We prove a strong optimal Hardy–Sobolev inequality for the twisted Laplacian on ℂn . The twisted Laplacian is the magnetic Laplacian for a system of n particles in the plane, corresponding to the constant magnetic field. The inequality we obtain is strong optimal in the sense that the weight cannot be improved. We also show that our result extends to a one-parameter family of weighted Sobolev spaces.

We consider semilinear elliptic problems on two-dimensional hyperbolic space. A model problem of our study is

where H 1(𝔹2) denotes the Sobolev space on the disc model of the hyperbolic space and f(x, t) denotes the function of critical growth in dimension 2. We first establish the Palais–Smale (PS) condition for the functional corresponding to the above equation, and using the PS condition we obtain existence of solutions. In addition, using a concentration argument, we also explore existence of infinitely many sign-changing solutions.

We investigate a system of singular–degenerate parabolic equations with non-local terms, which can be regarded as a spatially heterogeneous competition model of Lotka–Volterra type. Applying the Leray–Schauder fixed-point theorem, we establish the existence of coexistence periodic solutions to the problem, which, together with the existing literature, gives a complete picture for such a system for all parameters.

We consider an infinite planar straight strip perforated by small holes along a curve. In such a domain, we consider a general second-order elliptic operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic and satisfies rather weak assumptions, we describe all possible homogenized problems. Our main result is the norm-resolvent convergence of the perturbed operator to a homogenized one in various operator norms and the estimates for the rate of convergence. On the basis of the norm-resolvent convergence, we prove the convergence of the spectrum.

Let Ω be an open connected cone in ℝn with vertex at the origin. Assume that the Operator

is subcritical in Ω, where δΩ is the distance function to the boundary of Ω and μ ⩽ 1/4. We show that under some smoothness assumption on Ω the improved Hardy-type inequality

holds true, and the Hardy-weight λ(μ)|x|–2 is optimal in a certain definite sense. The constant λ(μ) > 0 is given explicitly.

In this paper we establish the existence and multiplicity of multi-bump nodal solutions for the class of problems

where λ ∈ (0, ∞), f is a continuous function with exponential critical growth and V : ℝ2→ ℝ is a continuous function verifying some hypotheses.

We consider the following equation:

where d(x) = d(x, ∂Ω), θ > –2 and Ω is a half-space. The existence and non-existence of several kinds of positive solutions to this equation when , f(u) = up (p > 1) and Ω is a bounded smooth domain were studied by Bandle, Moroz and Reichel in 2008. Here, we study exact the behaviour of positive solutions to this equation as d(x) → 0+ and d(x) → ∞, respectively, and the symmetry of positive solutions when , Ω is a half-space and f(u) is a more general nonlinearity term than up . Under suitable conditions for f, we show that the equation has a unique positive solution W, which is a function of x 1 only, and W satisfies

In this paper, we study the Cauchy problem for the semilinear heat and Schrödinger equations, with the nonlinear term $f(u)=\unicode[STIX]{x1D706}|u|^{\unicode[STIX]{x1D6FC}}u$. We show that low regularity of $f$ (i.e., $\unicode[STIX]{x1D6FC}>0$ but small) limits the regularity of any possible solution for a certain class of smooth initial data. We employ two different methods, which yield two different types of results. On the one hand, we consider the semilinear equation as a perturbation of the ODE $w_{t}=f(w)$. This yields, in particular, an optimal regularity result for the semilinear heat equation in Hölder spaces. In addition, this approach yields ill-posedness results for the nonlinear Schrödinger equation in certain $H^{s}$-spaces, which depend on the smallness of $\unicode[STIX]{x1D6FC}$ rather than the scaling properties of the equation. Our second method is to consider the semilinear equation as a perturbation of the linear equation via Duhamel’s formula. This yields, in particular, that if $\unicode[STIX]{x1D6FC}$ is sufficiently small and $N$ is sufficiently large, then the nonlinear heat equation is ill-posed in $H^{s}(\mathbb{R}^{N})$ for all $s\geqslant 0$.

The explosion probability before time t of a branching diffusion satisfies a nonlinear parabolic partial differential equation. This equation, along with the natural boundary and initial conditions, has only the trivial solution, i.e. explosion in finite time does not occur, provided the creation rate does not grow faster than the square power at ∞.

This paper is concerned with the asymptotic behaviour of the lifespan of solutions for a semilinear heat equation with initial datum λφ(x) in hyperbolic space. The growth rates for both λ → 0 and λ → ∞ are determined.

We study classical solutions of the Cauchy problem for a class of non-Lipschitz semilinear parabolic partial differential equations in one spatial dimension with sufficiently smooth initial data. When the nonlinearity is Lipschitz continuous, results concerning existence, uniqueness and continuous dependence on initial data are well established (see, for example, the texts of Friedman and Smoller and, in the context of the present paper, see also Meyer), as are the associated results concerning Hadamard well-posedness. We consider the situations when the nonlinearity is Hölder continuous and when the nonlinearity is upper Lipschitz continuous. Finally, we consider the situation when the nonlinearity is both Hölder continuous and upper Lipschitz continuous. In each case we focus upon the question of existence, uniqueness and continuous dependence on initial data, and thus upon aspects of Hadamard well-posedness.

We study a susceptible–infected–susceptible reaction–diffusion model with spatially heterogeneous disease transmission and recovery rates. A basic reproduction number is defined for the model. We first prove that there exists a unique endemic equilibrium if . We then consider the global attractivity of the disease-free equilibrium and the endemic equilibrium for two cases. If the disease transmission and recovery rates are constants or the diffusion rate of the susceptible individuals is equal to the diffusion rate of the infected individuals, we show that the disease-free equilibrium is globally attractive if , while the endemic equilibrium is globally attractive if .

We study the following coupled nonlinear Schr¨odinger system in ℝ3:

where μ1 > 0, μ2 > 0 and β ∈ ℝ is a coupling constant. Irrespective of whether the system is repulsive or attractive, we prove that it has nodal semi-classical segregated or synchronized bound states with clustered spikes for sufficiently small ε under some additional conditions on P(x), Q(x) and β. Moreover, the number of this type of solutions will go to infinity as ε → 0+.

We obtain all the solutions of types u(x, y) = f(x) + g(y) and u(x, y) = f(x)g(y) for three known mean-curvature-prescribed equations, namely, the capillary equation, the translating soliton equation and the two-dimensional analogue of the catenary.

In this paper we present the basic tools of a fractional function theory in higher dimensions by means of a fractional correspondence to the Weyl relations via fractional Riemann–Liouville derivatives. A Fischer decomposition, Almansi decomposition, fractional Euler and Gamma operators, monogenic projection, and basic fractional homogeneous powers are constructed. Moreover, we establish the fractional Cauchy–Kovalevskaya extension (FCK extension) theorem for fractional monogenic functions defined on ℝd . Based on this extension principle, fractional Fueter polynomials, forming a basis of the space of fractional spherical monogenics, i.e. fractional homogeneous polynomials, are introduced. We study the connection between the FCK extension of functions of the form xPl and the classical Gegenbauer polynomials. Finally, we present an example of an FCK extension.

In this paper, semidefinite optimization method is proposed to estimate bounds on linear functionals defined on solutions of linear ordinary differential equations (ODEs) with smooth coefficients. The method can get upper and lower bounds by solving two semidefinite programs, not solving ODEs directly. Its convergence theorem is proved. The theorem shows that the upper and lower bounds series of linear functionals discussed can approach their exact values infinitely. Numerical results show that the method is effective for the estimation problems discussed. In addition, in order to reduce calculation amount, Cheybeshev polynomials are applied to replace Taylor polynomials of smooth coefficients in computing process.

We consider the semilinear problem

where λ is a positive parameter and f has exponential critical growth. We first establish the existence of a non-zero weak solution. Then, by assuming that f is odd, we prove that the number of solutions increases when the parameter λ becomes large. In the proofs we apply variational methods in a suitable weighted Sobolev space consisting of functions with rapid decay at infinity.

We discuss the solvability of the periodic Navier problem for the plate equation with forced vibrations xtt (t, y)+Δ2 x(t, y)+l(t, y, x(t, y)) = 0 in higher dimensions with side lengths being irrational numbers and the nonlinearity being superlinear. We also derive a new dual variational method.

We improve a previous result about the local energy decay for the damped wave equation on $\mathbb{R}^{d}$. The problem is governed by a Laplacian associated with a long-range perturbation of the flat metric and a short-range absorption index. Our purpose is to recover the decay ${\mathcal{O}}(t^{-d+\unicode[STIX]{x1D700}})$ in the weighted energy spaces. The proof is based on uniform resolvent estimates, given by an improved version of the dissipative Mourre theory. In particular, we have to prove the limiting absorption principle for the powers of the resolvent with inserted weights.

The equation –ε2Δu + F(V(x), u) = 0 is studied over all of ℝn, and solutions are constructed that concentrate at an infinite set as ε → 0. The function V (x) is vector valued. This advances previous studies in which V (x) was scalar valued.