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In the case of oscillatory potentials, we give sufficient conditions for the oscillation of the forced nonlinear second order differential equations with delayed argument in the form x″(t)+q(t)x(τ(t))γsgnx(τ(t))=f(t) in the linear (γ=1) and the superlinear (γ>1) cases. [Copyright &y& Elsevier]

In this paper we study and relate several invariants connected to the solving degree of a polynomial system. This provides a rigorous framework for estimating the complexity of solving a system of polynomial equations via Gröbner bases methods. Our main results include a connection between the solving degree and the last fall degree and one between the degree of regularity and the Castelnuovo–Mumford regularity. [ABSTRACT FROM AUTHOR]

For an integer k ≥ 2 , let { F n (k) } n ≥ 2 − k be the k –generalized Fibonacci sequence which starts with 0 , ... , 0 , 1 (a total of k terms) and for which each term afterwards is the sum of the k preceding terms. In this paper, for an integer d ≥ 2 which is square-free, we show that there is at most one value of the positive integer x participating in the Pell equation x 2 − d y 2 = ± 1 , which is a k –generalized Fibonacci number, with a couple of parametric exceptions which we completely characterize. This paper extends previous work from [18] for the case k = 2 and [17] for the case k = 3. [ABSTRACT FROM AUTHOR]

This paper establishes the global regularity of the two-dimensional (2D) Boussinesq equations and micropolar equations with partial dissipation. Our first result is the global regularity of the 2D Boussinesq equations with fractional vertical dissipation in the horizontal velocity, horizontal dissipation in the vertical velocity and zero thermal diffusion, which is shown by taking advantage of the nice structure of the 2D Boussinesq equations and several refined commutator estimates. The second goal of this paper is to consider a system of the 2D incompressible micropolar equations with vertical dissipation in the horizontal velocity equation, horizontal dissipation in the vertical velocity equation and the fractional Λ α dissipation in the micro-rotation velocity. In order to overcome the difficulty caused by the lack of full Laplacian diffusion in the velocity equations, we fully exploit the nice structure of the corresponding equations to show that this equations with arbitrarily small α > 0 always possesses a unique global classical solution. [ABSTRACT FROM AUTHOR]

The current paper is devoted to the Cauchy problem for the white noise driven Ostrovsky equation with positive dispersion. We obtain the local well-posedness for the initial data u 0 (⋅ , ω) ∈ H s (R) (a.e. ω ∈ Ω) which is F 0 -measurable with s > − 3 4 and Φ ∈ L 2 0 , s ∩ L 2 0 (L 2 (R) , H ˙ s , − 1 2 + ϵ (R)) , and the globally well-posedness for the initial data u 0 (⋅ , ω) ∈ L 2 (R) (a.e. ω ∈ Ω) which is F 0 -measurable and Φ ∈ L 2 0 , 0 ∩ L 2 0 (L 2 (R) , H ˙ 0 , − 1 2 + ϵ (R)). The key ingredients that we used in this paper are some bilinear estimates in X s , b -type spaces given in subsection 2.1 , Itô formula, the BDG inequality and stopping time techniques as well as frequency truncated technique. Our method can be applied to the case γ = 0 , which is just the KdV equation, thus, our result improves the local well-posedness result of A. de Bouard, A. Debussche and Y. Tsutsumi (1999) [3]. [ABSTRACT FROM AUTHOR]

In this paper we study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. Despite its physical importance, this equation has not received a lot of mathematical attention we think due to the extreme complexity of the relativistic structure of the kernel of the collision operator. In this paper we first largely decompose the structure of the relativistic Landau collision operator. After that we prove the global Entropy dissipation estimate. Then we prove the propagation of any polynomial moment for a weak solution. Lastly we prove the existence of a true weak solution for a large class of initial data. [ABSTRACT FROM AUTHOR]

Abstract The present paper is devoted to the investigation of the following nonlocal dispersal equation u t (t , x) = D σ m [ ∫ Ω J σ (x − y) u (t , y) d y − u (t , x) ] + f (t , x , u (t , x)) , t > 0 , x ∈ Ω ‾ , where Ω ⊂ R N is a bounded and connected domain with smooth boundary, m ∈ [ 0 , 2) is the cost parameter, D > 0 is the dispersal rate, σ > 0 characterizes the dispersal range, J σ = 1 σ N J (⋅ σ) is the scaled dispersal kernel, and f is a time-periodic nonlinear function of generalized KPP type. This paper is a continuation of the works of Berestycki et al. [3,4] , where f was assumed to be time-independent. We first study the principal spectral theory of the linear operator associated to the linearization of the equation at u ≡ 0. We establish an easily verifiable, general and sharp sufficient condition for the existence of the principal eigenvalue as well as important sup-inf characterizations of the principal eigenvalue. Next, we study the influences of the principal spectrum point on the global dynamics and confirm that the principal spectrum point being zero is critical. It is followed by the investigation of the effects of the dispersal rate D and the dispersal range characterized by σ on the principal spectrum point and the positive time-periodic solution. In particular, we prove various limiting properties of the principal spectrum point and the positive time-periodic solution as D , σ → 0 + or ∞. To achieve these, we develop new techniques to overcome fundamental difficulties caused by the lack of the usual L 2 variational formula for the principal eigenvalue, the lack of the regularizing effects of the semigroup generated by the nonlocal dispersal operator, and the presence of the time-dependence of the nonlinearity f. Finally, we establish the maximum principle for time-periodic nonlocal dispersal operators. [ABSTRACT FROM AUTHOR]

Abstract Let n be a non-negative integer and put p n (x) = ∏ i = 0 n (x + i). In the first part of the paper, for given n , we study the existence of integer solutions of the Diophantine equation y m = p n (x) + ∑ i = 1 k p a i (x) , where m ∈ N ≥ 2 and a 1 < a 2 < ... < a k < n. This equation can be considered as a generalization of the Erdős–Selfridge Diophantine equation y m = p n (x). We present some general finiteness results concerning the integer solutions of the above equation. In particular, if n ≥ 2 with a 1 ≥ 2 , then our equation has only finitely many solutions in integers. In the second part of the paper we study the equation y m = ∑ i = 1 k p a i (x i) , for m = 2 , 3 , which can be seen as an additive version of the equation considered by Erdős and Graham. In particular, we prove that if m = 2 , a 1 = 1 or m = 3 , a 2 = 2 , then for each k − 1 tuple of positive integers (a 2 , ... , a k) there are infinitely many solutions in integers. [ABSTRACT FROM AUTHOR]

Abstract Hyperovals in PG (2 , GF (q)) with even q are maximal arcs and an interesting research topic in finite geometries and combinatorics. Hyperovals in PG (2 , GF (q)) are equivalent to [ q + 2 , 3 , q ] MDS codes over GF (q) , called hyperoval codes, in the sense that one can be constructed from the other. Ovals in PG (2 , GF (q)) for odd q are equivalent to [ q + 1 , 3 , q − 1 ] MDS codes over GF (q) , which are called oval codes. In this paper, we investigate the binary subfield codes of two families of hyperoval codes and the p -ary subfield codes of the conic codes. The weight distributions of these subfield codes and the parameters of their duals are determined. As a byproduct, we generalize one family of the binary subfield codes to the p -ary case and obtain its weight distribution. The codes presented in this paper are optimal or almost optimal in many cases. In addition, the parameters of these binary codes and p -ary codes seem new. [ABSTRACT FROM AUTHOR]

In this paper, we study quasilinear elliptic equations with the nonlinearity modelled after the p ( x ) -Laplacian on nonsmooth domains and obtain sharp Calderón–Zygmund type estimates in the variable exponent setting. In a recent work of [12] , the estimates obtained were strictly above the natural exponent and hence there was a gap between the natural energy estimates and estimates above p ( x ) , see (1.3) and (1.4) . Here, we bridge this gap to obtain the end point case of the estimates obtained in [12] , see (1.5) . In order to do this, we have to obtain significantly improved a priori estimates below p ( x ) , which is the main contribution of this paper. We also improve upon the previous results by obtaining the estimates for a larger class of domains than what was considered in the literature. [ABSTRACT FROM AUTHOR]

Text Let k , l , a , b , c be positive integers such that gcd ⁡ ( k a , l b ) = 1 , min ⁡ { a , b , c } > 1 , a ≠ 3 , b ≠ 3 and 2 ∤ c . In this paper, we prove that there are at most four solutions in positive integers ( x , y , z ) to the equation k a x + l b y = c z and at most two solutions when 2 ∤ ( u ( l / k ) ) , where u ( m ) is the least positive integer t with m t ≡ 1 ( mod c ) . Video For a video summary of this paper, please visit https://youtu.be/Dt3Y7TDMxlg . [ABSTRACT FROM AUTHOR]

In this paper, we characterize the coefficients of f ( x ) = x + a 1 x q ( q − 1 ) + 1 + a 2 x 2 ( q − 1 ) + 1 in F q 2 [ x ] for even q that lead f ( x ) to be a permutation of F q 2 . We transform the problem into studying some low-degree equations with variable in the unit circle, which are intensively investigated with some parameterization techniques. From the numerical results, the coefficients that lead f ( x ) to be a permutation appear to be completely characterized in this paper. It is also demonstrated that some permutations proposed in this paper are quasi-multiplicative (QM) inequivalent to the previously known permutation trinomials. [ABSTRACT FROM AUTHOR]

This paper is a continuation of the works in [35] and [37] , where the authors have established the global existence of smooth compressible flows in infinitely expanding balls for inviscid gases and viscid gases, respectively. In this paper, we are concerned with the global existence and large time behavior of compressible Boltzmann gases in an infinitely expanding ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time. Due to the conservation of mass, the fluid in the expanding ball becomes rarefied and eventually tends to a vacuum state meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. In the present paper, we will confirm this physical phenomenon for the Boltzmann equation by obtaining the exact lower and upper bound on the macroscopic density function. [ABSTRACT FROM AUTHOR]

The isotropic Grassmannian parametrizes isotropic subspaces of a vector space equipped with a quadratic form. In this paper, we show that any maximal isotropic Grassmannian in its Plücker embedding can be defined by pulling back the equations of G r iso (3 , 7) or G r iso (4 , 8). [ABSTRACT FROM AUTHOR]

Text Let K be a complete, algebraically closed, non-Archimedean valued field, and let ϕ ∈ K ( z ) with deg ⁡ ( ϕ ) ≥ 2 . In this paper we consider the family of functions ord Res ϕ n ( x ) , which measure the resultant of ϕ n at points x in P K 1 , the Berkovich projective line, and show that they converge locally uniformly to the diagonal values of the Arakelov–Green's function g μ ϕ ( x , x ) attached to the canonical measure of ϕ . Following this, we are able to prove an equidistribution result for Rumely's crucial measures ν ϕ n , each of which is a probability measure supported at finitely many points whose weights are determined by dynamical properties of ϕ . Video For a video summary of this paper, please visit https://youtu.be/YCCZD1iwe00 . [ABSTRACT FROM AUTHOR]

In this paper, we investigate model assessment, estimation and hypothesis testing in a linear functional relationship for replicated data when the distribution of the measurement errors is a multivariate Student t distribution. For statistical inference, we adopt the unbiased estimating equations approach. The resulting estimator is consistent and asymptotically normal; a closed form expression is also given for its asymptotic covariance matrix. A simple graphical device for model checking is proposed. We also describe how to test some hypotheses of interest on the parameter vector using the Wald statistic. A simulation study is performed to gauge the performance of the estimators and of the Wald statistic. The methodology developed in the paper is illustrated with a real data set. [ABSTRACT FROM AUTHOR]

The paper investigates a new Gronwall-type lemma with parameter ϵ > 0 , and establishes the uniform-in-time estimates. As an application, the paper studies the asymptotic behavior of the Kirchhoff-type equations with a strong dissipation, and proves that the related solution semigroup possesses the global attractor. [ABSTRACT FROM AUTHOR]

The main object of study in this paper is the well-known Somos-4 recurrence. We prove a theorem that any sequence generated by this equation also satisfies Gale-Robinson one. The corresponding identity is written in terms of its companion elliptic sequence. An example of such relationship is provided by the second-order linear sequence which, as we prove using Wajda's identity, satisfies the Somos-4 recurrence with suitable coefficients. Also, we construct a class of solutions to Volterra lattice equation closely related to the second-order linear sequence. [ABSTRACT FROM AUTHOR]

In this paper, we propose several classes of permutation polynomials with the form ( x p m − x + δ ) s + L ( x ) over the finite field F p 2 m , where p is an odd prime, and L ( x ) is a linearized polynomial with coefficients in F p . The main method used in this paper is to determine the number of solutions of some equations over finite fields of odd characteristic. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the boundary pointwise C 1 , α and C 2 , α regularity for viscosity solutions of fully nonlinear elliptic equations. That is, if ∂Ω is C 1 , α (or C 2 , α) at x 0 ∈ ∂ Ω , the solution is C 1 , α (or C 2 , α) at x 0. Our results are new even for the Laplace equation. Moreover, our proofs are simple. [ABSTRACT FROM AUTHOR]

In this paper we consider a spectral problem for ordinary differential equations of fourth order with the spectral parameter contained in three of the boundary conditions. We study the oscillatory properties of the eigenfunctions and, using these properties, we obtain sufficient conditions for the system of eigenfunctions of the problem in question to form a basis in the space L p (0 , 1) , 1 < p < ∞ , after removing three functions. [ABSTRACT FROM AUTHOR]

In this paper, we study the hydrodynamic limit for the inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations in a two or three dimensional bounded domain when the initial density is bounded away from zero. The proof relies on the relative entropy argument to obtain the strong convergence of macroscopic density of the particles n ϵ in L ∞ (0 , T ; L 1 (Ω)) , which extends the works of Goudon-Jabin-Vasseur [15] and Mellt-Vasseur [26] to inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations. Precisely, the relative entropy estimates in [15] and [26] give the strong convergence of u ϵ and n ϵ , ρ ϵ and n ϵ , respectively. However, we only obtain the strong convergence of n ϵ and u ϵ from the relative entropy estimate, and we use another way to obtain the strong convergence of ρ ϵ via the convergence of u ϵ. Furthermore, when the initial density may vanish, taking advantage of compactness result L M ↪ ↪ H − 1 of Orlicz spaces in 2D, we obtain the convergence of n ϵ in L ∞ (0 , T ; H − 1 (Ω)) , which is used to obtain the relative entropy estimate, thus we also show the hydrodynamic limit for 2D inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations when there is initial vacuum. [ABSTRACT FROM AUTHOR]

Staged trees or coloured probability tree models are statistical models coding conditional independence between events depicted in a tree graph. They include the very important class of Bayesian networks as a special case and provide a straightforward graphical tool for handling additional context-specific relationships. In this paper, we study the algebraic properties of their ideal of model invariants. We hereby find that the tree also provides a straightforward combinatorial tool to generalise the existing geometric characterisation of decomposable graphical models and Bayesian networks. In particular, from a staged tree we can directly understand the interplay between local and global sum-to-one conditions, read the generators of that ideal, and determine conditions under which the model is a toric variety intersected with the probability simplex. [ABSTRACT FROM AUTHOR]

Let L a be a Schrödinger operator with inverse square potential a | x | − 2 on R n , n ≥ 3. The main aim of this paper is to develop the theory of new Besov and Triebel–Lizorkin spaces associated to L a based on the new space of distributions. As applications, we apply the theory to study some problems on the parabolic equation associated to L a. [ABSTRACT FROM AUTHOR]

In this paper, we go beyond what was proposed in theory by Melnikov ([15]) to introduce a practical method to calculate the high order splitting distances of stable and unstable manifold in time-periodic equations. Not only we derive integral formula for splitting distances of all orders, but also we develop an analytic theory to evaluate the acquired multiple integrals. We reveal that the dominance of the exponentially small Poincaré/Melnikov function for equations of high frequency perturbation is caused by a certain symmetry embedded in the kernel functions of high order Melnikov integrals. This symmetry is beheld by many non-Hamiltonian equations. [ABSTRACT FROM AUTHOR]

This paper concerns the Liouville type result for the general semilinear elliptic equation (S) a i j (x) ∂ i j u (x) + K q i (x) ∂ i u (x) + f (x , u (x)) = 0 a.e. in R N , where f is of the KPP-monostable nonlinearity, as a continuation of the previous works of the second author [31,32]. The novelty of this work is that we allow f s (x , 0) to be sign-changing and to decay fast up to a Hardy potential near infinity. First, we introduce a weighted generalized principal eigenvalue and use it to characterize the Liouville type result for Eq. (S) that was proposed by H. Berestycki. Secondly, if (a i j) is the identity matrix and q is a compactly supported divergence-free vector field, we find a condition that Eq. (S) admits no positive solution for K > K ⋆ , where K ⋆ is a certain positive threshold. To achieve this, we derive some new techniques, thanks to the recent results on the principal spectral theory for elliptic operators [14] , to overcome some fundamental difficulties arising from the lack of compactness in the domain. This extends a nice result of Berestycki-Hamel-Nadirashvili [5] on the limit of eigenvalues with large drift to the case without periodic condition. Lastly, the well-posedness and long time behavior of the evolution equation corresponding to (S) are further investigated. The main tool of our work is based on the maximum principle for elliptic and parabolic equations however it is far from being obvious to see if the comparison principle for (S) holds or not. [ABSTRACT FROM AUTHOR]

This paper is concerned with the semilinear transmission of wave-plate system with source term on Riemannian manifold. We prove the existence of weak solutions by using Faedo-Galerkin's method. Furthermore, by introducing nonlinear boundary feedbacks acting only on plate, we establish the explicit and general decay rates of the system. Our proofs are based on the geometric multiplier method and the Riemannian geometry method. [ABSTRACT FROM AUTHOR]

This paper is devoted to the investigation of generalized Abel equation x ˙ = S (x , t) = ∑ i = 0 m a i (t) x i , where a i ∈ C ∞ ([ 0 , 1 ]). A solution x (t) is called a periodic solution if x (0) = x (1). In order to estimate the number of isolated periodic solutions of the equation, we propose a hypothesis (H) which is only concerned with S (x , t) on m straight lines: There exist m real numbers λ 1 < ⋯ < λ m such that either (− 1) i ⋅ S (λ i , t) ≥ 0 for i = 1 , ⋯ , m , or (− 1) i ⋅ S (λ i , t) ≤ 0 for i = 1 , ⋯ , m. By means of Lagrange interpolation formula, we prove that the equation has at most m isolated periodic solutions (counted with multiplicities) if hypothesis (H) holds, and the upper bound is sharp. Furthermore, this conclusion is also valid under some weaker geometric hypotheses. Applying our main result for the trigonometrical generalized Abel equation with coefficients of degree one, we give a criterion to obtain the upper bound for the number of isolated periodic solutions. This criterion is "almost equivalent" to hypothesis (H) and can be much more effectively checked. [ABSTRACT FROM AUTHOR]

The goal of this paper is to show existence of short-time classical solutions to the so called Master Equation of first order Mean Field Games, which can be thought of as the limit of the corresponding master equation of a stochastic mean field game as the individual noises approach zero. Despite being the equation of an idealistic model, its study is justified as a way of understanding mean field games in which the individual players' randomness is negligible; in this sense it can be compared to the study of ideal fluids. We restrict ourselves to mean field games with smooth coefficients but do not impose any monotonicity conditions on the running and initial costs, and we do not require convexity of the Hamiltonian, thus extending the result of Gangbo and Swiech to a considerably broader class of Hamiltonians. [ABSTRACT FROM AUTHOR]

In this paper, we construct the global large solution to the three-dimensional incompressible Hall-MHD equations with a class of initial data. Here the "large solution" means that the L ∞ norm can be arbitrarily large initially. [ABSTRACT FROM AUTHOR]

By proving the existence of non-monotone and non-oscillating wavefronts for the Nicholson's blowflies diffusive equation (the NDE), we answer an open question from [16]. Surprisingly, wavefronts of such a kind can be observed even for arbitrarily small delays. Similarly to the pushed fronts, obtained waves are not linearly determined. In contrast, a broader family of eventually monotone wavefronts for the NDE is indeed determined by properties of the spectra of the linearized equations. Our proofs use essentially several specific characteristics of the blowflies birth function (its unimodal form and the negativity of its Schwarz derivative, among others). One of the key auxiliary results of the paper shows that the Mallet-Paret–Cao–Arino theory of super-exponential solutions for scalar equations can be extended for some classes of second order delay differential equations. For the new type of non-monotone waves to the NDE, our numerical simulations also confirm their stability properties established by Mei et al. [ABSTRACT FROM AUTHOR]

This paper studies the internal control of the Korteweg–de Vries–Burgers (KdVB) equation on a bounded domain. The diffusion coefficient is time-dependent and the boundary conditions are mixed in the sense that homogeneous Dirichlet and periodic Neumann boundary conditions are considered. The exact controllability to the trajectories is proven for a linearized system by using duality and getting a new Carleman estimate. Then, using an inversion theorem we deduce the local exact controllability to the trajectories for the original KdVB equation, which is nonlinear. [ABSTRACT FROM AUTHOR]

In this paper, we prove the energy and cross-helicity conservation of weak solutions to the three-dimensional ideal MHD equations in a bounded domain under the interior Besov regularity conditions which are exactly same as the three-dimensional periodic domain case in [3] , and the boundedness and the Besov-type continuity conditions for both the velocity and magnetic fields near the boundary, which seem crucial for the bounded domain case due to the boundary effect. Note that our Besov-type continuity conditions near the boundary are consistent with the interior Besov regularity conditions. [ABSTRACT FROM AUTHOR]

In this paper, the Choquard equation with an inverse-square potential and both focusing and defocusing nonlinearities in the energy-subcritical regime is investigated. For all the cases, the local well-posedness result in H 1 (R N) is established. Moreover, the global existence result for arbitrary initial values is proved in the defocusing case while a global existence/blowup dichotomy below the ground state is established in the focusing case. [ABSTRACT FROM AUTHOR]

In this paper, we study some overdetermined boundary value problems for anisotropic elliptic PDEs in a bounded domain Ω in a convex cone of R n. By using some integral identities and maximum principle, we prove the corresponding Wulff shape characterizations, which includes the classical Serrin's overdetermined boundary value problem. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the propagation dynamics of a nonlocal dispersal Fisher-KPP equation in a time-periodic shifting habitat. We first show that this equation admits a periodic forced wave with the speed at which the habitat is shifting by using the monotone iteration method combined with a pair of generalized super- and sub-solutions. Then we establish the nonexistence, uniqueness and global exponential stability of periodic forced waves by applying the sliding technique and the comparison argument. Finally, we obtain the spreading properties for a large class of solutions. [ABSTRACT FROM AUTHOR]

In this paper we study the Cauchy problem for the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups when the time-dependent non-negative propagation speed is regular, Hölder, and distributional. For Hölder coefficients we derive the well-posedness in the spaces of ultradistributions associated to Rockland operators on graded groups. In the case when the propagation speed is a distribution, we employ the notion of "very weak solutions" to the Cauchy problem, that was already successfully used in similar contexts in [12] and [20]. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique "very weak solution" in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the time dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or p -evolution equations for higher order operators on R n or on groups, the results already being new in all these cases. [ABSTRACT FROM AUTHOR]

In this paper, the initial value problem for a plate equation with a memory is considered. Under weaker assumptions on the memory function, we obtain the decay estimates of solutions which is of the regularity-loss type by applying the energy method in Fourier space. Moreover, if the memory kernel is an exponential function, we derive the large-time behavior and asymptotic profile of solutions in the one-dimensional space by virtue of the technique of spectral analysis. [ABSTRACT FROM AUTHOR]

In this paper we deal with the controllability problem for some Sobolev type equations. We show that the equations cannot be driven to zero if the control region is strictly supported within the domain. Nevertheless, we also prove that it is possible to control the equations using controls which have a moving support, under some assumptions on their movement. [ABSTRACT FROM AUTHOR]

In this paper we mainly investigate the initial value problem of the periodic Euler-Poincaré equations. We first present a new blow-up result to the system for a special class of smooth initial data by using the rotational invariant properties of the system. Then, we prove that the periodic Euler-Poincaré equations are ill-posed in critical Besov spaces by a contradiction argument. Finally, we verify the system possesses a class of peakon solutions in the sense of distributions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with well-posedness of an anisotropic parabolic equation with the convection term. When some diffusion coefficients are degenerate on the boundary ∂Ω and the others are positive on Ω ‾ , we propose a novel partial boundary value condition to study the stability of the solutions for the anisotropic parabolic equation. A new concept, the general characteristic function of the domain Ω, is introduced and applied. The existence and stability of the solutions is established under the given partial boundary value conditions. [ABSTRACT FROM AUTHOR]

The paper is concerned with higher order Calderón-Zygmund estimates for the p -Laplace equation − div (A (∇ u)) : = − div (| ∇ u | p − 2 ∇ u) = − div F , 1 < p < ∞. We are able to transfer local interior Besov and Triebel-Lizorkin regularity up to first order derivatives from the force term F to the flux A (∇ u). For p ≥ 2 we show that F ∈ B ϱ , q s implies A (∇ u) ∈ B ϱ , q s for any s ∈ (0 , 1) and all reasonable ϱ , q ∈ (0 , ∞ ] in the planar case. The result fails for p < 2. In case of higher dimensions and systems we have a smallness restriction on s. The quasi-Banach case 0 < min ⁡ { ϱ , q } < 1 is included, since it has important applications in the adaptive finite element analysis. As an intermediate step we prove new linear decay estimates for p -harmonic functions in the plane for the full range 1 < p < ∞. [ABSTRACT FROM AUTHOR]

In this paper, two kinds of asymptotic limits to the isentropic compressible viscous magnetohydrodynamic equations in a three-dimensional bounded domain Ω with Navier-slip boundary conditions are discussed. One is the incompressible limit with ill-prepared initial data, and the other is the combined inviscid and incompressible limit with well-prepared initial data. In the first case, we show that the weak solutions of the compressible viscous magnetohydrodynamic equations converge weakly to the weak solutions of the incompressible viscous magnetohydrodynamic equations provided the index of the fluid friction coefficient α 1 ≥ 1 as the Mach number goes to 0. Moreover, the convergence of the velocity in L 2 (0 , ∞ ; L 2 (Ω)) is indeed strong under some geometrical assumptions on the domain Ω and α 1 ≥ 1 2. In the second case, we show that the weak solution of the compressible viscous magnetohydrodynamic equations converges to the local strong solution of the ideal incompressible magnetohydrodynamic equations. Furthermore, the convergence rates are also obtained. [ABSTRACT FROM AUTHOR]

In this paper we consider the spreading phenomena in the Fisher-KPP equation in a high dimensional cone with Dirichlet boundary condition. We show that any solution starting from a nonnegative and compact supported initial data spreads and converges to the unique positive steady state. Moreover, the asymptotic spreading speeds of the front in all directions pointing to the opening are c 0 (which is the minimal speed of the traveling wave solutions of the 1-dimensional Fisher-KPP equation). Surprisingly, they do not depend on the shape of the cone, the propagating directions and the boundary condition. [ABSTRACT FROM AUTHOR]

In this paper we study the Cauchy problem for a semilinear pseudo-parabolic equation with initial data non-rarefied at ∞. Our interest lies in the discussion of the effect of the non-rarefied factors on the bow-up phenomenon and the life span of the solution. [ABSTRACT FROM AUTHOR]

In this paper we consider the Liouville equation Δ u + λ 2 e u = 0 with Dirichlet boundary conditions in a two dimensional, doubly connected domain Ω. We show that there exists a simple, closed curve γ ⊂ Ω such that for a sequence λ n → 0 and a sequence of solutions u n it holds u n log ⁡ 1 λ n → H , where H is a harmonic function in Ω ∖ γ and λ n 2 log ⁡ 1 λ n ∫ Ω e u n d x → 8 π c Ω , where c Ω is a constant depending on the conformal class of Ω only. [ABSTRACT FROM AUTHOR]

In this paper, we study the periodic Cauchy problem for a mathematical model of the equatorial water waves propagating mainly in one direction with the weak Coriolis effect due to the Earth's rotation, which reduces to the Camassa-Holm equation as the Coriolis effect vanishes. We first prove that this model is well-posed in the sense of Hadamard when initial data belongs to the Sobolev space H s (T) with s > 3 / 2. Then we show that the well-posedness is sharp in the sense that the continuity of the data-to-solution map is not better than continuous, whose proof rests upon the method of approximate solutions and the well-posedness estimate. However, if H s (T) is equipped with a weaker H r -topology for 0 ≤ r < s , we demonstrate that the data-to-solution map is Hölder continuous with the exponent depending on r and s. Finally, we establish a Cauchy-Kowalevski type theorem for this model. [ABSTRACT FROM AUTHOR]

In this paper, we prove some rigidity theorems for the entire 2-convex solutions of 2-Hessian equation in Euclidean space. As an application, we obtain a Bernstein type theorem for global special Lagrangian graphs. [ABSTRACT FROM AUTHOR]

In this paper we consider a five-parameter equation including the Camassa-Holm and the Dullin-Gottwald-Holm equations, among others. We prove the existence and uniqueness of solutions of the Cauchy problem using Kato's approach. Conservation laws of the equation, up to second order, are also investigated. From these conservation laws we establish some properties for the solutions of the equation and we also find a quadrature for it. The quadrature obtained is of capital importance in a classification of bounded travelling wave solutions. We also find some explicit solutions, given in terms of elliptic integrals. Finally, we classify the members of the equation describing pseudo-spherical surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following semi-linear complex heat equation ∂ t u = Δ u + u p , u ∈ C in R n , with an arbitrary power p , p > 1. We construct for this equation a complex solution u = u 1 + i u 2 , which blows up in finite time T and only at one blowup point a. Moreover, we also describe the asymptotics of the solution by the following final profiles: u (x , T) ∼ [ (p − 1) 2 | x − a | 2 8 p | ln ⁡ | x − a | | ] − 1 p − 1 , u 2 (x , T) ∼ 2 p (p − 1) 2 [ (p − 1) 2 | x − a | 2 8 p | ln ⁡ | x − a | | ] − 1 p − 1 1 | ln ⁡ | x − a | | > 0 , as x → a. In addition to that, since we also have u 1 (0 , t) → + ∞ and u 2 (0 , t) → − ∞ as t → T , the blowup in the imaginary part shows a new phenomenon unknown for the standard heat equation in the real case: a non constant sign near the singularity, with the existence of a vanishing surface for the imaginary part, shrinking to the origin. In our work, we have succeeded to extend for any power p where the non linear term u p is not continuous if p is not integer. In particular, the solution which we have constructed has a positive real part. We study our equation as a system of the real part and the imaginary part u 1 and u 2. Our work relies on two main arguments: the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to get the conclusion. [ABSTRACT FROM AUTHOR]