American Institute of Mathematical Sciences

We are concerned with the coexistence states of the diffusive Lotka-Volterra system of two competing species when the growth rates of the two species depend periodically on the spacial variable. For the one-dimensional problem, we employ the generalized Jacobi elliptic function method to find semi-exact solutions under certain conditions on the parameters. In addition, we use the sine function to construct a pair of upper and lower solutions and obtain a solution of the above-mentioned system. Next, we provide a sufficient condition for the existence of pulsating fronts connecting two semi-trivial states by applying the abstract theory regarding monotone semiflows. Some numerical simulations are also included.

In this paper, we consider the subsolutions of the Monge-Ampère type equations \begin{document}$ {\det}^{\frac{1}{n}}(D^2u+\alpha I) = f(u) $\end{document} in \begin{document}$ \mathbb{R}^{n} $\end{document}. We obtain the necessary and sufficient condition of the existence of subsolutions.

This paper deals with changes of variables, and the exact bifurcation diagrams for a class of self-similar equations. Our first result is a change of variables which transforms radial \begin{document}$ k $\end{document}-Hessian equations into radial \begin{document}$ p $\end{document}-Laplace equations. Then, in another direction, we generalize the classical results of D.D. Joseph and T.S. Lundgren [10] by using the method we developed in [13] and [14]. We provide a considerably simpler approach, which yields additional information on the Morse index of solutions.

We introduce hybrid fractals as a class of fractals constructed by gluing several fractal pieces in a specific manner and study energy forms and Laplacians on them. We consider in particular a hybrid based on the \begin{document}$ 3 $\end{document}-level Sierpinski gasket, for which we construct explicitly an energy form with the property that it does not "capture" the \begin{document}$ 3 $\end{document}-level Sierpinski gasket structure. This characteristic type of energy forms that "miss" parts of the structure of the underlying space is investigated in the more general framework of finitely ramified cell structures. The spectrum of the associated Laplacian and its asymptotic behavior in two different hybrids is analyzed theoretically and numerically. A website with further numerical data analysis is available at http://www.math.cornell.edu/~harry970804/.

A delayed reaction-diffusion virus model with a general incidence function and spatially dependent parameters is investigated. The basic reproduction number for the model is derived, and the uniform persistence of solutions and global attractively of the equilibria are proved. We also show the global attractivity of the positive equilibria via constructing Lyapunov functional, in case that all the parameters are spatially independent. Numerical simulations are finally conducted to illustrate these analytical results.

Let \begin{document}$ \Omega\subset\mathbb{R}^n $\end{document} be a bounded domain. Let \begin{document}$ F: \mathbb{R}^n\rightarrow[0, +\infty) $\end{document} be a convex function of class \begin{document}$ C^2(\mathbb{R}^n\setminus\{0\}) $\end{document}, which is even and positively homogeneous of degree \begin{document}$ 1 $\end{document}. For such a function \begin{document}$ F $\end{document}, there exist two positive constants \begin{document}$ a_1\leq a_2 $\end{document} such that \begin{document}$ a_1|\xi|\leq F(\xi)\leq a_2|\xi|\; (\forall\xi\in\mathbb{R}^n) $\end{document}. Therefore, \begin{document}$ (\int_\Omega F(\nabla u)^n dx)^{1/n} $\end{document} and \begin{document}$ (\int_{\mathbb{R}^n}(F(\nabla u)^n+\tau |u|^n)dx)^{1/n} $\end{document} \begin{document}$ (\tau>0) $\end{document} are equivalent with the standard norms on \begin{document}$ W^{1, n}_0(\Omega) $\end{document} and \begin{document}$ W^{1, n}(\mathbb{R}^n) $\end{document} respectively. In this paper, we prove that

\begin{document}$ \begin{align*} \sup\limits_{u\in W^{1, n}_0(\Omega), \int_\Omega F(\nabla u)^n dx\leq1}\int_\Omega \frac{e^{\lambda|u|^{\frac{n}{n-1}}}}{F^0(x)^{\beta}}dx<+\infty \Leftrightarrow\frac{\lambda}{\lambda_n}+\frac{\beta}{n}\leq1 \end{align*} $\end{document}

and

\begin{document}$ \begin{align*} \sup\limits_{u\in W^{1, n}(\mathbb{R}^n), \int_{\mathbb{R}^n}(F(\nabla u)^n+\tau |u|^n)dx\leq1}\int_{\mathbb{R}^n}\frac{e^{\lambda|u|^{\frac{n}{n-1}}}-\sum_{k = 0}^{n-2}\frac{\lambda^k|u|^{\frac{nk}{n-1}}}{k!}}{F^0(x)^{\beta}}dx<\infty\nonumber\ \\ \Leftrightarrow\frac{\lambda}{\lambda_n}+\frac{\beta}{n}\leq1, \end{align*} $\end{document}

where \begin{document}$ F^0 $\end{document} is the polar function of \begin{document}$ F $\end{document}, \begin{document}$ \lambda>0 $\end{document}, \begin{document}$ \beta\in[0, n) $\end{document}, \begin{document}$ \tau>0 $\end{document}, \begin{document}$ \lambda_n = n^{\frac{n}{n-1}}\kappa_n^{\frac{1}{n-1}} $\end{document} and \begin{document}$ \kappa_n $\end{document} is the volume of the unit Wulff ball. Extremal functions for above two supremums are also considered.

We obtain estimates for sums of eigenvalues of the free plate under tension in terms of the dimension of the ambient space, the volume of the domain, and the tension parameter. We consequently obtain similar estimates for the eigenvalues. Our results generalize those of Kröger for the free membrane contained in [16].

We are concerned with nonlinear fractional Choquard equations involving critical growth in the sense of the Hardy-Littlewood-Sobolev inequality. Without the Ambrosetti-Rabinowitz condition or monotonicity condition on the nonlinearity, we establish the existence of radially symmetric ground state solutions.

We show the existence and \begin{document}$ C^{k, \gamma} $\end{document} smoothness of local integral manifolds at an equilibrium point for nonautonomous and ill-posed equations with sectorially dichotomous operator, provided that the nonlinearities are \begin{document}$ C^{k, \gamma} $\end{document} smooth with respect to the state variable. \begin{document}$ C^{k, \gamma} $\end{document} local unstable integral manifold follows from \begin{document}$ C^{k, \gamma} $\end{document} local stable integral manifold by reversing time variable directly. As an application, an elliptic PDE in infinite cylindrical domain is discussed.

We address small volume-fraction asymptotic properties of a nonlocal isoperimetric functional with a confinement term, derived as the sharp interface limit of a variational model for self-assembly of diblock copolymers under confinement by nanoparticle inclusion. We introduce a small parameter \begin{document}$ \eta $\end{document} to represent the size of the domains of the minority phase, and study the resulting droplet regime as \begin{document}$ \eta\to 0 $\end{document}. By considering confinement densities which are spatially variable and attain a unique nondegenerate maximum, we present a two-scale asymptotic analysis wherein a separation of length scales is captured due to competition between the nonlocal repulsive and confining attractive effects in the energy. A key role is played by a parameter \begin{document}$ M $\end{document} which gives the total volume of the droplets at order \begin{document}$ \eta^3 $\end{document} and its relation to existence and non-existence of Gamow's Liquid Drop model on \begin{document}$ \mathbb{R}^3 $\end{document}. For large values of \begin{document}$ M $\end{document}, the minority phase splits into several droplets at an intermediate scale \begin{document}$ \eta^{1/3} $\end{document}, while for small \begin{document}$ M $\end{document} minimizers form a single droplet converging to the maximum of the confinement density.

The morbidostat is a bacteria culture device that progressively increases antibiotic drug concentration and maintains a constant challenge for study of evolutionary pathway. The operation of a morbidostat under serial transfer has been analyzed previously. In this work, the global dynamics for the operation of a morbidostat under continuous dilution is analyzed. The device switches between drug on and drug off modes according to a simple threshold algorithm. We prove the extinction and uniform persistence of all species with both forward and backward mutations. Numerical simulations for the case of logistic growth and the Hill function for drug inhibition are also presented.

We are concerned with the existence and multiplicity of component-wise positive solutions for nonlinear system of Hammerstein integral equations with the weighted functions and the associated nonlinear eigenvalue problem. Our discussions are based on the product formula of fixed point index on product cones and the fixed point index theory. Moreover, we establish the existence and multiplicity of component-wise positive solutions for the associated nonlinear systems of second-order ordinary differential equations under the mixed boundary value conditions.

We prove the existence of positive classical solutions for the \begin{document}$ p $\end{document}-Laplacian problem

\begin{document}$ \begin{equation*} \left\{ \begin{array}{c} -(r(t)\phi (u^{\prime }))^{\prime } = -\frac{\lambda }{u^{\delta }}+f(t,u),\ t\in (0,1), \\ u(0) = u(1) = 0,\end{array}\right. \end{equation*} $\end{document}

where \begin{document}$ 0<\delta <1 $\end{document}, \begin{document}$ \phi (s) = |s|^{p-2}s $\end{document}, \begin{document}$ p>1 $\end{document}, \begin{document}$ f:(0,1)\times \lbrack 0,\infty )\rightarrow \mathbb{R} $\end{document} is a Carathéodory function satisfying \begin{document}$ \limsup\limits_{z\rightarrow 0^{+}}\frac{f(t,z)}{z^{p-1}}<\lambda _{1}<\liminf\limits_{z\rightarrow \infty }\frac{f(t,z)}{z^{p-1}} $\end{document} uniformly for a.e. \begin{document}$ t $\end{document} \begin{document}$ \in (0,1), $\end{document} where \begin{document}$ \lambda_{1} $\end{document} denotes the principal eigenvalue of \begin{document}$ -(r(t)\phi (u^{\prime }))^{\prime } $\end{document} with zero boundary conditions, and \begin{document}$ \lambda $\end{document} is a small nonnegative parameter.

The existence, non-existence and qualitative properties of time periodic pyramidal traveling front solutions for the time periodic Lotka-Volterra competition-diffusion system have already been studied in \begin{document}$ \Bbb{R}^{N} $\end{document} with \begin{document}$ N\geq 3 $\end{document}. In this paper, we continue to study the uniqueness and asymptotic stability of such time-periodic pyramidal traveling front in the three-dimensional whole space. For any given admissible pyramid, we show that the time periodic pyramidal traveling front is uniquely determined and it is asymptotically stable under the condition that given perturbations decay at infinity. Moreover, the time periodic pyramidal traveling front is uniquely determined as a combination of two-dimensional periodic V-form waves on the edges of the pyramid.

In this paper we prove the existence and multiplicity of subharmonic solutions for nonlinear equations involving the singular \begin{document}$ \phi $\end{document}-Laplacian. Such equations are in particular motivated by the one-dimensional mean curvature problems and by the acceleration of a relativistic particle of mass one at rest moving on a straight line. Our approach is based on phase-plane analysis and an application of the Poincaré-Birkhoff twist theorem.

This work studies the effects of rapid oscillations (with respect to time) of the forcing term on the long-time behaviour of the solutions of the Swift-Hohenberg equation. More precisely, we establish three kinds of averaging principles for the Swift-Hohenberg equation, they are averaging principle in a time-periodic problem, averaging principle on a finite time interval and averaging principle on the entire axis.

This paper is devoted to study the following systems of coupled elliptic equations with quadratic nonlinearity

\begin{document}$ \begin{equation*} \begin{cases} -\varepsilon^{2}\Delta v+P(x)v = \mu vw, &x\in{\mathbb{R}}^{N},\\ -\varepsilon^{2}\Delta w+Q(x)w = \frac{\mu}{2} v^{2}+\gamma w^{2}, &x\in{\mathbb{R}}^{N}, \end{cases} \end{equation*} $\end{document}

which arises from second- harmonic generation in quadratic optical media. We assume that the potential functions \begin{document}$ P(x) $\end{document} and \begin{document}$ Q(x) $\end{document} are positive functions and have a strict local maxima at \begin{document}$ x_{0} $\end{document}. Applying the finite dimensional reduction method, for any integer \begin{document}$ 1\leq k\leq N+1 $\end{document}, we prove the existence of positive solutions which have \begin{document}$ k $\end{document} local maximum points that concentrate at \begin{document}$ x_{0} $\end{document} simultaneously when \begin{document}$ \varepsilon $\end{document} is small.

In this paper we are interested in the following critical coupled Hartree system

\begin{document}$ \left\{\begin{array}{l} (-\Delta)^{s} \tilde{u}+\lambda_{1}\tilde{u} = \alpha_{1}\int_{\Omega}\frac{|\tilde{u}(z)|^{2_{\mu}^{\ast}} }{|x-z|^{\mu}}dz|\tilde{u}|^{2_{\mu}^{\ast}-2}\tilde{u}\\ \quad +\beta\int_{\Omega}\frac{|\tilde{v}(z)|^{2_{\mu}^{\ast}}} {|x-z|^{\mu}}dz|\tilde{u}|^{2_{\mu}^{\ast}-2}\tilde{u}, \ \ &&\mbox{in} \ \Omega,\\ (-\Delta)^{s} \tilde{v}+\lambda_{2}\tilde{v} = \alpha_{2}\int_{\Omega}\frac{|\tilde{v}(z)|^{2_{\mu}^{\ast}} }{|x-z|^{\mu}}dz|\tilde{v}|^{2_{\mu}^{\ast}-2}\tilde{v}\\ \quad +\beta\int_{\Omega}\frac{|\tilde{u}(z)|^{2_{\mu}^{\ast}}} {|x-z|^{\mu}}dz|\tilde{v}|^{2_{\mu}^{\ast}-2}\tilde{v}, \ \ &&\mbox{in} \ \Omega,\\ \tilde{u} = \tilde{v} = 0, \ \ && \mbox{on} \ \partial \Omega, \end{array} \right. $\end{document}

where \begin{document}$ 0<s<1 $\end{document}, \begin{document}$ \alpha_{1}, \alpha_{2}>0 $\end{document}, \begin{document}$ \beta\neq0 $\end{document}, \begin{document}$ 4s<\mu<N $\end{document}, \begin{document}$ 2_{\mu}^{\ast} = (2N-\mu)/(N-2s) $\end{document}, \begin{document}$ \Omega\subset\mathbb{R}^N(N\geq3) $\end{document} is a smooth bounded domain, \begin{document}$ -\lambda_{1}(\Omega)<\lambda_{1}, \lambda_{2}<0 $\end{document} with \begin{document}$ \lambda_{1}(\Omega) $\end{document} the first eigenvalue of \begin{document}$ (-\Delta)^{s} $\end{document} under the Dirichlet boundary condition. Assume that the nonlinearity and the coupling terms are both of the upper critical growth due to the Hardy–Littlewood–Sobolev inequality, by applying the Dirichlet-to-Neumann map, we are able to obtain the existence of the ground state solution of the critical coupled Hartree system.

In this paper we define a more general convolution product (associated with Gaussian processes) of functionals on the function space \begin{document}$ C_{a, b}[0, T] $\end{document}. The function space \begin{document}$ C_{a, b}[0, T] $\end{document} is induced by a generalized Brownian motion process. Thus the Gaussian processes used in this paper are non-centered processes. We then develop the fundamental relationships between the generalized Fourier–Feynman transform associated with the Gaussian process and the convolution product.

This paper is concerned with a class of biharmonic elliptic differential inclusion in \begin{document}$ \mathbb R^N $\end{document}. Under various growth conditions on the nonlinearity, new results on existence and multiplicity of solutions are derived. The main tools used in this paper are the nonsmooth version of mountain pass theorem and the generalized gradient for locally Lipschitz functionals.

Recently, in [9] is characterized the analytic integrability problem around a nilpotent singularity for differential systems in the plane under generic conditions. In this work we solve the remaining case completing the analytic integrability problem for such singularity.

We consider the initial value problem (IVP) associated with the Schrödinger-Debye system posed on a $d$-dimensional compact Riemannian manifold \begin{document}$M $\end{document} and prove the local well-posedness result for given data \begin{document}$ (u_0, v_0)\in H^s(M)\times (H^s(M)\cap L^{\infty}(M))$\end{document} whenever \begin{document}$s>\frac{d}2-\frac12 $\end{document}, \begin{document}$d\geq 2 $\end{document}. For \begin{document}$d=2 $\end{document}, we apply a sharp version of the Gagliardo-Nirenberg inequality in compact manifold to derive an a priori estimate for the \begin{document}$H^1 $\end{document}-solution and use it to prove the global well-posedness result in this space.

This paper is devoted to the study of a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Based on the Faedo-Galerkin method and standard arguments of density corresponding to the regularity of initial conditions, we first establish two local existence theorems of weak solutions. By the construction of a suitable Lyapunov functional, we next prove a blow up result and a decay result of global solutions.

We prove the weak maximum principle for second-order elliptic and parabolic equations in divergence form with the conormal derivative boundary conditions when the lower-order coefficients are unbounded and domains are beyond Lipschitz boundary regularity. In the elliptic case we consider John domains and lower-order coefficients in \begin{document}$ L_n $\end{document} spaces (\begin{document}$ a^i, b^i \in L_q $\end{document}, \begin{document}$ c \in L_{q/2} $\end{document}, \begin{document}$ q = n $\end{document} if \begin{document}$ n \geq 3 $\end{document} and \begin{document}$ q > 2 $\end{document} if \begin{document}$ n = 2 $\end{document}). For the parabolic case, the lower-order coefficients \begin{document}$ a^i $\end{document}, \begin{document}$ b^i $\end{document}, and \begin{document}$ c $\end{document} belong to \begin{document}$ L_{q,r} $\end{document} spaces (\begin{document}$ a^i,b^i, |c|^{1/2} \in L_{q,r} $\end{document} with \begin{document}$ n/q+2/r \leq 1 $\end{document}), \begin{document}$ q \in (n,\infty] $\end{document}, \begin{document}$ r \in [2,\infty] $\end{document}, \begin{document}$ n\ge 2 $\end{document}. We also consider coefficients in \begin{document}$ L_{n,\infty} $\end{document} with a smallness condition for parabolic equations.

We consider the boundary value problem

\begin{document}$\begin{equation*} \begin{cases} -{\rm div}_G(w_1\nabla_G u) = w_2f(u) &\text{ in } \Omega,\\ u=0 &\text{ on } \partial\Omega, \end{cases}\end{equation*}$ \end{document}

where \begin{document}$\Omega$\end{document} is a bounded or unbounded \begin{document}$C^1$\end{document} domain of \begin{document}$\mathbb{R}^N$\end{document}, \begin{document}$w_1, w_2 \in L^1_{\rm loc}(\Omega)\setminus\{0\}$\end{document} are nonnegative functions, \begin{document}$f$\end{document} is an increasing function, \begin{document}$\nabla_G$\end{document} and \begin{document}${\rm div}_G$\end{document} are Grushin gradient and Grushin divergence, respectively. We prove some Liouville theorems for stable weak solutions of the problem under suitable assumptions on \begin{document}$\Omega$\end{document}, \begin{document}$w_1$\end{document}, \begin{document}$w_2$\end{document} and \begin{document}$f$\end{document}. We also show the sharpness of our results when \begin{document}$\Omega=\mathbb{R}^N$\end{document} and \begin{document}$f$\end{document} has power or exponential growth.

Let \begin{document}$ u \in L_{sp} \cap C^{1, 1}_{\rm loc}(\mathbb{R}^n\setminus\{0\}) $\end{document} be a positive solution, which may blow up at zero, of the equation

\begin{document}$ (-\Delta)^s_p u = \left(\frac{1}{|x|^{n-\beta }} * \frac{u^q}{|x|^\alpha}\right) \frac{u^{q-1 }}{|x|^\alpha} \quad\text{ in } \mathbb{R}^n \setminus \{0\}, $\end{document}

where \begin{document}$ 0 < s < 1 $\end{document}, \begin{document}$ 0 < \beta < n $\end{document}, \begin{document}$ p>2 $\end{document}, \begin{document}$ q\ge 1 $\end{document} and \begin{document}$ \alpha>0 $\end{document}. We prove that if \begin{document}$ u $\end{document} satisfies some suitable asymptotic properties, then \begin{document}$ u $\end{document} must be radially symmetric and monotone decreasing about the origin. In stead of using equivalent fractional systems, we exploit a direct method of moving planes for the weighted Choquard nonlinearity. This method allows us to cover the full range \begin{document}$ 0 < \beta < n $\end{document} in our results.

This paper focuses on almost-periodic time-dependent perturbations of a class of almost-periodically forced systems near non-hyperbolic equilibrium points in two cases: (a) elliptic case, (b) degenerate case (including completely degenerate). In elliptic case, it is shown that, under suitable hypothesis of analyticity, nonresonance and nondegeneracy with respect to perturbation parameter \begin{document}$ \epsilon, $\end{document} there exists a Cantor set \begin{document}$ \mathcal{E}\subset (0, \epsilon_0) $\end{document} of positive Lebesgue measure with sufficiently small \begin{document}$ \epsilon_0 $\end{document} such that for each \begin{document}$ \epsilon\in\mathcal{E} $\end{document} the system has an almost-periodic response solution. In degenerate case, we prove that, firstly, the almost-periodically perturbed degenerate system in one-dimensional case admits an almost-periodic response solution under nonzero average condition on perturbation and some weak non-resonant condition; Secondly, imposing further restriction on smallness of the perturbation besides nonzero average, we prove the almost-periodically forced degenerate system in \begin{document}$ n $\end{document}-dimensional case has an almost-periodic response solution under small perturbation without any non-resonant condition; Finally, almost-periodic response solution can still be obtained with weakened nonzero average condition by used Herman method but non-resonant condition should be strengthened. Some proofs of main results are based on a modified Pöschel-Rüssmann KAM method, our results show that Pöschel-Rüssmann KAM method can be applied to study the existence of almost-periodic solutions for almost-periodically forced non-conservative systems. Our results generalize the works in [14,13,23,20] from quasi-periodic case to almost-periodic case and also give rise to the reducibility of almost-periodic perturbed linear differential systems.

In this paper we consider the semi-classical solutions of a massive Dirac equations in presence of a critical growth nonlinearity

\begin{document}$ -i\hbar \sum\limits_{k = 1}^{3}\alpha_k\partial_k w+a\beta w+V(x)w = f(|w|)w. $\end{document}

Under a local condition imposed on the potential \begin{document}$ V $\end{document}, we relate the number of solutions with the topology of the set where the potential attains its minimum. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

We prove a sharp resolvent estimate in scale invariant norms of Amgon–Hörmander type for a magnetic Schrödinger operator on \begin{document}$ \mathbb{R}^{n} $\end{document}, \begin{document}$ n\ge3 $\end{document}

\begin{document}$ \begin{equation*} L = -(\partial+iA)^{2}+V \end{equation*} $\end{document}

with large potentials \begin{document}$ A, V $\end{document} of almost critical decay and regularity.

The estimate is applied to prove sharp smoothing and Strichartz estimates for the Schrödinger, wave and Klein–Gordon flows associated to \begin{document}$ L $\end{document}.