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 μ 1 > 0, μ 2 > 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.

We consider the ergodic (or additive eigenvalue) problem for the Neumann-type boundary-value problem for Hamilton–Jacobi equations and the corresponding discounted problems. Denoting by uλ the solution of the discounted problem with discount factor λ > 0, we establish the convergence of the whole family to a solution of the ergodic problem as λ → 0, and give a representation formula for the limit function via the Mather measures and Peierls function. As an interesting by-product, we introduce Mather measures associated with Hamilton–Jacobi equations with the Neumann-type boundary conditions. These results are variants of the main results in a recent paper by Davini et al., who study the same convergence problem on smooth compact manifolds without boundary.

In this paper, we study the following Kirchhoff-type equation:where a, b are positive constants and N = 1, 2, 3. Under appropriate assumptions on V, K and g, we obtain a ground-state solution by using the approach developed by Szulkin and Weth in 2010.

We study the indefinite Kirchhoff-type problemwhere Ω is a smooth bounded domain in and . We require that f is sublinear at the origin and superlinear at infinity. Using the mountain pass theorem and Ekeland variational principle, we obtain the multiplicity of non-trivial non-negative solutions. We improve and extend some recent results in the literature.

This paper concerns the quenching phenomena of a reaction–diffusion equation $u_{t}=u_{xx}+1/(1-u)$ in a one dimensional varying domain $[g(t),h(t)]$ , where $g(t)$ and $h(t)$ are two free boundaries evolving by a Stefan condition. We prove that all solutions will quench regardless of the choice of initial data, and we also show that the quenching set is a compact subset of the initial occupying domain and that the two free boundaries remain bounded.

In this paper we study the multiplicity of non-trivial solutions to a class of nonlinear boundary-value problems of Kirchhoff type. We prove existence results when the problem has nonlinearities with subcritical and with critical Caffarelli–Kohn–Nirenberg exponent.

We examine the regularity of the extremal solution of the nonlinear eigenvalue problem

on a general bounded domain Ω in ℝN, with Navier boundary condition u = Δu on ∂Ω. Firstly, we prove the extremal solution is smooth for any p > 1 and N ⩽ 4, which improves the result of Guo and Wei (Discrete Contin. Dynam. Syst. A 34 (2014), 2561–2580). Secondly, if p = 3, N = 3, we prove that any radial weak solution of this nonlinear eigenvalue problem is smooth in the case Ω = 𝔹, which completes the result of Dávila et al. (Math. Annalen348 (2009), 143–193). Finally, we also consider the stability of the entire solution of Δ2 u = –l/up in ℝN with u > 0.

We develop an algebraic approach to constructing short-time asymptotic expansions of solutions of a class of abstract semilinear evolution equations. The expansions are typically valid for both the solution of the equation and its gradient. We apply a perturbation approach based on the symbolic calculus of pseudo-differential operators and heat kernel methods. The construction is explicit and can be done to arbitrary order. All results are rigorously formulated in terms of Banach algebras. As an application we obtain a novel approach to finding approximate solutions of Markovian backward stochastic differential equations.

We prove a family of sharp bilinear space–time estimates for the half-wave propagator $\text{e}^{\text{i}t\sqrt{-\unicode[STIX]{x1D6E5}}}$ . As a consequence, for radially symmetric initial data, we establish sharp estimates of this kind for a range of exponents beyond the classical range.

For a non-negative and non-trivial real-valued continuous function hΩ × [0, ∞) such that h(x, 0) = 0 for all x ∈ Ω, we study the boundary-value problem

where Ω ⊆ ℝN , N ⩾ 2, is a bounded smooth domain and Δp := div(|Du|p–2DDu) is the p-Laplacian. This work investigates growth conditions on h(x, t) that would lead to the existence or non-existence of distributional solutions to (BVP). In a major departure from past works on similar problems, in this paper we do not impose any special structure on the inhomogeneous term h(x, t), nor do we require any monotonicity condition on h in the second variable. Furthermore, h(x, t) is allowed to vanish in either of the variables.