American Institute of Mathematical Sciences

In the perfect conductivity problem arising from composites, the electric field may become arbitrarily large as \begin{document}$ \varepsilon $\end{document}, the distance between the inclusions and the matrix boundary, tends to zero. In this paper, by making clear the singular role of the blow-up factor \begin{document}$ Q[\varphi] $\end{document} introduced in [27] for some special boundary data of even function type with \begin{document}$ k $\end{document}-order growth, we prove the optimality of the blow-up rate in the presence of \begin{document}$ m $\end{document}-convex inclusions close to touching the matrix boundary in all dimensions. Finally, we give closer analysis in terms of the singular behavior of the concentrated field for eccentric and concentric core-shell geometries with circular and spherical boundaries from the practical application angle.

We consider a Gierer-Meinhardt system on a surface coupled with a parabolic PDE in the bulk, the domain confined by this surface. Such a model was recently proposed and analyzed for two-dimensional bulk domains by Gomez, Ward and Wei (SIAM J. Appl. Dyn. Syst. 18, 2019). We prove the well-posedness of the bulk-surface system in arbitrary space dimensions and show that solutions remain uniformly bounded in parabolic Hölder spaces for all times. The cytosolic diffusion is typically much larger than the lateral diffusion on the membrane. This motivates to a corresponding asymptotic reduction, which consists of a nonlocal system on the membrane. We prove the convergence of solutions of the full system towards unique solutions of the reduction.

In this paper, we study the following fractional Schrödinger-Poiss-on system

\begin{document}$ \begin{equation*} \begin{cases} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u = g(u) & \hbox{in $\mathbb{R}^3$,} \\ \varepsilon^{2t}(-\Delta)^t\phi = u^2,\,\, u>0& \hbox{in $\mathbb{R}^3$,} \end{cases} \end{equation*} $\end{document}

where \begin{document}$ s,t\in(0,1) $\end{document}, \begin{document}$ \varepsilon>0 $\end{document} is a small parameter. Under some local assumptions on \begin{document}$ V(x) $\end{document} and suitable assumptions on the nonlinearity \begin{document}$ g $\end{document}, we construct a family of positive solutions \begin{document}$ u_{\varepsilon}\in H_{\varepsilon} $\end{document} which concentrate around the global minima of \begin{document}$ V(x) $\end{document} as \begin{document}$ \varepsilon\rightarrow0 $\end{document}.

In this paper we study the three-marginal optimal mass transportation problem for the Coulomb cost on the plane \begin{document}$ \mathbb R^2 $\end{document}. The key question is the optimality of the so-called Seidl map, first disproved by Colombo and Stra. We generalize the partial positive result obtained by Colombo and Stra and give a necessary and sufficient condition for the radial Coulomb cost to coincide with a much simpler cost that corresponds to the situation where all three particles are aligned. Moreover, we produce an infinite class of regular counterexamples to the optimality of this family of maps.

We study the Cauchy problem for the surface quasi-geostrophic equation with the critical dissipation in the two dimensional half space under the homogeneous Dirichlet boundary condition. We show the global existence, the uniqueness and the analyticity of solutions, and the real analyticity up to the boundary is obtained. We will show a natural ways to estimate the nonlinear term for functions satisfying the Dirichlet boundary condition.

We consider the semi-relativistic Hartree equation with combined Hartree-type nonlinearities given by

\begin{document}$ i\partial_t \psi = \sqrt{-\triangle+m^2}\, \psi+\beta(\frac{1}{|x|^\alpha}\ast |\psi|^2)\psi-(\frac{1}{|x|}\ast |\psi|^2)\psi\ \ \ \text{on $\mathbb{R}^3$.} $\end{document}

where \begin{document}$ 0<\alpha<1 $\end{document} and \begin{document}$ \beta>0 $\end{document}. Firstly we study the existence and stability of the maximal ground state \begin{document}$ \psi_\beta $\end{document} at \begin{document}$ N = N_c $\end{document}, where \begin{document}$ N_c $\end{document} is a threshold value and can be regarded as "Chandrasekhar limiting mass". Secondly, we analyse blow-up behaviours of maximal ground states \begin{document}$ \psi_\beta $\end{document} when \begin{document}$ \beta\rightarrow 0^+ $\end{document}, and the optimal blow-up rate with respect to \begin{document}$ \beta $\end{document} will be calculated.

This paper introduce a novel optimization procedure to reduce mixture of Gaussian and impulse noise from images. This technique exploits a non-convex PDE-constrained characterized by a fractional-order operator. The used non-convex term facilitated the impulse component approximation controlled by a spatial parameter \begin{document}$ \gamma $\end{document}. A non-convex and non-smooth bi-level optimization framework with a modified projected gradient algorithm is then proposed in order to learn the parameter \begin{document}$ \gamma $\end{document}. Denoising tests confirm that the non-convex term and learned parameter \begin{document}$ \gamma $\end{document} lead in general to an improved reconstruction when compared to results of convex norm and manual parameter \begin{document}$ \lambda $\end{document} choice.

We give a general theory on well-posedness and time asymptotics for growth fragmentation equations in \begin{document}$ L^{1} $\end{document} spaces. We prove first generation of \begin{document}$ C_{0} $\end{document}-semigroups governing them for unbounded total fragmentation rate and fragmentation kernel \begin{document}$ b(.,.) $\end{document} such that \begin{document}$ \int_{0}^{y}xb(x,y)dx = y-\eta (y)y $\end{document} (\begin{document}$ 0\leq \eta (y)\leq 1 $\end{document} expresses the mass loss) and continuous growth rate \begin{document}$ r(.) $\end{document} such that \begin{document}$ \int_{0}^{\infty }\frac{1}{r(\tau )}d\tau = +\infty . $\end{document}This is done in the spaces of finite mass or finite mass and number of agregates. Generation relies on unbounded perturbation theory peculiar to positive semigroups in \begin{document}$ L^{1} $\end{document} spaces. Secondly, we show that the semigroup has a spectral gap and asynchronous exponential growth. The analysis relies on weak compactness tools and Frobenius theory of positive operators. A systematic functional analytic construction is provided.

In this paper we consider the critical quasilinear Schrödinger-Poisson system

\begin{document}$ \begin{eqnarray*} \left \{\begin{array}{ll} -\Delta_p u+|u|^{p-2}u+\mu\phi |u|^{p-2}u = \lambda|u|^{q-2}u+|u|^{p^*-2}u,&\mathrm{in} \ \mathbb{R}^3,\\ -\Delta \phi = |u|^p, &\mathrm{in}\ \mathbb{R}^3, \end{array} \right. \end{eqnarray*} $\end{document}

where \begin{document}$ \frac{3}{2}<p<3 $\end{document}, \begin{document}$ \Delta_p u = \hbox{div}(|\nabla u|^{p-2}\nabla u) $\end{document}, \begin{document}$ p<q<p^*: = \frac{3p}{3-p} $\end{document} and \begin{document}$ \mu,\lambda>0 $\end{document}. Based upon the variational approach, the ground state solutions and the nontrivial solutions are obtained depending on the parameters \begin{document}$ q $\end{document}, \begin{document}$ \mu $\end{document} and \begin{document}$ \lambda $\end{document}.

In this paper we prove, using a duality method, existence of solutions for the nonlinear elliptic mean field games type system

\begin{document}$ \left\{ \begin{array}{cl} -{\mathop{{{\rm{div}}}}}(M(x)\,{\nabla} u) + u - {\mathop{{{\rm{div}}}}}(u\,A(x)\,{\nabla}\psi) = f(x) & {\rm{in \; \Omega ,}}\\ -{\mathop{{{\rm{div}}}}}(M(x)\,{\nabla}\psi) + \psi + A(x)\,{\nabla}\psi \cdot {\nabla} \psi = u^{p-1} & {\rm{in \; \Omega ,}} \\ u = 0 = \psi & {\rm{on \; \partial\Omega ,}} \end{array} \right. $\end{document}

under different assumptions on \begin{document}$ p > 1 $\end{document}, and the function \begin{document}$ f(x) $\end{document} in Lebesgue spaces.

In this paper, the periodic solution and extinction in a periodic chemostat model with delay in microorganism growth are investigated. The positivity and ultimate boundedness of solutions are firstly obtained. Next, the necessary and sufficient conditions on the existence of positive \begin{document}$ \omega $\end{document}-periodic solutions are established by constructing Poincaré map and using the Whyburn Lemma and Leray-Schauder degree theory. Furthermore, according to the implicit function theorem, the uniqueness of the positive periodic solution is obtained when delay \begin{document}$ \tau $\end{document} is small enough. Finally, the necessary and sufficient conditions for the extinction of microorganism species are established.

We study the regularity properties of the second order linear operator in \begin{document}$ {{\mathbb {R}}}^{N+1} $\end{document}:

\begin{document}$ \begin{equation*} \mathscr{L} u : = \sum\limits_{j,k = 1}^{m} a_{jk}\partial_{x_j x_k}^2 u + \sum\limits_{j,k = 1}^{N} b_{jk}x_k \partial_{x_j} u - \partial_t u, \end{equation*} $\end{document}

where \begin{document}$ A = \left( a_{jk} \right)_{j,k = 1, \dots, m}, B = \left( b_{jk} \right)_{j,k = 1, \dots, N} $\end{document} are real valued matrices with constant coefficients, with \begin{document}$ A $\end{document} symmetric and strictly positive. We prove that, if the operator \begin{document}$ {\mathscr{L}} $\end{document} satisfies Hörmander's hypoellipticity condition, and \begin{document}$ f $\end{document} is a Dini continuous function, then the second order derivatives of the solution \begin{document}$ u $\end{document} to the equation \begin{document}$ {\mathscr{L}} u = f $\end{document} are Dini continuous functions as well. We also consider the case of Dini continuous coefficients \begin{document}$ a_{jk} $\end{document}'s. A key step in our proof is a Taylor formula for classical solutions to \begin{document}$ {\mathscr{L}} u = f $\end{document} that we establish under minimal regularity assumptions on \begin{document}$ u $\end{document}.

In this paper we consider the linear quasi-periodic system

\begin{document}$ \begin{equation*} \dot{x} = (A+\epsilon P(t)) x, x\in \mathbb{R}^{d}, \end{equation*} $\end{document}

where \begin{document}$ A $\end{document} is a \begin{document}$ d\times d $\end{document} constant matrix with elliptic type, \begin{document}$ P(t) $\end{document} is analytic quasi-periodic with respect to \begin{document}$ t $\end{document} with basic frequencies \begin{document}$ \omega = (1, \alpha), $\end{document} with \begin{document}$ \alpha $\end{document} being irrational, and \begin{document}$ \epsilon $\end{document} is a small perturbation parameter. If some suitable non-resonant conditions and non-degeneracy conditions hold, and the basic frequencies satisfy that \begin{document}$ 0\leq \beta(\alpha) < r, $\end{document} where \begin{document}$ \beta(\alpha) = \limsup\limits_{n\rightarrow \infty}\frac{\ln q_{n+1}}{q_{n}}, $\end{document} \begin{document}$ q_{n} $\end{document} is the sequence of denominations of the best rational approximations for \begin{document}$ \alpha \in \mathbb{R} \setminus\mathbb{Q}, $\end{document} \begin{document}$ r $\end{document} is the initial radius of analytic domain, it is proved that for most sufficiently small \begin{document}$ \epsilon, $\end{document} this system can be reduced to a constant system \begin{document}$ \dot{x} = A^{*}x, x\in \mathbb{R}^{d}, $\end{document} where \begin{document}$ A^{*} $\end{document} is a constant matrix close to \begin{document}$ A. $\end{document} As some applications, we apply our results to quasi-periodic Schrödinger equations with an external parameter to study the Lyapunov stability of the equilibrium and the existence of quasi-periodic solutions.

In this paper, we consider a homogeneous Neumann initial-boundary value problem (IBVP) for the following two-species and two-stimuli chemotaxis model with both paracrine and autocrine loops:

\begin{document}$ \begin{equation*} \label{IBVP} \left\{ \begin{aligned} &u_t = \nabla\cdot(D_1(u)\nabla u-S_1(u)\nabla v), &\qquad x\in\Omega, \, t>0, \\ & \tau_1 v_t = \Delta v- v+w, &\qquad x\in\Omega, \, t>0, \\ &w_t = \nabla\cdot(D_2(w)\nabla w-S_2(w)\nabla z-S_3(w)\nabla v), &\qquad x\in\Omega, \, t>0, \\ & \tau_2 z_t = \Delta z- z+ u, &\qquad x\in\Omega, \, t>0, \end{aligned} \right. \end{equation*} $\end{document}

where \begin{document}$ u(t, x) $\end{document} and \begin{document}$ w(t, x) $\end{document} denote the density of macrophages and tumor cells at time \begin{document}$ t $\end{document} and location \begin{document}$ x\in \Omega, $\end{document} respectively, \begin{document}$ v(t, x) $\end{document} and \begin{document}$ z(t, x) $\end{document} represent the concentration of colony stimulating factor 1 (CSF-1) secreted by the tumor cells and epidermal growth factor (EGF) secreted by macrophages at time \begin{document}$ t $\end{document} and location \begin{document}$ x\in \Omega, $\end{document} respectively. \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} is a bounded region with smooth boundary, \begin{document}$ \tau_i\ge 0 \; (i = 1, 2) $\end{document}, \begin{document}$ D_i(s)\ge d_i(s+1)^{m_i-1} $\end{document} with parameters \begin{document}$ m_i\ge 1 \; (i = 1, 2) $\end{document} and \begin{document}$ S_j(s)\lesssim (s+1)^{q_j} $\end{document} with parameters \begin{document}$ q_j>0 \;(j = 1, 2, 3) $\end{document}. For the case without autocrine loop (i.e., \begin{document}$ S_3(w) = 0 $\end{document}), it is shown that when \begin{document}$ q_j\le 1 \; (j = 1, 2) $\end{document}, if one of \begin{document}$ q_j $\end{document} is smaller than one or one of \begin{document}$ m_i $\end{document} is larger than one, then the IBVP has a global classical solution which is uniformly bounded. Moreover, when \begin{document}$ m_1 = m_2 = q_1 = q_2 = 1 $\end{document}, an inequality involving the product \begin{document}$ d_1d_2 $\end{document} and the product of the two species' initial mass is obtained which guarantees the existence of global bounded classical solutions. More specifically, it allows one of \begin{document}$ d_i $\end{document} to be small or one of the species initial mass to be large. For the case with autocrine loop (i.e \begin{document}$ S_3(w)\ne 0 $\end{document}), similar results hold only if \begin{document}$ q_3<1 $\end{document}. If \begin{document}$ q_3 = 1 $\end{document}, solutions to the IBVP exist globally only when \begin{document}$ d_2 $\end{document} is suitably large or the mass of species \begin{document}$ w $\end{document} is suitably small.

We consider ground states of the following time-independent nonlinear \begin{document}$ L^2 $\end{document}-critical Schrödinger equation

\begin{document}$ -\Delta u(x)+V(x)u(x)-a|x|^{-b}|u|^{\frac{4-2b}{N}}u(x) = \mu u(x)\,\ \hbox{in}\,\ {\mathbb{R}}^N, $\end{document}

where \begin{document}$ \mu\!\in\! {\mathbb{R}} $\end{document}, \begin{document}$ a\!>\!0 $\end{document}, \begin{document}$ N\!\geq\! 1 $\end{document}, \begin{document}$ 0\!<\!b\!<\!\min\{2,N\} $\end{document}, and \begin{document}$ V(x)\!\geq\! 0 $\end{document} is an external potential. We get ground states of the above equation by solving the associated constrained minimization problem. In this paper, we prove that there is a threshold \begin{document}$ a^*\!>\!0 $\end{document} such that minimizer exists for \begin{document}$ 0\!<\!a\!<\!a^* $\end{document}, and minimizer does not exist for any \begin{document}$ a\!>\!a^* $\end{document}. However if \begin{document}$ a\! = \!a^* $\end{document}, it is showed that whether minimizer exists depends sensitively on the value of \begin{document}$ V(0) $\end{document}. Moreover if \begin{document}$ V(0)\! = \!0 $\end{document}, we prove that minimizers must concentrate at the origin as \begin{document}$ a\nearrow a^* $\end{document} and give a detailed concentration behavior of minimizers as \begin{document}$ a\nearrow a^* $\end{document}, based on which we finally prove that there is a unique minimizer when \begin{document}$ a $\end{document} is close enough to \begin{document}$ a^* $\end{document}.

The parametric estimation of drift parameter for distribution - dependent stochastic differential delay equations with a small diffusion is presented. The principle technique of our investigation is to construct an appropriate contrast function and carry out a limiting type of argument to show the consistency and convergence rate of the least squares estimator of the drift parameter via interacting particle systems. In addition, two examples are constructed to demonstrate the effectiveness of our work.