In this paper we deal with a linear integral transform, defined on a vectorial L2-space, whose kernel arises from a one-dimensional system of Dirac operators. Unlike the regular Sturm–Liouville transform, which is associated with a regular Sturm–Liouville problem, the range of this transform is a whole Paley–Wiener space. As a consequence, some results for the Paley–Wiener space are derived; in particular, the sampling formula associated with a regular Dirac operator. Finally, we obtain an inversion formula by means of a continuous measure for suitable Sobolev spaces in the initial L2-space.

We present an L–A pair for the Hess–Apel'rot case of a heavy rigid three-dimensional body. Using it, we give an algebro-geometric integration procedure. Generalizing this L–A pair, we obtain a new completely integrable case of the Euler–Poisson equations in dimension four. Explicit formulae for integrals that are in involution are given. This system is a counterexample to one of Ratiu's theorems. A corrected version of this classification theorem is proved.

We consider here the radial Stefan problem with Gibbs–Thomson law, which is a classical model describing growth or melting of a spherical crystal in a surrounding liquid. We shall specialize to the cases of two and three space dimensions and discuss the asymptotic behaviour of a melting crystal near its dissolution time t* > 0. We prove here that, when the interface shrinks monotonically, the asymptotics near t = t* is of the form Here, R(t) denotes the radius of the crystal, σ is a surface tension parameter and u(r, t) represents the field temperature. An important point to be noticed is that (*) exhibits no dependence on the space dimension N, in sharp contrast with results known for the case σ = 0.

We are concerned with the following minimization problems, where Ω ⊂ RN, N > 4, is a smooth bounded domain, qc = 2N/(N − 4), ϕ ∈ C(Ω) ∩ Lqc(Ω) and . We show that, for ϕ ≢ 0, each infimum is achieved. Under suitable conditions on ϕ, we establish the following gap phenomenon, for q ≤ qc.

Moreover, we study the limit behaviour of the minimizers, as q goes to qc, in the case ϕ ∈ H(Ω).

The Brezis-Nirenberg equation and the scalar field equation on the three-dimensional unit ball are studied. Under the Robin condition, we show the existence and uniqueness of radial solutions in a unified way. In particular, it is shown that the global structure of solutions changes qualitatively when a parameter in the boundary condition exceeds a certain critical value.

In this paper we construct new classes of stationary solutions for the Cahn–Hilliard equation by a novel approach.

One of the results is as follows. Given a positive integer K and a (not necessarily non-degenerate) local minimum point of the mean curvature of the boundary, then there are boundary K-spike solutions whose peaks all approach this point. This implies that for any smooth and bounded domain there exist boundary K-spike solutions.

The central ingredient of our analysis is the novel derivation and exploitation of a reduction of the energy to finite dimensions (lemma 3.5), where the variables are closely related to the peak locations.

In this paper we prove the existence and analyticity of solitary waves of generalized Kadomtsev–Petviashvili equations satisfying a set of conditions on linear and nonlinear terms which determine their behaviour at infinity and around 0.

We define a new class of admissible solutions for the parabolic problems where the theory of viscosity solutions does not apply. A typical example is the porous medium equation complemented by lower-order terms, nonlinear with respect to the gradient. In the case, the nonlinearity in the equation fails to be proper in the sense of the theory of viscosity solutions. The admissible solutions satisfy a very general comparison principle and, consequently, the corresponding initial-boundary value problems are well-posed in this class. They coincide with the standard viscosity solutions provided the nonlinearity in the equation is proper.

This paper concerns the reaction-diffusion equation ut = uxx + u2(1 − u). Previous numerical solutions of this equation have demonstrated various different types of wave front solutions, generated by different initial conditions. In this paper, the authors use a phase-plane form of comparison theorems for partial differential equations (PDEs) to confirm analytically these numerical results. In particular, they show that initial conditions with an exponentially decaying tail evolve to the unique exponentially decaying travelling wave, while initial conditions with algebraically decaying tails evolve either to an algebraically decaying travelling wave, or to the exponentially decaying wave, or to a perpetually accelerating wave, dependent upon the exact form of the decay of the initial conditions. We then focus on the case of accelerating waves and investigate their form in more detail, by approximating the full equation in this case with a hyperbolic PDE, which we solve using the method of characteristics. We use this approximate solution to derive a leading-order approximation to the wave speed.

We consider asymptotic problems for coupled equations modelling interactions between particles and a viscous fluid. The particles are driven by a Vlasov-like equation, involving the velocity of the fluid. We obtain, as certain parameters tend to 0, hydrodynamic equations for the macroscopic density and the velocity.

We approximate functionals depending on the gradient of u and on the behaviour of u near the discontinuity points by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise convergence, Γ-convergence and a compactness result, which implies, in particular, the convergence of minima and minimizers.

We prove limiting imbeddings of spaces with dominating mixed derivatives into the spaces of almost Lipschitz continuous functions.

Giving as answer to Bergman's question, Cohen and Montgomery proved that, for every finite group G with identity e and each G-graded ring R = ⊕g∈GRg, the Jacobson radical J(Re) of the initial component Re is equal to Re ∩ J(R). We describe all semigroups S, which satisfy the following natural analogue of this property: J(Re) = Re ∩ J(R) for each S-graded ring R = ⊕s∈SRs and every idempotent e ∈ S.

We discuss the three-space problem on discreteness for the Jacobson topology on the spectrum of a C*-algebra in detail. As an application, it is shown that a C*-algebra A is a dual C*-algebra if and only if a closed ideal I of A and the quotient A/I are dual C*-algebras and the open central projection, in the second dual of A, corresponding to I is a multiplier for A.

The Kadomtsev–Petviashvili (KP) equation can be formally derived as an envelope equation for three-dimensional unidirectional water waves in the limit of long waves. As a first step towards a mathematical justification, we consider here a two-dimensional Boussinesq equation, which is a realistic model for three-dimensional water waves. Using rigorous estimates, we show that part of the dynamics of the KP equation can be found approximately in the two-dimensional Boussinesq equation. On the other hand, there exist initial data for the KP equation such that the corresponding solutions of the two-dimensional Boussinesq equation behave in no way according to the KP prediction. We expect that similar results hold for the three-dimensional water wave problem too.

A parabolic system with an unknown boundary is considered. The boundary condition at the unknown boundary is concerned with its mean curvature. The physical prototype is superconductivity of type I. Existence of classical solutions is proved by the use of fixed-point theorem, uniqueness is obtained as well.

We prove nonlinear stability of planar shock fronts for certain relaxation systems in two spatial dimensions. If the subcharacteristic condition is assumed and the initial perturbation is sufficiently small and the mass carried by the perturbations is not necessarily finite, then the solution converges to a shifted planar shock front solution as time t ↑ ∞. The asymptotic phase shift of shock fronts is, in general, non-zero and governed by a similarity solution to the heat equation. The asymptotic decay rate to the shock front is proved to be t−1/4 in L∞(R2) without imposing extra decay rates in space for the initial perturbations. The proofs are based on an elementary weighted energy analysis to the error equation.

In this paper we prove a certain regularity property of configurations of immiscible fluids, filling a bounded container Ω and locally minimizing the interface energy ∑i<jcij‖Sij‖, where Sij represents the interface between fluid i and fluid j, ‖·‖ stands for area or more general area-type functional, and cij is a positive coefficient. More precisely, we show that, under strict triangularity of the cij, no infiltrations of other fluids are allowed between two main ones. A remarkable consequence of this fact is the almost-everywhere regularity of the interfaces. Our analysis is performed in general dimension n ≥ 2 and with volume constraints on fluids.

We obtain some new exact multiplicity results for the Dirichlet boundary-value problem on a unit ball Bn in Rn. We consider several classes of nonlinearities f(u), including both positive and sign-changing cases. A crucial part of the proof is to establish positivity of solutions for the corresponding linearized problem. As an application we obtain exact multiplicity results for the Holling-Tanner population model.

The global solution, stability and evolution of regularities are studied for the initial-boundary-value problem of a viscous compressible gas flow. With global a priori estimates and careful analysis, the existence of global classical solutions in Hölder space is established.