we establish existence, uniqueness and continuous dependence of solutions on the boundary data \begin{document}$ f $\end{document} in the function space \begin{document}$ \mathbf{X}^{q}_{\infty} $\end{document} where

\begin{document}$ \|u\|_{ \mathbf{X}^{q}_{\infty}} = \sup\limits_{t>0}t^{\frac{n+1}{q}-1}\|u(t)\|_{L^{\infty}( \mathbb{R}^{n})}+\|u\|_{L^{\frac{q(m+1)}{2}, \infty}( \mathbb{R}^{n+1}_{+})}+\|\nabla u\|_{L^{q, \infty}( \mathbb{R}^{n+1}_{+})}. $\end{document}

As a direct consequence, we obtain the local regularity property \begin{document}$ C^{1, \nu}_{loc} $\end{document}, \begin{document}$ \nu\in (0, 1) $\end{document} of these solutions as well as energy estimates for certain values of \begin{document}$ m $\end{document}. Boundary values decaying faster than \begin{document}$ |x|^{-(m+1)/(m-1)} $\end{document}, \begin{document}$ x\in \mathbb{R}^{n}\setminus\{0\} $\end{document} yield solvability and this decay property is shown to be sharp for positive nonlinearities.

Moreover, we are able to show that solutions inherit qualitative features of the boundary data such as positivity, rotational symmetry with respect to the \begin{document}$ (n+1) $\end{document}-axis, radial monotonicity in the tangential variable and homogeneity. When \begin{document}$ a, b>0 $\end{document}, the critical exponent \begin{document}$ m_c $\end{document} for the existence of positive solutions is identified, \begin{document}$ m_c = (n+1)/(n-1) $\end{document}.

We examine the following degenerate elliptic system:

\begin{document}$ -\Delta_{s} u \! = \! v^p, \quad -\Delta_{s} v\! = \! u^\theta, \;\; u, v>0 \;\;\mbox{in }\; \mathbb{R}^N = \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}, \quad\mbox{where}\;\; s \geq 0\;\; \mbox{and} \;\;p, \theta \!>\!0. $\end{document}

We prove that the system has no stable solution provided \begin{document}$ p, \theta >0 $\end{document} and \begin{document}$ N_s: = N_1+(1+s)N_2< 2 + \alpha + \beta, $\end{document} where

\begin{document}$ \alpha = \frac{2(p+1)}{p\theta - 1} \quad\mbox{and} \quad \beta = \frac{2(\theta +1)}{p\theta - 1}. $\end{document}

This result is an extension of some results in [15]. In particular, we establish a new integral estimate for \begin{document}$ u $\end{document} and \begin{document}$ v $\end{document} (see Proposition 1.1), which is crucial to deal with the case \begin{document}$ 0 < p < 1. $\end{document}

In this paper, we mainly study several problems on the weakly dissipative generalized Camassa-Holm equation. We first establish the local well-posedness of solutions by Kato's semigroup theory. We then derive the necessary and sufficient condition of the blow-up of solutions and a criteria to guarantee occurrence of wave breaking. Moreover, when the solution blows up, we obtain the precise blow-up rate. We finally show that the equation has a unique global solution provided the momentum density associated with their initial datum satisfies appropriate sign conditions.

The main goal of this paper is to study the asymptotic behavior of a coupled Klein-Gordon-Schrödinger system in three dimensional unbounded domain. We prove the existence of a global attractor of the systems of the nonlinear Klein-Gordon-Schrödinger (KGS) equations in \begin{document}$ H^1({\mathbb R}^3)\times H^1({\mathbb R}^3)\times L^2({\mathbb R}^3) $\end{document} and more particularly that this attractor is in fact a compact set of \begin{document}$ H^2({\mathbb R}^3)\times H^2({\mathbb R}^3)\times H^1({\mathbb R}^3) $\end{document}.

In this paper, we investigate a system of pseudoparabolic equations with Robin-Dirichlet conditions. First, the local existence and uniqueness of a weak solution are established by applying the Faedo-Galerkin method. Next, for suitable initial datum, we obtain the global existence and decay of weak solutions. Finally, using concavity method, we prove blow-up results for solutions when the initial energy is nonnegative or negative, then we establish here the lifespan for the equations via finding the upper bound and the lower bound for the blow-up times.

This paper concerns the existence of solutions of the following supercritical PDE: \begin{document}$ (P_\varepsilon) $\end{document}: \begin{document}$ -\Delta u = K|u|^{\frac{4}{n-2}+\varepsilon}u\; \mbox{ in }\Omega, \; u = 0 \mbox{ on }\partial\Omega, $\end{document} where \begin{document}$ \Omega $\end{document} is a smooth bounded domain in \begin{document}$ \mathbb{R}^n $\end{document}, \begin{document}$ n\geq 3 $\end{document}, \begin{document}$ K $\end{document} is a \begin{document}$ C^3 $\end{document} positive function and \begin{document}$ \varepsilon $\end{document} is a small positive real. Our method is inspired from the work of Bahri-Li-Rey. It consists to reduce the existence of a critical point to a finite dimensional system. Using a fixed-point theorem, we are able to construct positive solutions of \begin{document}$ (P_\varepsilon) $\end{document} having the form of two bubbles with non comparable speeds and which have only one blow-up point in \begin{document}$ \Omega $\end{document}. That means that this blow-up point is non simple. This represents a new phenomenon compared with the subcritical case.

In this paper, we study the following coupled nonlinear Schrödinger system of the form

\begin{document}$ \left\{\begin{array}{l} -\Delta u_i-\kappa_iu_i = g_i(u_i)+\lambda\partial_iF(\vec{u}), \\ \vec{u} = (u_1,u_2,\cdots,u_m), u_i\in D_0^{1,2}(\Omega), \end{array}\right. $\end{document}

for \begin{document}$ m = 2,3 $\end{document}, where \begin{document}$ \Omega\subset \mathbb{R}^N $\end{document} is a bounded domain or \begin{document}$ \mathbb{R}^N $\end{document}, \begin{document}$ N\geq 3 $\end{document}, \begin{document}$ F(t_1,t_2\cdots,t_m)\in C^1(\mathbb{R}^m,\mathbb{R}) $\end{document}, \begin{document}$ \kappa_i\in\mathbb{R} $\end{document}, \begin{document}$ g_i\in C(\mathbb{R}) \ (i = 1,2,\cdots,m) $\end{document} and \begin{document}$ \lambda>0 $\end{document} is large enough. In this work we mainly focus on the existence of fully nontrivial ground-state solutions and synchronized ground-state solutions under certain conditions.

This paper is concerned with the following quasilinear Schrödinger system in the entire space \begin{document}$ \mathbb R^{N}(N\geq3) $\end{document}:

\begin{document}$ \left\{\begin{aligned} &-\Delta u+A(x)u+\frac{k}{2}\triangle(u^{2})u = \frac{2\alpha }{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},\\ &-\Delta v+Bv+\frac{k}{2}\triangle(v^{2})v = \frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,\\ & u(x)\to 0,\ \ v(x)\to 0\ \ \hbox{as}\ |x|\to \infty,\end{aligned}\right. $\end{document}

where \begin{document}$ \alpha,\beta>1 $\end{document}, \begin{document}$ 2<\alpha+\beta<2^* = \frac{2N}{N-2} $\end{document} and \begin{document}$ k >0 $\end{document} is a parameter. By using the principle of symmetric criticality and the moser iteration, for any given integer \begin{document}$ \xi\geq2 $\end{document}, we construct a non-radially symmetrical nodal solution with its \begin{document}$ 2\xi $\end{document} nodal domains. Our results can be looked on as a generalization to results by Alves, Wang and Shen (Soliton solutions for a class of quasilinear Schrödinger equations with a parameter. J. Differ. Equ. 259 (2015) 318-343).

This paper deals with the doubly degenerate nutrient taxis system

\begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{ll} u_t = (uv u_x)_x - (u^2 vv_x)_x + \ell uv, \qquad & x\in \Omega, \ t>0, \\ v_t = v_{xx} -uv, \qquad & x\in \Omega, \ t>0, \end{array} \right. \end{eqnarray*} $\end{document}

in an open bounded interval \begin{document}$ \Omega\subset \mathbb{R} $\end{document}, with \begin{document}$ \ell \ge0 $\end{document}, which has been proposed to model the formation of diverse morphological aggregation patterns observed in colonies of Bacillus subtilis growing on the surface of thin agar plates.

It is shown that under the mere assumption that

\begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{l} u_0\in W^{1,\infty}( \Omega) \mbox{ is nonnegative with } u_0\not\equiv 0 \qquad \mbox{and} \\ v_0\in W^{1,\infty}( \Omega) \mbox{ is positive in } \overline{\Omega}, \end{array} \right. \qquad \qquad (\star) \end{eqnarray*} $\end{document}

an associated no-flux initial boundary value problem possesses a globally defined and continuous weak solution \begin{document}$ (u,v) $\end{document}, where \begin{document}$ u\ge 0 $\end{document} and \begin{document}$ v>0 $\end{document} in \begin{document}$ \overline{\Omega}\times [0,\infty) $\end{document}, and that moreover there exists \begin{document}$ u_\infty\in C^0( \overline{\Omega}) $\end{document} such that the solution \begin{document}$ (u,v) $\end{document} approaches the pair \begin{document}$ (u_\infty,0) $\end{document} in the large time limit with respect to the topology \begin{document}$ (L^{\infty}( \Omega)) ^2 $\end{document}. This extends comparable results recently obtained in [17], the latter crucially relying on the additional requirement that \begin{document}$ \int_\Omega \ln u_0>-\infty $\end{document}, to situations involving nontrivially supported initial data \begin{document}$ u_0 $\end{document}, which seems to be of particular relevance in the addressed application context of sparsely distributed populations.

We study the structure of positive solutions to steady state ecological models of the form:

\begin{document}$ \begin{array}{l} \left\{ \begin{split} -\Delta u& = \lambda uf(u)\; \; && {\rm{in}}\; \; \Omega,\\ \alpha(u)&\frac{\partial u}{\partial \eta}+[1-\alpha(u)]u = 0 &&\;\;\;{\rm{on}}\; \; \partial\Omega, \end{split} \right. \end{array} $\end{document}

where \begin{document}$ \Omega $\end{document} is a bounded domain in \begin{document}$ \mathbb{R}^n; $\end{document} \begin{document}$ n>1 $\end{document} with smooth boundary \begin{document}$ \partial\Omega $\end{document} or \begin{document}$ \Omega = (0,1) $\end{document}, \begin{document}$ \frac{\partial}{\partial\eta} $\end{document} represents the outward normal derivative on the boundary, \begin{document}$ \lambda $\end{document} is a positive parameter, \begin{document}$ f:[0,\infty)\to \mathbb{R} $\end{document} is a \begin{document}$ C^2 $\end{document} function such that \begin{document}$ \tfrac{f(s)}{k-s}>0 $\end{document} for some \begin{document}$ k>0 $\end{document}, and \begin{document}$ \alpha:[0,k]\to[0,1] $\end{document} is also a \begin{document}$ C^2 $\end{document} function. Here \begin{document}$ f(u) $\end{document} represents the per capita growth rate, \begin{document}$ \alpha(u) $\end{document} represents the fraction of the population that stays on the patch upon reaching the boundary, and \begin{document}$ \lambda $\end{document} relates to the patch size and the diffusion rate. In particular, we will discuss models in which the per capita growth rate is increasing for small \begin{document}$ u $\end{document}, and models where grazing is involved. We will focus on the cases when \begin{document}$ \alpha'(s)\geq 0 $\end{document}; \begin{document}$ [0,k] $\end{document}, which represents negative density dependent dispersal on the boundary. We employ the method of sub-super solutions, bifurcation theory, and stability analysis to obtain our results. We provide detailed bifurcation diagrams via a quadrature method for the case \begin{document}$ \Omega = (0,1) $\end{document}.

We consider certain kinds of weighted multi-linear fractional integral inequalities which can be regarded as extensions of the Hardy-Littlewood-Sobolev inequality. For a particular case, we characterize the sufficient and necessary conditions which ensure that the corresponding inequality holds. For the general case, we give some sufficient conditions and necessary conditions, respectively.