American Institute of Mathematical Sciences

In this paper, we consider the Cauchy problem of \begin{document}$ d $\end{document}-dimension Hartree type Dirac equation with nonlinearity \begin{document}$ c|x|^{-\gamma} * \langle \psi, \beta \psi\rangle $\end{document}, where \begin{document}$ c\in \mathbb R\setminus\{0\} $\end{document}, \begin{document}$ 0 < \gamma < d $\end{document}(\begin{document}$ d = 2,3 $\end{document}). Our aim is to show the local well-posedness in \begin{document}$ H^s $\end{document} for \begin{document}$ s > \frac{\gamma-1}2 $\end{document} with mass-supercritical cases(\begin{document}$ 1 < \gamma<d $\end{document}) and mass-critical case(\begin{document}$ {\gamma} = 1 $\end{document}) via bilinear estimates and angular decomposition for which we use the null structure of nonlinear term effectively. We also provide the flow of Dirac equations cannot be \begin{document}$ C^3 $\end{document} at the origin for \begin{document}$ H^s $\end{document} with \begin{document}$ s < \frac{\gamma-1}2 $\end{document}.

In this paper, we establish blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime with nonlinearity of derivative type. Our approach is based on the integral representation formula for the solution to the corresponding linear problem in the one-dimensional case, that we will determine through Yagdjian's Integral Transform approach. As upper bound for the exponent of the nonlinear term, we discover a Glassey-type exponent which depends both on the space dimension and on the Lorentzian metric in the generalized Einstein-de Sitter spacetime.

We are concerned with the following nonlinear fractional Schrödinger equation:

\begin{document}$\begin{equation} (-\Delta)^s u+V(x)u+\omega u = |u|^{p-2}u\quad {\rm{in}}\,\,{\mathbb{R}}^N,\;\;\;\;\;\;({\textbf{P}})\end{equation}$ \end{document}

where \begin{document}$ s\in(0,1) $\end{document} and \begin{document}$ p\in\left(2+4s/N,2^*_s\right) $\end{document}, that is, the mass supercritical and Sobolev subcritical. Under certain assumptions on the potential \begin{document}$ V:{\mathbb{R}}^N\rightarrow {\mathbb{R}} $\end{document}, positive and vanishing at infinity including potentials with singularities (which is important for physical reasons), we prove that there exists at least one \begin{document}$ L^2 $\end{document}-normalized solution \begin{document}$ (u,\omega)\in H^s({\mathbb{R}}^N)\times{\mathbb{R}}^+ $\end{document} of equation (P). In order to overcome the lack of compactness, the proof is based on a new min-max argument and splitting lemma for nonlocal version.

This paper is concerned with the study of the pullback dynamics of a piezoelectric system with magnetic and thermal effects and subjected to small perturbations of non-autonomous external forces with a parameter \begin{document}$ \epsilon $\end{document}. The existence of pullback exponential attractors and the existence of pullback attractors for the associated non-autonomous dynamical system are proved. Finally, the upper-semicontinuity of pullback attractors as \begin{document}$ \epsilon\to0 $\end{document} is shown.

This paper is concerned with the existence and multiplicity of constant sign solutions for the following fully nonlinear equation

\begin{document}$ \begin{equation*} \left\{ \begin{array}{l} -\mathcal{M}_\mathcal{C}^{\pm}(D^2u) = \mu f(u) \ \ \ \ \text{in} \ \ \Omega,\\ u = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{on}\ \partial\Omega, \end{array} \right. \end{equation*} $\end{document}

where \begin{document}$ \Omega\subset\mathbb{R}^N $\end{document} is a bounded regular domain with \begin{document}$ N\geq3 $\end{document}, \begin{document}$ \mathcal{M}_\mathcal{C}^{\pm} $\end{document} are general Hamilton-Jacobi-Bellman operators, \begin{document}$ \mu $\end{document} is a real parameter. By using bifurcation theory, we determine the range of parameter \begin{document}$ \mu $\end{document} of the above problem which has one or multiple constant sign solutions according to the behaviors of \begin{document}$ f $\end{document} at \begin{document}$ 0 $\end{document} and \begin{document}$ \infty $\end{document}, and whether \begin{document}$ f $\end{document} satisfies the signum condition \begin{document}$ f(s)s>0 $\end{document} for \begin{document}$ s\neq0 $\end{document}.

In this work, we are dealing with a non-linear eikonal system in one dimensional space that describes the evolution of interfaces moving with non-signed strongly coupled velocities. We prove a global existence result in the framework of continuous viscosity solution. The approach is made by adding a viscosity term and passing to the limit for vanishing viscosity, relying on a new gradient entropy and \begin{document}$ BV $\end{document} estimates. A uniqueness result is also proved through a comparison principle property.

The following degenerate chemotaxis system with flux limitation and nonlinear signal production

\begin{document}$ \begin{equation*} \begin{cases} u_t = \nabla\cdot(\frac{u\nabla u}{\sqrt {u^{2}+|\nabla u|^{2}}})-\chi\nabla\cdot(\frac{u\nabla v}{\sqrt {1+|\nabla v|^{2}}}) \quad &in\quad B_{R}\times(0, +\infty), \\ 0 = \Delta v-\mu (t)+u^{\kappa}, \quad \mu(t): = \frac{1}{|\Omega|}\int_{\Omega}u^{\kappa}(\cdot, t) \quad &in\quad B_{R}\times(0, +\infty) \end{cases} \end{equation*} $\end{document}

is considered in balls \begin{document}$ B_R = B_R(0)\subset \mathbb{R}^n $\end{document} for \begin{document}$ n\geq 1 $\end{document} and \begin{document}$ R>0 $\end{document} with no-flux boundary conditions, where \begin{document}$ \chi>0, \kappa>0 $\end{document}. We obtained local existence of unique classical solution and extensibility criterion ruling out gradient blow-up, and moreover proved global existence and boundedness of solutions under some conditions for \begin{document}$ \chi, \kappa $\end{document} and \begin{document}$ \int_{B_R}u_{0} $\end{document}.

In this paper, we study higher-order Hardy-Hénon elliptic systems with weights. We first prove a new theorem on regularities of the positive solutions at the origin, then study equivalence between the higher-order Hardy-Hénon elliptic system and a proper integral system, and we obtain a new and interesting Liouville-type theorem by methods of moving planes and moving spheres for integral system. We also use this Liouville-type theorem to prove the Hénon-Lane-Emden conjecture for polyharmonic system under some conditions.

This paper aims at establishing fine bounds for subcritical best Sobolev constants of the embeddings

\begin{document}$ W_{0}^{1,p}(\Omega)\hookrightarrow L^{q}(\Omega),\quad 1\leq q< \begin{cases} \frac{Np}{N-p},& 1\leq p<N\\ \infty,& p = N \end{cases} $\end{document}

where \begin{document}$ N\geq p\geq1 $\end{document} and \begin{document}$ \Omega $\end{document} is a bounded smooth domain in \begin{document}$ \mathbb{R}^{N} $\end{document} or the whole space. The Sobolev limiting case \begin{document}$ p = N $\end{document} is also covered by means of a limiting procedure.

In this paper we study the maximum number of limit cycles bifurcating from the periodic orbits of the center \begin{document}$ \dot x = -y((x^2+y^2)/2)^m, \dot y = x((x^2+y^2)/2)^m $\end{document} with \begin{document}$ m\ge0 $\end{document} under discontinuous piecewise polynomial (resp. polynomial Hamiltonian) perturbations of degree \begin{document}$ n $\end{document} with the discontinuity set \begin{document}$ \{(x, y)\in\mathbb{R}^2: xy = 0\} $\end{document}. Using the averaging theory up to any order \begin{document}$ N $\end{document}, we give upper bounds for the maximum number of limit cycles in the function of \begin{document}$ m, n, N $\end{document}. More importantly, employing the higher order averaging method we provide new lower bounds of the maximum number of limit cycles for several types of piecewise polynomial systems, which improve the results of the previous works. Besides, we explore the effect of 4-star-symmetry on the maximum number of limit cycles bifurcating from the unperturbed periodic orbits. Our result implies that 4-star-symmetry almost halves the maximum number.

In this paper we consider an initial-boundary value problem of a semilinear regularity-loss-type plate equation with memory in a bounded domain of \begin{document}$ \mathbb{R}^n $\end{document} (\begin{document}$ n = 1,2,\cdots $\end{document}). By using the Faedo-Galërkin method and some theories of ordinary differential equations, we obtain the local existence and uniqueness of weak solutions. Then, we study the dynamics of the weak solutions, such as global existence and finite time blow-up, by energy estimation and some ordinary differential inequalities. Moreover, the upper bound of blow-up time for the blow-up solutions is also considered.

In this paper, we are concerned with the threshold dynamics of a diffusive cholera model incorporating latency and bacterial hyperinfectivity. Our model takes the form of spatially nonlocal reaction-diffusion system associated with zero-flux boundary condition and time delay. By studying the associated eigenvalue problem, we establish the threshold dynamics that determines whether or not cholera will spread. We also confirm that the threshold dynamics can be determined by the basic reproduction number. By constructing Lyapunov functional, we address the global attractivity of the unique positive equilibrium whenever it exists. The theoretical results are still hold for the case when the constant parameters are replaced by strictly positive and spatial dependent functions.

We consider a damped Klein-Gordon equation with a variable diffusion coefficient. This problem is challenging because of the equation's unbounded nonlinearity. First, we study the nonlinearity's continuity properties. Then the existence and the uniqueness of the solutions is established. The main result is the continuity of the solution map on the set of admissible parameters. Its application to the parameter identification problem is considered.

We study a structural robustness of the complete oscillator death state in the Winfree model with random inputs and adaptive couplings. For this, we present a sufficient framework formulated in terms of initial data, natural frequencies and adaptive coupling strengths. In our proposed framework, we derive propagation of infinitesimal variations in random space and asymptotic disappearance of random effects which exhibits the robustness of the complete oscillator death state for the random Winfree model.

We establish the existence of solutions to a weakly-coupled competitive system of polyharmonic equations in \begin{document}$ \mathbb{R}^N $\end{document} which are invariant under a group of conformal diffeomorphisms, and study the behavior of least energy solutions as the coupling parameters tend to \begin{document}$ -\infty $\end{document}. We show that the supports of the limiting profiles of their components are pairwise disjoint smooth domains and solve a nonlinear optimal partition problem of \begin{document}$ \mathbb R^N $\end{document}. We give a detailed description of the shape of these domains.

This paper is a continuation of the authors earlier work on stability of Current Density Impedance Imaging (CDII) [R. Lopez, A. Moradifam, Stability of Current Density Impedance Imaging, SIAM J. Math. Anal. (2020).] We show that CDII is stable with respect to errors in both measurement of the magnitude of the current density vector field in the interior and the measurement of the voltage potential on the boundary. This completes the authors study of the stability of Current Density Independence Imaging which was previously shown only by numerical simulations.