Existence of weak solutions is proved for a phase field model describing an interface in an elastically deformable solid, which moves by diffusion of atoms along the interface. The volume of the different regions separated by the interface is conserved, since no exchange of atoms across the interface occurs. The diffusion is driven only by reduction of the bulk free energy. The evolution of the order parameter in this model is governed by a degenerate parabolic fourth-order equation. If a regularizing parameter in this equation tends to zero, then solutions tend to solutions of a sharp interface model for interface diffusion. The existence proof is valid only for a 1½-dimensional situation.

In this paper we construct solutions which develop two negative spikes as $\varepsilon\to0^+$ for the problem $-\Delta u=|u|^{4/(N-2)}u+\varepsilon f(x)$ in $\varOmega$, $u=0$ on $\partial\varOmega$, where $\varOmega\subset\mathbb{R}^N$ is a bounded smooth domain exhibiting a small hole, with $f\geq0$, $f\not\equiv0$. This result extends a recent work of Clapp et al. in the sense that no symmetry assumptions on the domain are required.

In this paper we prove that the best constant in the Sobolev trace embedding $H^1(\varOmega)\hookrightarrow L^q(\partial\varOmega)$ in a bounded smooth domain can be obtained as the limit as $\varepsilon\to0$ of the best constant of the usual Sobolev embedding $H^1(\varOmega) \hookrightarrow L^q(\omega_\varepsilon,\mathrm{d} x/\varepsilon)$, where $\omega_\varepsilon=\{x\in\varOmega:\mathrm{dist}(x,\partial\varOmega)<\varepsilon\}$ is a small neighbourhood of the boundary. We also analyse symmetry properties of extremals of the latter embedding when $\varOmega$ is a ball.

Consider the two-dimensional autonomous systems of differential equations

$$ \dot{x}=-y+\lambda x+P(x,y),\qquad\dot{y}=x+\lambda y+Q(x,y), $$

where $\lambda$ is a real constant and $P$ and $Q$ are $\mathcal{C}^{\infty}$-functions of order greater than or equal to two. These systems, so-called centre-focus-type systems, have either a centre or a focus at the origin. In this work, we give necessary and sufficient conditions of isochronicity using normal forms. We characterize the systems which have either an isochronous centre or an isochronous focus at the origin by means of the existence of a commutator of the field. Moreover, we prove that the maximum order of a weak isochronous focus for quadratic systems is two, and that for systems with cubic nonlinearities is three.

We continue the research on reiteration results between interpolation methods associated to polygons and the real method. Applications are given to N-tuples of function spaces, of spaces of bounded linear operators and Banach algebras.

In this article, we show that the magnetohydrodynamic system in $\mathbb{R}^N$ with variable density, variable viscosity and variable conductivity has a local weak solution in the Besov space $\dot{B}^{N/p_1}_{p_1,1}(\mathbb{R}^N)\times\dot{B}^{(N/p_2)-1}_{p_2,1}(\mathbb{R}^N) \times\dot{B}^{(N/p_2)-1}_{p_2,1}(\mathbb{R}^N)$ for all $1<p_2<+\infty$ and some $1<p_1\leq2N/3$ if the initial density approaches a positive constant. Moreover, this solution is unique if we impose the restrictive condition $1<p_2\leq2N$. We also prove that the constructed solution exists globally in time if the initial data are small. In particular, this allows us to work in the framework of Besov space with negative regularity indices and this fact is particularly important when the initial data are strongly oscillating.

In the analysis of stability in bifurcation problems it is often assumed that the (appropriate reduced) equations are in normal form. In the presence of symmetry, the truncated normal form is an equivariant polynomial map. Therefore, the determination of invariants and equivariants of the group of symmetries of the problem is an important step. In general, these are hard problems of invariant theory and, in most cases, they are tractable only through symbolic computer programs. Nevertheless, it is desirable to obtain some of the information about invariants and equivariants without actually computing them, for example, the number of linearly independent homogeneous invariants or equivariants of a certain degree. Generating functions for these dimensions are generally known as ‘Molien functions'.

We obtain formulae for the number of linearly independent homogeneous invariants or equivariants for Hopf bifurcation in terms of characters. We also show how to construct Molien functions for invariants and equivariants for Hopf bifurcation. Our results are then applied to the computation of the number of invariants and equivariants for Hopf bifurcation for several finite groups and the continuous group $\mathbb{O}(3)$.

We consider the stationary equations of a general viscous fluid in an infinite (periodic) slab supplemented with Navier's boundary condition with a friction term on the upper part of the boundary. In addition, we assume that the upper part of the boundary is described by a graph of a function φε, where φε oscillates in a specific direction with amplitude proportional to ε. We identify the limit problem when ε → 0, in particular, the effective boundary conditions.

Every finite group G acts as an automorphism group of some non-orientable Klein surfaces without boundary. The minimal genus of these surfaces is called the symmetric cross-cap number and denoted by . This number is related to other parameters defined on surfaces as the symmetric genus and the strong symmetric genus.

The systematic study of the symmetric cross-cap number was begun by C. L. May, who also calculated it for certain finite groups. Here we obtain the symmetric cross-cap number for the groups Cm × Dn. As an application of this result, we obtain arithmetic sequences of integers which are the symmetric cross-cap number of some group. Finally, we recall the several different genera of the groups Cm × Dn.

We establish the persistence of the asymptotic stability of a linear equation $v'=A(t)v$ in a Banach space under sufficiently small perturbations, when the linear equation admits a non-uniform exponential contraction or a non-uniform exponential dichotomy. Moreover, we obtain optimal estimates for the decay of solutions of the perturbed equation, that in general may depend on the initial time.

In this paper, we are concerned with the existence of unbounded orbits of the mapping

\begin{align*} \theta_1&=\theta+2\pi+\frac{1}{\rho}\mu(\theta)+o(\rho^{-1}), \\ \rho_1&=\rho+c-\mu'(\theta)+o(1),\quad\rho\to\infty, \end{align*}

where $c$ is a constant and $\mu(\theta)$ is $2\pi$-periodic. Assume that $c\not=0$, that $\mu(\theta)$ is non-negative (or non-positive) and that $\mu(\theta)$ has finitely many degenerate zeros in $[0,2\pi]$. We prove that every orbit of the given mapping tends to infinity in the future or in the past for sufficiently large $\rho$. On the basis of this conclusion, we further prove that the equation $x''+f(x)x'+V'(x)+\phi(x)=p(t)$ has unbounded solutions provided that $V$ is an isochronous potential at resonance and $F(x)$ ($F(x)=\int_0^xf(s)\,\mathrm{d} s$) and $\phi(x)$ satisfy some limit conditions. Meanwhile, we also obtain the existence of $2\pi$-periodic solutions of this equation.

We present some transference results for a convolution operator with kernel $K$ which is bounded from $L^{p_0}(w_0)$ into $L^{p_1}(w_1)$. Our results are a natural extension of the classical results of Coifman and Weiss and their further extensions. We shall give several applications in the setting of restriction of Fourier multipliers.

Using Leray–Schauder degree theory, a theorem of upper and lower solutions and a strong maximum principle for the telegraph equation we prove an Ambrosetti–Prodi-type result for periodic solutions of the telegraph equation.

We study the cyclicity of limit periodic sets that occur in families of vector fields of slow–fast type. The limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point. At this turning point a stability change takes place: on one side of the turning point the dynamics point strongly towards the curve of singularities; on the other side the dynamics point away from the curve of singularities. The presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point, i.e. orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve. All existing results deal with a non-zero slow movement, permitting a good estimate of the cyclicity by considering the slow-divergence integral along the curve of singularities. In this paper we study what happens when the slow dynamics exhibit singularities. In particular, our study includes the cyclicity of the slow–fast two-saddle cycle, formed by a regular saddle connection (the fast part) and a part of the curve of singularities (the slow part). We see that the relevant information is no longer merely contained in the slow-divergence integral.

The usual asymptotic expansion for the solutions of an elliptic linear problem with oscillatory periodic coefficients is known to not be accurate near the boundary. In order to obtain a better approximation it is necessary to add to this expansion a boundary-layer term. This term has been obtained by other authors in the case of a plane boundary, such that its normal is proportional to some period. We consider the case where the normal is arbitrary.špace{-8pt}

We study the evolution of solutions to the initial-boundary-value problem

\begin{alignat*}{3} u_t&=(u^m)_{xx}+\lambda(u^q)_x, & \quad x&>0,&\quad t&\in(0,T), \\[2pt] -(u^m)_x(0,t)&=u^p(0,t), & &&t&\in(0,T), \\[2pt] u(x,0)&=u_0(x), & \quad x&>0, \end{alignat*}

and give a rather complete characterization, in terms of the parameters $m\ge1$, $p,q>0$ and $\lambda>0$, of whether all solutions are global in time or, on the contrary, there exist blow-up solutions. We show that the presence of the convective term has a preventive effect on the blow-up (with respect to the case $\lambda=0$) and gives rise to a collapse of the region where all solutions blow up in this case. On the other hand, a new Fujita-type phenomenon takes place at the level $p=q$ and $0<\lambda<1$.

We study the existence and non-existence of classical solutions to a general Gierer—Meinhardt system with Dirichlet boundary condition. The main feature of this paper is that we are concerned with a model in which both the activator and the inhibitor have different sources given by general nonlinearities. Under some additional hypotheses and in the case of pure powers in nonlinearities, regularity and uniqueness of the solution in one dimension is also presented.

For m > 0 and p ∈ (1, (N + 2)/(N − 2)), we show the uniqueness and a linearized non-degeneracy of solutions for the following problem:

In this work we describe the isoperimetric regions in complete symmetric annuli of revolution with Gauss curvature non-decreasing from the shortest parallel. This description allows us to complete the classification of isoperimetric regions in quadrics of revolution.

We consider the periodic solutions of the equation

satisfying −1 ≤ x ≤ 1 ẋ(t−) = −ẋ(t+) if x(t) = ±1, where f is a continuous function, periodic in t. A sequence of periodic solutions is obtained via the variational method.