The existence and uniqueness of travelling-wave solutions is investigated for a system of two reaction–diffusion equations where one diffusion constant vanishes. The system arises in population dynamics and epidemiology. Travelling-wave solutions satisfy a three-dimensional system about (u, u′, ν), whose equilibria lie on the u-axis. Our main result shows that, given any wave speed c > 0, the unstable manifold at any point (a, 0, 0) on the u-axis, where a ∈ (0, γ) and γ is a positive number, provides a travelling-wave solution connecting another point (b, 0, 0) on the u-axis, where b:= b(a) ∈ (γ, ∞), and furthermore, b(·): (0, γ) → (γ, ∞) is continuous and bijective

In this paper we study Mumford–Shah-type functionals associated with doubling metric measures or strong A∞ weights in the setting of the perimeter theory in the sense of Ambrosio and Miranda in metric spaces. We prove an existence theorem in a suitably defined class of special BV functions.

Conical shock waves are generated as sharp conical solid projectiles fly supersonically in the air. We study such conical shock waves in steady supersonic flow using an isentropic Euler system. The stability of such attached conical shock waves for non-symmetrical conical projectiles and non-uniform incoming supersonic flow is established. Meanwhile, the existence of the solution to the Euler system with such attached conical shock as a free boundary is also proved for solid projectiles close to a regular solid cone.

The existence and uniqueness of travelling-wave solutions is investigated for a system of two reaction–diffusion equations where one diffusion constant vanishes. The system arises in population dynamics and epidemiology. Travelling-wave solutions satisfy a three-dimensional system about (u, u′, ν), whose equilibria lie on the u-axis. Our main result shows that, given any wave speed c > 0, the unstable manifold at any point (a, 0, 0) on the u-axis, where a ∈ (0, γ) and γ is a positive number, provides a travelling-wave solution connecting another point (b, 0, 0) on the u-axis, where b:= b(a) ∈ (γ, ∞), and furthermore, b(·): (0, γ) → (γ, ∞) is continuous and bijective

In this work we consider the Cauchy problem associated with dissipative perturbations of infinite-dimensional Hamiltonian systems. we describe abstract conditions under which the problem is locally and globally well posed. Moreover, we establish the existence of global attractor. Finally, we present several applications of the theory.

In unbounded cylindrical domains, new Phragmén–Lindelöf theorems ‘at infinity’ are established for solutions of Dirichlet problems for elliptic equations whose operators belong to a large class to which the existing literature does not apply.

We study a Ginzburg–Landau functional for a vector-valued order parameter that carries a spin and couples directly to the magnetic field. We verify that spin coupling reduces the lower critical field Hc1, the smallest value of the applied field strength at which vortices penetrate the superconductor, and discuss the related phenomena of a ‘spontaneous vortex phase’ for sufficiently strong spin coupling, predicted by physicists. The method uses energy methods and a priori bounds to confine the vortices and spin transitions to very small balls

In unbounded cylindrical domains, new Phragmén–Lindelöf theorems ‘at infinity’ are established for solutions of Dirichlet problems for elliptic equations whose operators belong to a large class to which the existing literature does not apply.

This paper describes the characterization of optimal constants for some coercivity inequalities in W1,p(Ω), 1 < p < ∞. A general result involving inequalities of p-homogeneous forms on a reflexive Banach space is first proved. The constants are shown to be the least eigenvalues of certain eigenproblems with equality holding for the corresponding eigenfunctions. This result is applied to three different classes of coercivity results on W1,p(Ω). The inequalities include very general versions of the Friedrichs and Poincaré inequalities. Scaling laws for the inequalities are also described.

We obtain spatial and temporal decay rates of weak solutions of the Navier–Stokes equations, and for strong solutions. For the spatial decay rate of the weak solutions, the power of the weight given by He and Xin in 2001 does not exceed 3/2;. However, we show the power can be extended up to 5/2;.

By looking for critical points of functionals defined in some subspaces of , invariant under some subgroups of O (N), we prove the existence of many positive non-radial solutions for the following semilinear elliptic problem involving critical Sobolev exponent on an annulus, where 2* − 1 := (N + 2)/(N − 2) (N ≥ 4), the domain is an annulus and f : R+ × R+ → R is a C1 function, which is a subcritical perturbation.

We prove uniqueness of solutions to scalar conservation laws with space discontinuous fluxes. To do so, we introduce a partial adaptation of Kružkov's entropies which naturally takes into account the space dependency of the flux. The advantage of this approach is that the proof turns out to be a simple variant of the original method of Kružkov. In particular, we do not need traces, interface conditions, bounded variation assumptions (neither on the solution nor on the flux), or convex fluxes. However, we use a special ‘local uniform invertibility’ structure of the flux, which applies to cases where different interface conditions are known to yield different solutions

We study curvature structures of compact hypersurfaces in the unit sphere Sn+1(1) with two distinct principal curvatures. First of all, we prove that the Riemannian product is the only compact hypersurface in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies where n(n − 1)r is the scalar curvature of hypersurfaces and c2 = (n − 2)/nr. This generalized the result of Cheng, where the scalar curvature is constant is assumed. Secondly, we prove that the Riemannian product is the only compact hypersurface with non-zero mean curvature in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies This gives a partial answer for the problem proposed by Cheng.

Let n ≥ 0 be an integer and let λ(n) be the median of the Gamma distribution of order n + 1 with parameter 1. In 1986, Chen and Rubin conjectured that n ↦ λ (n) − n (n = 0, 1, 2, …) is decreasing. We prove the following monotonicity theorem, which settles this conjecture.

Let α and β be real numbers. The sequence n ↦ λ (n) – αn (n = 0, 1, 2, …) is strictly decreasing if and only if α; ≥ 1. And n ↦ λ(n) − βn (n = 0, 1, 2, …) is strictly increasing if and only if β < λ(1) − log 2 = 0.98519….

This paper describes the characterization of optimal constants for some coercivity inequalities in W1,p(Ω), 1 < p < ∞. A general result involving inequalities of p-homogeneous forms on a reflexive Banach space is first proved. The constants are shown to be the least eigenvalues of certain eigenproblems with equality holding for the corresponding eigenfunctions. This result is applied to three different classes of coercivity results on W1,p(Ω). The inequalities include very general versions of the Friedrichs and Poincaré inequalities. Scaling laws for the inequalities are also described.

We consider a path of sectorial operators t ↦ A (t) ∈ Cα (R, L (D, X)), 0 < α < 1, in general Banach space X, with common domain D (A (t)) = D and with hyperbolic limits at ±∞. We prove that there exist exponential dichotomies in the half-lines (−∞, −T] and [T, +∞) for large T, and we study the operator (Lu)(t) = u′(t) − A(t)u(t) in the space Cα (R, D) ∩ C1+α (R, X). In particular, we give sufficient conditions in order that L is a Fredholm operator. In this case, the index of L is given by an explicit formula, which coincides to the well-known spectral flow formula in finite dimension. Such sufficient conditions are satisfied, for instance, if the embedding D ↪ X is compact.

In this paper we consider Beurling-type distributions in the Hankel setting. The Hankel transform and Hankel convolution are studied on Beurling-type distributions. We also introduce a class of ultra-differential operators that allows us to show a Hankel version of the second structure theorem of Komatsu and Braun. Necessary and sufficient conditions are established in order that a Beurling distribution generates a surjective Hankel convolution operator.

We deal with the initial-value problem for parabolic equations with discontinuous nonlinearities and establish the existence of its weak solution. Next, we show that for a suitable class of initial data, the weak solution is locally or globally unique in time. Lastly, we prove that there exist at least two different weak solutions in general if initial data do not belong to this class.

We consider the homogeneous Dirichlet problem δu = −f(u) ≤ 0 in Ω with u = 0 on ∂Ω. We are interested in the inverse problem of determining the nonlinear source f from knowledge of the normal derivative of u, ∂u/δn, on an open arc Γ of ∂Ω. It is well known that this fails if Ω is a ball. On the other hand, Beretta and Vogelius proved that an analytic source f is uniquely determined from knowledge of (∂u/∂n)|Γ if Γ has at least a true corner. In this paper we try to bridge the gap finding a class of smooth domains for which the determination of analytic f is possible

Let n ≥ 0 be an integer and let λ(n) be the median of the Gamma distribution of order n + 1 with parameter 1. In 1986, Chen and Rubin conjectured that n ↦ λ (n) − n (n = 0, 1, 2, …) is decreasing. We prove the following monotonicity theorem, which settles this conjecture.

Let α and β be real numbers. The sequence n ↦ λ (n) – αn (n = 0, 1, 2, …) is strictly decreasing if and only if α; ≥ 1. And n ↦ λ(n) − βn (n = 0, 1, 2, …) is strictly increasing if and only if β < λ(1) − log 2 = 0.98519….