American Institute of Mathematical Sciences

The purpose of this paper is to establish a Cameron-Storvick theorem for the analytic Feynman integral of functionals in non-stationary Gaussian processes on Wiener space. As interesting applications, we apply this theorem to evaluate the generalized analytic Feynman integral of certain polynomials in terms of Paley-Wiener-Zygmund stochastic integrals.

In this paper we consider the multiplicity and concentration behavior of positive solutions for the following fractional nonlinear Schrödinger equation

\begin{document}$\left\{ \begin{align} &{{\varepsilon }^{2s}}{{\left( -\Delta \right)}^{s}}u+V\left( x \right)u = f\left( u \right)\ \ \ \ \ \ x\in {{\mathbb{R}}^{N}}, \\ &u\in {{H}^{s}}\left( {{\mathbb{R}}^{N}} \right)\ \ \ \ \ \ \ \ u\left( x \right)>0, \\ \end{align} \right.$ \end{document}

where \begin{document}$\varepsilon$\end{document} is a positive parameter, \begin{document}$(-Δ)^{s}$\end{document} is the fractional Laplacian, \begin{document}$s ∈ (0,1)$\end{document} and \begin{document}$N> 2s$\end{document}. Suppose that the potential \begin{document}$V(x) ∈\mathcal{C}(\mathbb{R}^{N})$\end{document} satisfies \begin{document}$\text{inf}_{\mathbb{R}^{N}} V(x)>0$\end{document}, and there exist \begin{document}$k$\end{document} points \begin{document}$x^{j} ∈ \mathbb{R}^{N}$\end{document} such that for each \begin{document}$j = 1,···,k$\end{document}, \begin{document}$V(x^{j})$\end{document} are strict global minimum. When \begin{document}$f$\end{document} is subcritical, we prove that the problem has at least \begin{document}$k$\end{document} positive solutions for \begin{document}$\varepsilon>0$\end{document} small. Moreover, we establish the concentration property of the solutions as \begin{document}$\varepsilon$\end{document} tends to zero.

We study the combined effect of concave and convex nonlinearities on the numbers of positive solutions for a fractional equation involving critical Sobolev exponents. In this paper, we concerned with the following fractional equation

\begin{document}$ \left\{ \begin{array}{l}{\left( { - \Delta } \right)^s}u = \lambda f\left( x \right){\left| u \right|^{q - 2}}u + g\left( x \right){\left| u \right|^{2_s^* - 2}}u,\;\;\;x \in \Omega ,\\u = 0,\;\;x \in \partial \Omega ,\end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right) $ \end{document}

where \begin{document}$ 0<s<1 $\end{document}, \begin{document}$ λ>0 $\end{document}, \begin{document}$ 1≤q <2$\end{document}, \begin{document}$ 2_s^* = \frac{2N}{N-2s} $\end{document}, \begin{document}$ 0∈ Ω\subset \mathbb{R} ^N(N>4s) $\end{document} is a bounded domain with smooth boundary \begin{document}$ \partialΩ $\end{document}, and \begin{document}$ f,\,g $\end{document} are nonnegative continuous functions on \begin{document}$\bar{Ω} $\end{document}. Here \begin{document}$ (-Δ)^s $\end{document} denotes the fractional Laplace operator.

In this paper, we consider the nonstationary Navier-Stokes equations approximated by the pressure stabilization method. We can obtain the local in time existence theorem for the approximated Navier-Stokes equations. Moreover we can obtain the error estimate between the solution to the usual Navier-Stokes equations and the Navier-Stokes equations approximated by the pressure stabilization method.

In this paper, we investigate the Moser-Trudinger inequality when it involves a Finsler-Laplacian operator that is associated with functionals containing \begin{document}$F^2(\nabla u)$\end{document}. Here \begin{document}$F$\end{document} is convex and homogeneous of degree 1, and its polar \begin{document}$F^o$\end{document} represents a Finsler metric on \begin{document}$\mathbb{R}^n$\end{document}. We obtain an existence result on the extremal functions for this sharp geometric inequality.

In this paper, we construct Liouville theorem for the MHD system and apply it to study the potential singularities of its weak solution. And we mainly study weak axi-symmetric solutions of MHD system in \begin{document}$\mathbb{R}^3× (0, T)$\end{document}.

Many important index transforms can be constructed via the spectral theory of Sturm-Liouville differential operators. Using the spectral expansion method, we investigate the general connection between the index transforms and the associated parabolic partial differential equations.

We show that the notion of Yor integral, recently introduced by the second author, can be extended to the class of Sturm-Liouville integral transforms. We furthermore show that, by means of the Feynman-Kac theorem, index transforms can be used for studying Markovian diffusion processes. This gives rise to new applications of index transforms to problems in mathematical finance.

We establish a Liouville type theorem for nonnegative cylindrical solutions to the extension problem corresponding to a fractional CR covariant equation on the Heisenberg group by using the generalized CR inversion and the moving plane method.

In this paper, we prove the uniqueness and stability of viscosity solutions of the following initial-boundary problem related to the random game named tug-of-war with a transport term

\begin{document}$\left\{ \begin{array}{*{35}{l}} {{u}_{t}}-\Delta _{\infty }^{N}u-\langle \xi ,Du\rangle = f(x,t),\ \ \ \ \ \ \text{in}\ \ {{Q}_{T}}, \\ u = g,\ \ \ \ \ \ \ \ \text{on}\ \ \ \ \ {{\partial }_{p}}{{Q}_{T}}, \\\end{array} \right. $ \end{document}

where \begin{document}$ \Delta _{\infty }^{N}u = \frac{1}{{{\left| Du \right|}^{2}}}\sum\limits_{i,j = 1}^{n}{{{u}_{{{x}_{i}}}}}{{u}_{{{x}_{j}}}}{{u}_{{{x}_{i}}{{x}_{j}}}}$\end{document} denotes the normalized infinity Laplacian, \begin{document}$ ξ: Q_T\to R^n$\end{document} is a continuous vector field, \begin{document}$ f$\end{document} and \begin{document}$ g$\end{document} are continuous. When \begin{document}$ ξ$\end{document} is a fixed field and the inhomogeneous term \begin{document}$ f$\end{document} is constant, the existence is obtained by the approximate procedure. When \begin{document}$ f$\end{document} and \begin{document}$ ξ$\end{document} are Lipschitz continuous, we also establish the Lipschitz continuity of the viscosity solutions. Furthermore we establish the comparison principle of the generalized equation with the first order term with initial-boundary condition

\begin{document}${u_t}(x,t) -Δ _∞ ^N u (x,t) -H(x,t,Du(x,t)) = f(x,t),$ \end{document}

where \begin{document}$ H(x,t,p):Q_T× R^n\to R$\end{document} is continuous, \begin{document}$ H(x,t,0) = 0$\end{document} and grows at most linearly at infinity with respect to the variable \begin{document}$ p$\end{document}. And the existence result is also obtained when \begin{document}$ H(x,t,p) = H(p)$\end{document} and \begin{document}$ f$\end{document} is constant for the generalized equation.

This paper is devoted to the Boussinesq equations that models natural convection in a viscous fluid by coupling Navier-Stokes and heat equations via a zero order approximation. We consider the problem in \begin{document}$ \mathbb{R}^{n}$\end{document} and prove the existence of stationary solutions in critical Besov-Lorentz-Morrey spaces. For that, we prove some estimates for the product of distributions in these spaces, as well as Bernstein inequalities and Mihlin multiplier type results in our setting. Considering in particular the decoupled case, our existence result provides a new class of stationary solutions for the Navier-Stokes equations in critical spaces.

In this paper, we prove Perelman type \begin{document}$ \mathcal{W}$\end{document}-entropy formulae and global differential Harnack estimates for positive solutions to porous medium equation on the closed Riemannian manifolds with Ricci curvature bounded below. As applications, we derive Harnack inequalities and Laplacian estimates.

This work is concerned with a non-isothermal diffuse-interface model which describes the motion of a mixture of two viscous incompressible fluids. The model consists of modified Navier-Stokes equations coupled with a phase-field equation given by a convective Allen-Cahn equation, and energy transport equation for the temperature. This model admits a dissipative energy inequality. It is investigated the well-posedness of the problem in the two and three dimensional cases without any restriction on the size of the initial data. Moreover, regular and singular potentials for the phase-field equation are considered.

This paper deals with some existence results in Banach spaces for Hilfer and Hilfer-Hadamard fractional differential inclusions. The main tools used in the proofs are Mönch's fixed point theorem and the concept of a measure of noncompactness.

In this article we present several results concerning uniqueness of \begin{document}$C$\end{document}-viscosity and \begin{document}$L_{p}$\end{document}-viscosity solutions for fully nonlinear parabolic equations. In case of the Isaacs equations we allow lower order terms to have just measurable bounded coefficients. Higher-order coefficients are assumed to be Hölder continuous in \begin{document}$x$\end{document} with exponent slightly less than \begin{document}$1/2$\end{document}. This case is treated by using stability of maximal and minimal \begin{document}$L_{p}$\end{document}-viscosity solutions.

This paper is concerned with entire solutions for a two-dimensional periodic lattice dynamical system with nonlocal dispersal. In the bistable case, by applying comparison principle and constructing appropriate upper- and lowersolutions, two different types of entire solutions are constructed. The first type behaves like a monostable front merges with a bistable front and one chases another from the same side; while the other type can be represented by two monostable fronts merge and converge to a single bistable front. In the monostable case, we first establish the existence and properties of spatially periodic solutions which connect two steady states. Then new types of entire solutions are constructed by mixing a heteroclinic orbit of the spatially averaged ordinary differential equations with traveling wave fronts with different speeds. Further, for a class of special heterogeneous reaction function, we establish the uniqueness and continuous dependence of the entire solution on parameters, such as wave speeds and shifted variables.

We study the regularity of stable solutions to the problem

\begin{document}$\begin{align}\left\{ \begin{gathered} {\left( { - \Delta } \right)^s}&u = f\left( u \right)&{\text{in}}\;\;{B_1}, \hfill \\ &u \equiv 0&{\text{in}}\;\;{{\mathbb{R}}^n}\backslash {B_1}, \hfill \\ \end{gathered} \right.\end{align}$ \end{document}

where \begin{document}$s∈(0,1)$ \end{document}. Our main result establishes an \begin{document}$L^∞$ \end{document} bound for stable and radially decreasing \begin{document}$H^s$ \end{document} solutions to this problem in dimensions \begin{document}$2 ≤ n < 2(s+2+\sqrt{2(s+1)})$ \end{document}. In particular, this estimate holds for all \begin{document}$s∈(0,1)$ \end{document} in dimensions \begin{document}$2 ≤ n≤ 6$ \end{document}. It applies to all nonlinearities \begin{document}$f∈ C^2$ \end{document}.

For such parameters \begin{document}$s$ \end{document} and \begin{document}$n$ \end{document}, our result leads to the regularity of the extremal solution when \begin{document}$f$ \end{document} is replaced by \begin{document}$λ f$ \end{document} with \begin{document}$λ > 0$ \end{document}. This is a widely studied question for \begin{document}$s = 1$ \end{document}, which is still largely open in the nonradial case both for \begin{document}$s = 1$ \end{document} and \begin{document}$s < 1$ \end{document}.

In this paper, we study a free boundary problem for a class of parabolic-elliptic type chemotaxis model in high dimensional symmetry domain Ω. By using the contraction mapping principle and operator semigroup approach, we establish the existence of the solution for such kind of chemotaxis system in the domain Ω with free boundary condition. Besides, we get the explicit formula for the free boundary and show the chemotactic collapse for the solution of the system.

In this paper, we study the following Kirchhoff-type equation with critical growth

\begin{document}$-(a+b\int {_{\mathbb{R}^3}} |\nabla u|^2dx)\triangle u+V(x)u = λ f(x,u)+|u|^4u, \; x \; ∈\mathbb{R}^3,$ \end{document}

where a>0, b>0, λ>0 and f is a continuous superlinear but subcritical nonlinearity. When V and f are asymptotically periodic in x, we prove that the equation has a ground state solution for large λ by Nehari method. Moreover, we regard b as a parameter and obtain a convergence property of the ground state solution as \begin{document}$b\searrow 0$\end{document}.

We consider a parametric (p, q)-equation with competing nonlinearities in the reaction. There is a parametric concave term and a resonant Caratheordory perturbation. The resonance is with respect to the principal eigenvalue and occurs from the right. So the energy functional of the problem is indefinite. Using variational tools and truncation and comparison techniques we show that for all small values of the parameter the problem has at least two positive smooth solutions.

We prove that the Cauchy problem for the two-dimensional Zakharov system is locally well-posed for initial data which are localized perturbations of a line solitary wave. Furthermore, for this Zakharov system, we prove a weak convergence to a nonlinear Schrödinger equation.

The exterior Bernoulli free boundary problem is considered and reformulated into a shape optimization setting wherein the Neumann data is being tracked. The shape differentiability of the cost functional associated with the formulation is studied, and the expression for its shape derivative is established through a Lagrangian formulation coupled with the velocity method. Also, it is illustrated how the computed shape derivative can be combined with the modified $H^1$ gradient method to obtain an efficient algorithm for the numerical solution of the shape optimization problem.

A predator prey model with nonlinear harvesting (Holling type-Ⅱ) with both constant and distributed delay is considered. The boundeness of solutions is proved and some sufficient conditions ensuring the persistence of the two populations are established. Also, a detailed study of the bifurcation of positive equilibria is provided. All the results are illustrated by some numerical simulations.

In this paper the author studies the isoperimetric problem in \begin{document} ${\mathbb{R}}^n$ \end{document} with perimeter density \begin{document} $|x|^p$ \end{document} and volume density 1. We settle completely the case \begin{document} $n = 2$ \end{document}, completing a previous work by the author: we characterize the case of equality if \begin{document} $0≤p≤1$ \end{document} and deal with the case \begin{document} $-∞<p<-1$ \end{document} (with the additional assumption \begin{document} $0∈Ω$ \end{document}). In the case \begin{document} $n≥3$ \end{document} we deal mainly with the case \begin{document} $-∞<p<0$ \end{document}, showing among others that the results in 2 dimensions do not generalize for the range \begin{document} $-n+1<p<0.$ \end{document}

In this article we prove the existence and uniqueness of a (weak) solution \begin{document}$u$\end{document} in \begin{document}$L_p\left( (0, T); Λ_{γ+m}\right)$\end{document} to the Cauchy problem

\begin{document}$\begin{align}\notag&\frac{\partial u}{\partial t}(t, x) = ψ(t, i\nabla)u(t, x)+f(t, x), \;\;\;(t, x) ∈ (0, T) × {\bf{R}}^d \\& u(0, x) = 0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)\end{align}$ \end{document}

where \begin{document}$d ∈ \mathbb{N}$\end{document}, \begin{document}$p ∈ (1, ∞]$\end{document}, \begin{document}$γ, m ∈ (0, ∞)$\end{document}, \begin{document}$Λ_{γ+m}$\end{document} is the Lipschitz space on \begin{document}${\bf{R}}^d$\end{document} whose order is \begin{document}$γ+m$\end{document}, \begin{document}$f ∈ L_p\left( (0, T) ; Λ_{γ} \right)$\end{document}, and \begin{document}$ψ(t, i\nabla)$\end{document} is a time measurable pseudo-differential operator whose symbol is \begin{document}$ψ(t, ξ)$\end{document}, i.e.

\begin{document}$ψ(t, i\nabla)u(t, x) = \mathcal{F}^{-1}[ψ(t, ξ){\mathcal{F}}\left[u(t, ·)\right]\left(ξ)\right](x), $ \end{document}

with the assumptions

\begin{document}$\begin{align*}\Re[ψ(t, ξ)] ≤ -ν|ξ|^{γ}, \end{align*}$ \end{document}

and

\begin{document}$\begin{align*}|D_{ξ}^{α}ψ(t, ξ)|≤ν^{-1}|ξ|^{γ-|α|}.\end{align*}$ \end{document}

Furthermore, we show

\begin{document}$\begin{align}\int_0^T \|u(t, ·)\|^p_{Λ_{γ+m}} dt ≤ N \int_0^T \|f(t, ·)\|^p_{Λ_{m}} dt, \;\;\;\;\;\;\;\;\;\;(2)\end{align}$ \end{document}

where \begin{document}$N$\end{document} is a positive constant depending only on \begin{document}$d$\end{document}, \begin{document}$p$\end{document}, \begin{document}$γ$\end{document}, \begin{document}$ν$\end{document}, \begin{document}$m$\end{document}, and \begin{document}$T$\end{document},

The unique solvability of equation (1) in \begin{document}$L_p$\end{document}-Hölder space is also considered.More precisely, for any \begin{document}$f ∈ L_p((0, T);C^{n+α})$\end{document}, there exists a unique solution \begin{document}$u ∈ L_p((0, T);C^{γ+n+α}({\bf{R}}^d))$\end{document} to equation (1) and for this solution \begin{document}$u$\end{document},

\begin{document}$\begin{align}\int_0^T \|u(t, ·)\|^p_{C^{γ+n+α}}dt ≤N \int_0^T \|f(t, ·)\|^p_{C^{n+α}}dt, \;\;\;\;\;\;\;\;\;\;(3)\end{align}$ \end{document}

where \begin{document}$n ∈ \mathbb{Z}_+$\end{document}, \begin{document}$α ∈ (0, 1)$\end{document}, and \begin{document}$γ+α \notin \mathbb{Z}_+$\end{document}.

In this article we consider a special type of degenerate elliptic partial differential equations of second order in convex domains that satisfy the interior sphere condition. We show that any positive viscosity solution \begin{document} $u$ \end{document} of

\begin{document}$-|\nabla u|^α Δ^N_p u = 1$ \end{document}

has the property that \begin{document} $u^\frac{α+1}{α+2}$ \end{document} is a concave function. Secondly we consider positive solutions of the eigenvalue problem

\begin{document}$-|\nabla u|^α Δ^N_p u = λ |u|^α u, $ \end{document}

in which case \begin{document} $\log u$ \end{document} turns out to be concave. The methods provided include a weak comparison principle and a Hopf-type Lemma.

We establish the existence of a positive solution for semilinear elliptic equation in exterior domains

\begin{document}$\begin{array}{lc}-Δ u + V(x) u = f(u),\ \ {\rm{in}} \ \ Ω \subseteq \mathbb{R}^N &&&(P_V)\end{array}$\end{document}

where \begin{document}$N≥2$\end{document}, \begin{document}$Ω$\end{document} is an open subset of \begin{document}$\mathbb{R}^N$\end{document} and \begin{document}$ \mathbb{R}^N \setminus Ω $\end{document} is bounded and not empty but there is no restriction on its size, nor any symmetry assumption. The nonlinear term \begin{document}$f$\end{document} is a non homogeneous, asymptotically linear or superlinear function at infinity. Moreover, the potential Ⅴ is a positive function, not necessarily symmetric. The existence of a solution is established in situations where this problem does not have a ground state.

In this paper, first we survey the recent progress in usage of the critical point theory to study the existence of multiple periodic and subharmonic solutions in second order difference equations and discrete Hamiltonian systems with variational structure. Next, we propose a new topological method, based on the application of the equivariant version of the Brouwer degree to study difference equations without an extra assumption on variational structure. New result on the existence of multiple periodic solutions in difference systems (without assuming that they are their variational) satisfying a Nagumo-type condition is obtained. Finally, we also put forward a new direction for further investigations.

We study bounded, unbounded and blow-up solutions of a delay logistic equation without assuming the dominance of the instantaneous feedback. It is shown that there can exist an exponential (thus unbounded) solution for the nonlinear problem, and in this case the positive equilibrium is always unstable. We obtain a necessary and sufficient condition for the existence of blow-up solutions, and characterize a wide class of such solutions. There is a parameter set such that the non-trivial equilibrium is locally stable but not globally stable due to the co-existence with blow-up solutions.