American Institute of Mathematical Sciences

We construct Green's functions for second order parabolic operators of the form \begin{document}$ Pu = \partial_t u-{\rm div}({\mathbf A} \nabla u+ {\mathbf b}u)+ {\mathbf c} \cdot \nabla u+du $\end{document} in \begin{document}$ (-\infty, \infty) \times \Omega $\end{document}, where \begin{document}$ \Omega $\end{document} is an open connected set in \begin{document}$ \mathbb{R}^n $\end{document}. It is not necessary that \begin{document}$ \Omega $\end{document} to be bounded and \begin{document}$ \Omega = \mathbb{R}^n $\end{document} is not excluded. We assume that the leading coefficients \begin{document}$ \mathbf A $\end{document} are bounded and measurable and the lower order coefficients \begin{document}$ \boldsymbol{b} $\end{document}, \begin{document}$ \boldsymbol{c} $\end{document}, and \begin{document}$ d $\end{document} belong to critical mixed norm Lebesgue spaces and satisfy the conditions \begin{document}$ d-{\rm div} \boldsymbol{b} \ge 0 $\end{document} and \begin{document}$ {\rm div}(\boldsymbol{b}-\boldsymbol{c}) \ge 0 $\end{document}. We show that the Green's function has the Gaussian bound in the entire \begin{document}$ (-\infty, \infty) \times \Omega $\end{document}.

In this paper we continue the work that we began in [6]. Given \begin{document}$ 1<p<N $\end{document}, two measurable functions \begin{document}$ V\left(r \right)\geq 0 $\end{document} and \begin{document}$ K\left(r\right)> 0 $\end{document}, and a continuous function \begin{document}$ A(r) >0 $\end{document} (\begin{document}$ r>0 $\end{document}), we consider the quasilinear elliptic equation

\begin{document}$ -\mathrm{div}\left(A(|x| )|\nabla u|^{p-2} \nabla u\right) +V\left( \left| x\right| \right) |u|^{p-2}u = K(|x|) f(u) \quad \text{in }\mathbb{R}^{N}, $\end{document}

where all the potentials \begin{document}$ A,V,K $\end{document} may be singular or vanishing, at the origin or at infinity. We find existence of nonnegative solutions by the application of variational methods, for which we need to study the compactness of the embedding of a suitable function space \begin{document}$ X $\end{document} into the sum of Lebesgue spaces \begin{document}$ L_{K}^{q_{1}}+L_{K}^{q_{2}} $\end{document}. The nonlinearity has a double-power super \begin{document}$ p $\end{document}-linear behavior, as \begin{document}$ f(t) = \min \left\{ t^{q_1 -1}, t^{q_2 -1} \right\} $\end{document} with \begin{document}$ q_1,q_2>p $\end{document} (recovering the power case if \begin{document}$ q_1 = q_2 $\end{document}). With respect to [6], in the present paper we assume some more hypotheses on \begin{document}$ V $\end{document}, and we are able to enlarge the set of values \begin{document}$ q_1 , q_2 $\end{document} for which we get existence results.

In this paper, we develop a new method to study Rabinowitz's conjecture on the existence of periodic solutions with prescribed minimal period for second order even Hamiltonian system without any convexity assumptions. Specifically, we first study the associated homogenous Dirichlet boundary value problems for the discretization of the Hamiltonian system with given step length and obtain a sequence of nonnegative solutions corresponding to different step lengths by using discrete variational methods. Then, using the sequence of nonnegative solutions, we construct a sequence of continuous functions which can be shown to be precompact. Finally, by utilizing the limit function of convergent subsequence and the symmetry of the potential, we will obtain the desired periodic solution. In particular, we prove Rabinowitz's conjecture in the case when the potential satisfies a certain symmetric assumption. Moreover, our main result greatly improves the related results in the literature in the case where \begin{document}$ N = 1 $\end{document}.

Reactive transport processes in porous media including thin heterogeneous layers play an important role in many applications. In this paper, we investigate a reaction-diffusion problem with nonlinear diffusion in a domain consisting of two bulk domains which are separated by a thin layer with a periodic heterogeneous structure. The thickness of the layer, as well as the periodicity within the layer are of order \begin{document}$ \epsilon $\end{document}, where \begin{document}$ \epsilon $\end{document} is much smaller than the size of the bulk domains. For the singular limit \begin{document}$ \epsilon \to 0 $\end{document}, when the thin layer reduces to an interface, we rigorously derive a macroscopic model with effective interface conditions between the two bulk domains. Due to the oscillations within the layer, we have the combine dimension reduction techniques with methods from the homogenization theory. To cope with these difficulties, we make use of the two-scale convergence in thin heterogeneous layers. However, in our case the diffusion in the thin layer is low and depends nonlinearly on the concentration itself. The low diffusion leads to a two-scale limit depending on a macroscopic and a microscopic variable. Hence, weak compactness results based on standard a priori estimates are not enough to pass to the limit \begin{document}$ \epsilon \to 0 $\end{document} in the nonlinear terms. Therefore, we derive strong two-scale compactness results based on a variational principle. Further, we establish uniqueness for the microscopic and the macroscopic model.

We consider the nonlinear fractional elliptic system

\begin{document}$ \begin{equation*} \left\{\begin{array}{ll} (- \Delta)^{\frac{\alpha_1}{2}}u(x) = f(x, u, v), & \text{in}\, \, \, \Omega, \\ (- \Delta)^{\frac{\alpha_2}{2}}v(x) = g(x, u, v), & \text{in}\, \, \, \Omega, \\ u = v = 0, & \text{in}\, \, \, \mathbb{R}^n\setminus\Omega, \end{array} \right. \label{a-1.2} \end{equation*} $\end{document}

where \begin{document}$ 0<\alpha_1, \alpha_2<2 $\end{document} and \begin{document}$ \Omega $\end{document} is a bounded domain with \begin{document}$ C^2 $\end{document} boundary in \begin{document}$ \mathbb{R}^n $\end{document}. To overcome the technical difficulty due to the different fractional orders, we employ two distinct methods and derive the a priori estimates for \begin{document}$ 0<\alpha_1, \alpha_2<1 $\end{document} and \begin{document}$ 1<\alpha_1, \alpha_2 <2 $\end{document} respectively. Moreover, combining the a priori estimate with the topological degree theory, we prove the existence of positive solutions.

This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar \begin{document}$ \theta $\end{document} on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity \begin{document}$ u $\end{document} is of lower singularity, i.e., \begin{document}$ u = -\nabla^{\perp} \Lambda^{ \beta-2}p( \Lambda) \theta $\end{document}, where \begin{document}$ p $\end{document} is a logarithmic smoothing operator and \begin{document}$ \beta \in [0, 1] $\end{document}. We complete this study by considering the more singular regime \begin{document}$ \beta\in(1, 2) $\end{document}. The main tool is the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. We observe that this structure is ultimately crucial for obtaining continuity of the flow map. In particular, straightforward applications of previous methods for active transport equations fail to capture the more nuanced commutator structure of the equation in this more singular regime. The proposed linearized system nontrivially modifies the flux of the original system in such a way that it coincides with the original flux when evaluated along solutions of the original system. The requisite estimates are developed for this modified linear system to ensure its well-posedness.

In this paper we analyzed the integrability and asymptotic behavior of the positive solutions to the Euler-Lagrange system associated with a class of weighted Hardy-Littlewood-Sobolev inequality on the upper half space \begin{document}$ \mathbb{R}_+^n. $\end{document} We first obtained the optimal integrability for the solutions by the regularity lifting theorem. And then, with this integrability, we investigated the growth rate of the solutions around the origin and the decay rate near infinity.

Here, we consider positive singular solutions of

\begin{document}$ \begin{equation*} \left\{ \begin{array}{lcc} -\Delta u = |x|^\alpha |\nabla u|^p & \text{in}& \Omega \backslash\{0\},\\ u = 0&\text{on}& \partial \Omega, \end{array} \right. \end{equation*} $\end{document}

where \begin{document}$ \Omega $\end{document} is a small smooth perturbation of the unit ball in \begin{document}$ \mathbb{R}^N $\end{document} and \begin{document}$ \alpha $\end{document} and \begin{document}$ p $\end{document} are parameters in a certain range. Using an explicit solution on \begin{document}$ B_1 $\end{document} and a linearization argument, we obtain positive singular solutions on perturbations of the unit ball.

Marcinkiewicz integral operators on product domains defined by translates determined by twisted surfaces are introduced. Maximal functions along twisted surfaces are also introduced. Conditions on the underlined surfaces implying that the corresponding Marcinkiewicz integral operators map \begin{document}$ L^{p}\rightarrow L^{p} $\end{document} for \begin{document}$ 1<p<\infty $\end{document} are obtained. A general method involving lacunary families of multi-indices is developed.

We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic "phases" that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work "Nonlocal self-improving properties" Analysis & PDE, 8(1):57–114 for the specific nonlinear setting under investigation in this manuscript.

In this paper, we establish a large deviation principle for stochastic Burgers type equation with reflection perturbed by the small multiplicative noise. The main difficulties come from the highly non-linear coefficient and the singularity caused by the reflection. Here, we adopt a new sufficient condition for the weak convergence criteria, which is proposed by Matoussi, Sabbagh and Zhang [14].

In this paper we establish L^p-boundedness properties for variation operators defined by semigroups associated with Fourier-Bessel expansions.

We first prove that solutions of fractional p-Laplacian problems with nonlocal Neumann boundary conditions are bounded and then we apply such a result to study some resonant problems by means of variational tools and Morse theory.

In this work, we study the inverse scattering transform of a nonlocal Hirota equation in detail, and obtain the corresponding soliton solutions formula. Starting from the Lax pair of this equation, we obtain the corresponding infinite number of conservation laws and some properties of scattering data. By analyzing the direct scattering problem, we get a critical symmetric relation which is different from the local equations. A novel left-right Riemann-Hilbert problem is proposed to develop the inverse scattering theory. The potentials are recovered and the pure soliton solutions formula is obtained when the reflection coefficients are zero. Based on the zero types of scattering data, nine types of soliton solutions are obtained and three typical types are described in detail. In addition, some dynamic behaviors are given to illustrate the soliton characteristics of the space symmetric nonlocal Hirota equation.

The dynamical behavior of an SIRS epidemic reaction-diffusion model with frequency-dependent mechanism in a spatially heterogeneous environment is studied, with a chemotaxis effect that susceptible individuals tend to move away from higher concentration of infected individuals. Regardless of the strength of the chemotactic coefficient and the spatial dimension \begin{document}$ n $\end{document}, it is established the unique global classical solution which is uniformly-in-time bounded. The model still recognizes the threshold dynamics in terms of the basic reproduction number \begin{document}$ \mathcal{R}_{0} $\end{document} even in the case of chemotaxis effects: if \begin{document}$ \mathcal{R}_{0}<1 $\end{document}, the unique disease free equilibrium is globally stable; if \begin{document}$ \mathcal{R}_{0}>1 $\end{document}, the disease is uniformly persistent and there is at least one endemic equilibrium, which is globally stable in some special cases with weak chemotactic sensitivity. We also show the asymptotic profile of endemic equilibria (when exists) if the diffusion (migration) rate of the susceptible is small, which indicates that the disease always exists in the entire habitat in this case. Our results suggest that one cannot eradicate the SIRS disease model by only controlling the diffusion rate of susceptible individuals.

We prove the existence and multiplicity of subharmonic solutions for bounded coupled Hamiltonian systems. The nonlinearities are assumed to satisfy Landesman-Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is based on phase plane analysis and a higher dimensional version of the Poincaré-Birkhoff twist theorem by Fonda and Ureña. The results obtained generalize the previous works for scalar second-order differential equations or relativistic equations to higher dimensional systems.