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In this paper, we prove the existence of smooth initial data for the two-dimensional free boundary incompressible viscous magnetohydrodynamics (MHD) equations, for which the interface remains regular but collapses into a splash singularity (self-intersects in at least one point) in finite time. The existence of the splash singularities is guaranteed by a local existence theorem, in which we need suitable spaces for the modified magnetic field together with the modification of the velocity and pressure such that the modified initial velocity is zero, and a stability result which allows us to construct a class of initial velocities and domains for an arbitrary initial magnetic field. It turns out that the presence of the magnetic field does not prevent the viscous fluid from forming splash singularities for certain smooth initial data. [ABSTRACT FROM AUTHOR]

This paper is mainly devoted to the study of controlled sweeping processes with polyhedral moving sets in Hilbert spaces. Based on a detailed analysis of truncated Hausdorff distances between moving polyhedra, we derive new existence and uniqueness theorems for sweeping trajectories corresponding to various classes of control functions acting in moving sets. Then we establish quantitative stability results, which provide efficient estimates on the sweeping trajectory dependence on controls and initial values. Our final topic, accomplished in finite-dimensional state spaces, is deriving new necessary optimality and suboptimality conditions for sweeping control systems with endpoint constrains by using constructive discrete approximations. [ABSTRACT FROM AUTHOR]

This paper develops L p - L q decay estimates of the gradient of the Stokes semigroup (T (t)) t ≥ 0 generated by the negative of the Stokes operator in exterior Lipschitz domains Ω ⊂ R n , n ≥ 3. More precisely, the L p - L q estimates of ∇ T (t) with optimal rates are proved if p and q satisfy | 1 / p − 1 / 2 | < 1 / (2 n) + ε , | 1 / q − 1 / 2 | < 1 / (2 n) + ε , and p ≤ q ≤ n with some ε > 0 , which should be useful for the study of stability of the several physically relevant flows. As an application, we obtain the existence theorem of global-in-time strong solutions to the three-dimensional Navier–Stokes equations in the critical space L ∞ (0 , ∞ ; L σ 3 (Ω)) provided that the initial velocity is small in the L 3 -norm. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate a nonlinear optimal control problem governed by a complicated dynamic variational-hemivariational inequality (DVHVI, for short) with history-dependent operators in the framework of an evolution triple of spaces. First, we prove a generalized existence and uniqueness theorem to a class of dynamic variational-hemivariational inequalities, which extends the recent results established by Han-Migórski-Sofonea [12, Theorem 9] and Migórski-Bai [23, Theorem 5]. Then, an optimal control problem driven by a (DVHVI) is studied and an existence theorem is delivered. Finally, we explore the sensitivity properties of the optimal control problem under the consideration depending on the initial data ξ ∈ H and some further parameter λ ∈ E. [ABSTRACT FROM AUTHOR]

In this paper, we study the planar L p -Minkowski problem (0.1) u θ θ + u = f u p − 1 , θ ∈ S 1 for all p ∈ R , which was introduced by Lutwak [21]. A detailed exploration of (0.1) on solvability will be presented. More precisely, we will prove that for p ∈ (0 , 2) , there exists a positive function f ∈ C α (S 1) , α ∈ (0 , 1) such that (0.1) admits a nonnegative solution vanishes somewhere on S 1. In case p ∈ (− 1 , 0 ] , a surprising a-priori upper/lower bound for solution was established, which implies the existence of positive classical solution to each positive function f ∈ C α (S 1). When p ∈ (− 2 , − 1 ] , the existence of some special positive classical solution has already been known using the Blaschke-Santalo inequality [7]. Upon the final case p ≤ − 2 , we show that there exist some positive functions f ∈ C α (S 1) such that (0.1) admits no solution. Our results clarify and improve largely the planar version of Chou-Wang's existence theorem [7] for p < 2. At the end of this paper, some new uniqueness results will also be shown. [ABSTRACT FROM AUTHOR]

In this paper, we consider the one-dimensional Cauchy problem of the generalized viscous Burgers equations with a time delay and analyze a delay effect. We prove the existence of the global in time solution when the product of the size of the delay parameter and the one of the initial history is suitably small. Moreover we show theorem concerning with the regularity of the global in time solution. [ABSTRACT FROM AUTHOR]

In this paper, we study distribution dependent stochastic differential equations driven simultaneously by fractional Brownian motion with Hurst index H > 1 2 and standard Brownian motion. We first establish the existence and uniqueness theorem for solutions of the distribution dependent stochastic differential equations by utilising the Carathéodory approximation. Then under certain averaging condition, we show that the solutions of distribution dependent stochastic differential equations can be approximated by the solutions of the associated averaged distribution dependent stochastic differential equations in the sense of the mean square convergence. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with almost periodic sequences. We study various types of almost periodic sequences arising in the literature and adapt their notions to the parametric framework. Then, similarly to the continuous case, we aim to prove variational principles. These variational principles are used to obtain some structural results and some theorems of existence in a Hilbert framework. We give, in particular, a discrete version of Amerio's criterion, and adapt a known method to the discrete time case in order to get an existence result on Hilbert space. The condition we obtain allows to find a classical known condition in the case where the system is linear with constant coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we study a non-local coupled system arising in the modeling of the dynamics of dislocation densities in crystals. For this system, the global existence and uniqueness are available only for continuous viscosity solutions. In the present paper, we investigate the global time existence of this system by considering BV initial data. Based on a fundamental uniform BV estimate and the finite speed of propagation property of this system, we show, in a particular setting, the global existence of discontinuous viscosity solutions of this problem. [ABSTRACT FROM AUTHOR]

We prove existence for an initial value problem featuring a conservation law whose flux has a discontinuous spatial dependence. For the type of flux considered here, where the spatial dependence occurs in a very general form, a uniqueness result is known, but the existence question was open until the recent work of Piccoli and Tournus (2018) [22]. Piccoli and Tournus proved existence using approximate solutions generated by the wave front tracking algorithm. A concavity assumption plays a simplifying role in their analysis. The main contribution of the present paper is an extension of this existence theorem in the absence of the concavity assumption. We accomplish this via finite difference approximations. [ABSTRACT FROM AUTHOR]

We investigate a version of one velocity Baer-Nunziato type system with dissipation describing the motion of a mixture of two compressible fluids. We define for this system weak solutions on one hand and dissipative weak solutions on the other hand, and recall the theorem about their existence on a large time interval. We investigate strong solutions and show their existence on a short time interval. Finally, we prove that any weak solution satisfies a relative energy inequality and prove for this system the weak-strong uniqueness principle. This is the main result of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we establish the global C 2 estimates of the Neumann problem of special Lagrangian equations with supercritical phase and the existence theorem by the method of continuity. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the construction of periodic solutions of the nonlocal problem (− Δ) s u = f (u) in R , where (− Δ) s stands for the s -Laplacian, s ∈ (0 , 1). We introduce a suitable framework which allows, by means of regularity, to link the searching of such solutions into the existence of the ones of a semilinear problem in a suitable Hilbert space. Then by a bifurcation theory from eigenvalues of odd multiplicity and also variational method that avoid the constant solutions we get existence theorems which are lately enlightened with the analysis of some examples. In particular, multiplicity results for generalized Benjamin-Ono equation are obtained. [ABSTRACT FROM AUTHOR]

This paper deals with the two-species chemotaxis-Navier–Stokes system with Lotka–Volterra competitive kinetics { ( n 1 ) t + u ⋅ ∇ n 1 = Δ n 1 − χ 1 ∇ ⋅ ( n 1 ∇ c ) + μ 1 n 1 ( 1 − n 1 − a 1 n 2 ) in Ω × ( 0 , ∞ ) , ( n 2 ) t + u ⋅ ∇ n 2 = Δ n 2 − χ 2 ∇ ⋅ ( n 2 ∇ c ) + μ 2 n 2 ( 1 − a 2 n 1 − n 2 ) in Ω × ( 0 , ∞ ) , c t + u ⋅ ∇ c = Δ c − ( α n 1 + β n 2 ) c in Ω × ( 0 , ∞ ) , u t + ( u ⋅ ∇ ) u = Δ u + ∇ P + ( γ n 1 + δ n 2 ) ∇ ϕ , ∇ ⋅ u = 0 in Ω × ( 0 , ∞ ) under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R 2 with smooth boundary. This system consists of two models which were attracting mathematical interests. One is a chemotaxis-Navier–Stokes system which is known as a challenging model. The other is a two-species chemotaxis system with Lotka–Volterra competitive kinetics which has a complicated form. Both systems were recently well investigated; however, the above mixed system seems to be not studied yet. This paper gives results for global existence, boundedness and stabilization of solutions to the above system. [ABSTRACT FROM AUTHOR]

In this paper we study the space-homogeneous Boltzmann equation in a spatially flat FLRW spacetime. We consider Israel particles, which are the relativistic counterpart of the Maxwellian particles, and obtain global-in-time existence and the asymptotic behaviour of solutions. The main argument of the paper is to use the energy method of Guo, and we observe that the method can be applied to study small solutions in a cosmological case. It is the first result of this type where a physically well-motivated scattering kernel is considered for the general relativistic Boltzmann equation. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate the weak solutions of the three-dimensional incompressible Euler equations with helical symmetry in the whole space when the helical swirl vanishes. Specifically, we establish the global existence of weak solutions when the initial vorticity lies in L 1 ∩ L p with p > 1 . Our result extends the previous work [2] , where the initial vorticity is compactly supported and belongs to L p with p > 4 / 3 . The key ingredient in this paper involves the explicit analysis of Biot–Savart law with helical symmetry in domain R 2 × [ − π , π ] via the theories of singular integral operators and second order elliptic equations. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study the solvability of the rotationally symmetric centroaffine Minkowski problem. By delicate blow-up analyses, we remove a technical condition in the existence result obtained by Lu and Wang [30]. [ABSTRACT FROM AUTHOR]

Abstract This paper links at the formal level the entropy structure of a multi-species cross-diffusion system of Shigesada–Kawasaki–Teramoto (SKT) type (cf. [1]) satisfying the detailed balance condition with the entropy structure of a reversible microscopic many-particle Markov process on a discretised space. The link is established by first performing a mean-field limit to a master equation over discretised space. Then the spatial discretisation limit is performed in a completely rigorous way. This by itself provides a novel strategy for proving global existence of weak solutions to a class of cross-diffusion systems. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present a general mathematical framework for integrating smooth vector fields in the vicinity of a fixed point with a spiral saddle. We restrict our study to the three-dimensional setting, where the stable manifold is of spiral type (and thus two-dimensional), and the unstable manifold is one-dimensional. The aim is to produce a general purpose set of bounds that can be applied to any system of this type. The existence (and explicit computation) of such bounds is important when integrating along the flow near the spiral saddle fixed point. As an application, we apply our work to a concrete situation: the cubic Chua's equations. Here, we present a computer assisted proof of the existence of a trapping region for the flow. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study the Palais–Smale sequences of the conformal Dirac–Einstein problem. After we characterize the bubbling phenomena, we prove an Aubin type result leading to the existence of a positive solution. Then we show the existence of infinitely many solutions to the problem provided that the underlying manifold exhibits certain symmetries. [ABSTRACT FROM AUTHOR]

In this paper we consider the global existence and large-time behavior of solutions of one-dimensional viscous radiative and reactive gas with large initial data. In precedent studies the case for the initial-boundary value problem in bounded domain has been investigated and our main purpose focuses on the corresponding problem in unbounded domain. If the initial data is a large perturbation of a non-vacuum constant equilibrium state and is assumed to be without vacuum, mass concentrations, or vanishing temperatures, then we can show that its Cauchy problem admits a unique global smooth non-vacuum solution which tends time-asymptotically to such an equilibrium state. The key point in the analysis is to deduce the uniform positive lower and upper bounds on the specific volume and the absolute temperature. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the fractional order static Hartree equations with critical nonlocal nonlinearity. We prove that the positive solutions are radially symmetric about some point in R d and must assume the certain explicit forms. The arguments used in our proof is a variant (for nonlocal nonlinearity) of the direct moving plane method for fractional Laplacians in [6] . The main ingredients are the variants (for nonlocal nonlinearity) of the maximum principles, i.e., Decay at infinity and Narrow region principle ( Theorem 2.1 and 2.6 ). [ABSTRACT FROM AUTHOR]

In this paper, we study a system of Schrödinger–Poisson equation { − ε 2 Δ v + V ( x ) v + K ( x ) ϕ v = | v | p − 2 v , x ∈ R 3 , − ε 2 Δ ϕ = K ( x ) v 2 , x ∈ R 3 , where p ∈ ( 4 , 6 ) , the potentials V , K ∈ C ( R 3 , R + ) and ε > 0 is a parameter. Under the critical frequency assumptions on V and K , we investigate the existence and multiplicity of semi-classical solutions for this system and exhibit the concentration behavior that such solutions converge to the least energy solutions of the associated limit problem as ε → 0 . [ABSTRACT FROM AUTHOR]

We consider a time-periodic competition system with a free boundary, which can be used to describe two species that are struggling on the free boundary to obtain their own habitats and the final competition result is the regional partition of two species. In [15] , we have studied the (classical) traveling wave solutions of the temporal homogeneous problems. In this paper, we consider the time-periodic case and study periodic traveling wave solutions of the problem. More precisely, we will prove the existence and uniqueness of the periodic traveling wave solution, and show that the speed of the periodic traveling wave is uniquely determined by and monotonically dependent on the time-periodic terms in the system. Moreover, we will present some sufficient conditions to guarantee that the wave travels with a positive speed. [ABSTRACT FROM AUTHOR]

In this paper we consider a semilinear parabolic equation in an infinite cylinder. The spatially varying nonlinearity is such that it connects two (spatially independent) bistable nonlinearities in a compact set in space. We prove that, given such a setting, a traveling wave obeying the equation with the one bistable nonlinearity and starting at the respective side of the cylinder, will converge to a traveling wave solution prescribed by the nonlinearity on the other side. [ABSTRACT FROM AUTHOR]

This paper is a continuation of the work in [8] , where the authors established the global existence of smooth small data solutions to the general 3-D quasilinear wave equation ∑ i , j = 0 3 g i j ( u , ∂ u ) ∂ i j 2 u = 0 when the weak null condition holds. In the present paper, we show that the smooth small data solutions of equation ∑ i , j = 0 3 g i j ( u , ∂ u ) ∂ i j 2 u = 0 will blow up in finite time when the weak null condition does not hold and a generic nondegenerate condition on the initial data is satisfied, moreover, a precise blowup time is completely determined. Therefore, collecting the main results in this paper and [8] , we have given a basically complete study on the blowup or global existence of small data solutions to the 3-D quasilinear wave equation ∑ i , j = 0 3 g i j ( u , ∂ u ) ∂ i j 2 u = 0 . [ABSTRACT FROM AUTHOR]

We are interested in almost global existence cases in the general theory for nonlinear wave equations, which are caused by critical exponents of nonlinear terms. Such situations can be found in only three cases in the theory, cubic terms in two space dimensions, quadratic terms in three space dimensions and quadratic terms including a square of unknown functions itself in four space dimensions. Except for the last case, criteria to classify nonlinear terms into the almost global, or global existence case, are well-studied and known to be so-called null condition and non-positive condition. Our motivation of this work is to find such a kind of the criterion in four space dimensions. In our previous paper, an example of the non-single term for the almost global existence case is introduced. In this paper, we show an example of the global existence case. These two examples have nonlinear integral terms which are closely related to derivative loss due to high dimensions. But it may help us to describe the final form of the criterion. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of an almost invariant symplectic slow manifold for analytic Hamiltonian slow–fast systems with finitely many slow degrees of freedom for which the error field is exponentially small. We allow for infinitely many fast degrees of freedom. The method we use is motivated by a paper of MacKay from 2004. The method does not notice resonances, and therefore we do not pose any restrictions on the motion normal to the slow manifold other than it being fast and analytic. We also present a stability result and obtain a generalization of a result of Gelfreich and Lerman on an invariant slow manifold to (finitely) many fast degrees of freedom. [ABSTRACT FROM AUTHOR]

This paper proves the global existence and boundedness of solutions to a general reaction–diffusion predator–prey system with prey-taxis defined on a smooth bounded domain with no-flux boundary condition. The result holds for domains in arbitrary spatial dimension and small prey-taxis sensitivity coefficient. This paper also proves the existence of a global attractor and the uniform persistence of the system under some additional conditions. Applications to models from ecology and chemotaxis are discussed. [ABSTRACT FROM AUTHOR]

We study the expansion of a wedge of rest gas into a vacuum. When the rest gas is a van der Waals gas, the gas away from the sharp corner of the wedge may expand into the vacuum as symmetrical planar rarefaction waves, planar fan-jump composite waves, or planar fan-jump-fan composite waves. So, in order to solve the expansion problem we need to study the interactions of these elementary waves. In a recent paper [17] , we obtained the existence of global in time classical solution to the interaction of the planar rarefaction waves. This paper studies the interaction of the planar fan-jump composite waves. To construct the flow in the interaction region of the fan-jump composite waves, we consider a discontinuous Goursat problem for the two-dimensional self-similar Euler system. The existence of global solution to the discontinuous Goursat problem is obtained constructively by using the characteristic method. [ABSTRACT FROM AUTHOR]

This paper deals with the initial-boundary value problem (denoted by (CGL) ) for the complex Ginzburg–Landau type equation ∂ u ∂ t − ( λ + i α ) Δ u + ( κ + i β ) | u | q − 1 u − γ u = 0 with initial data u 0 ∈ L p ( Ω ) in the case 1 < q < 1 + 2 p / N , where Ω is bounded or unbounded in R N , λ > 0 , α , β , γ , κ ∈ R . There are a lot of studies on local and global existence of solutions to (CGL) including the physically relevant case q = 3 and κ > 0 . This paper gives existence results with precise properties of solutions and rigorous proof from a mathematical point of view. The physically relevant case can be considered as a special case of the results. Moreover, in the case κ < 0 , local and global existence of solutions with the decay estimate ‖ u ( t ) ‖ L p ( Ω ) ≤ c 1 e − c 2 t ( c 1 , c 2 are positive constants) is obtained under some conditions. The key to the local existence is to construct a semigroup { e t [ ( λ + i α ) Δ ] } and its L p - L q estimate. On the other hand, the key to the global existence is to derive estimates for solutions by using a kind of interpolation inequality with Re 〈 | v | p − 2 v , − ( λ + i α ) Δ v 〉 . [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the existence of weak solutions for a perturbed nonlinear elliptic equation driven by the fractional p -Laplacian operator as follows: ( − Δ ) p s u + V ( x ) | u | p − 2 u = λ a ( x ) | u | r − 2 u − b ( x ) | u | q − 2 u in R N , where λ is a real parameter, ( − Δ ) p s is the fractional p -Laplacian operator with 0 < s < 1 < p < ∞ , p < r < min ⁡ { q , p s ⁎ } and V , a , b : R N → ( 0 , ∞ ) are three positive weights. Using variational methods, we obtain nonexistence and multiplicity results for the above-mentioned equations depending on λ and according to the integrability properties of the ratio a q − p / b r − p . Our results extend the previous work of Autuori and Pucci (2013) [5] to the fractional p -Laplacian setting. Furthermore, we weaken one of the conditions used in their paper. Hence the results of this paper are new even in the fractional Laplacian case. [ABSTRACT FROM AUTHOR]

In an earlier paper, [1] , the authors treated a family of Allen–Cahn model problems for which 0 and 1 are solutions and further solutions were found that are near 1 on a prescribed set, T + Ω , where T ⊂ Z n , and near 0 on ( Z n ∖ T ) + Ω . Here Ω ⊂ ( 0 , 1 ) n . In this paper, a more general class of potentials is treated for which the pair, { 0 , 1 } , is replaced by Z and the existence of a far richer structure of shadowing solutions, including unbounded ones, is established. [ABSTRACT FROM AUTHOR]

In this paper the non-linear wave equation with a spatial inhomogeneity is considered. The inhomogeneity splits the unbounded spatial domain into three or more intervals, on each of which the non-linear wave equation is homogeneous. In such setting, there often exist multiple stationary fronts. In this paper we present a necessary and sufficient stability criterion in terms of the length of the middle interval(s) and the energy associated with the front in these interval(s). To prove this criterion, it is shown that critical points of the length function and zeros of the linearisation have the same order. Furthermore, the Evans function is used to identify the stable branch. The criterion is illustrated with an example which shows the existence of bi-stability: two stable fronts, one of which is non-monotonic. The Evans function also gives a sufficient instability criterion in terms of the derivative of the length function. [ABSTRACT FROM AUTHOR]

This paper is concerned with a coupled system of quasilinear parabolic equations where the effect of time delays is taken into consideration in the reaction functions of the system. The partial differential operators in the system may be degenerate and the reaction functions possess some mixed quasimonotone property, including quasimonotone nondecreasing functions. The aim of the paper is to show the existence and uniqueness of a global solution to the parabolic system, the existence of positive quasisolutions or maximal–minimal solutions of the corresponding elliptic system, and the asymptotic behavior of the solution of the parabolic system in relation to the quasisolutions or maximal–minimal solutions of the elliptic system. Applications are given to three reaction–diffusion models arising from mathematical biology and ecology where the diffusion coefficients are density dependent and are degenerate. This degenerate density-dependent diffusion leads to some interesting distinct asymptotic behavior of the time-dependent solution when compared with density-independent diffusion. [ABSTRACT FROM AUTHOR]

In this paper we study a two-dimensional toxin-determined functional response model (TDFRM). The toxin-determined functional response explicitly takes into consideration the reduction in the consumption of plants by herbivore due to chemical defense, which generates more complex dynamics of the plant–herbivore interactions. The purpose of the present paper is to analyze the existence of limit cycles and bifurcations of the model. By applying the theories of rotated vector fields and the extended planar termination principle, we establish the conditions for the existence of limit cycles and homoclinic loop. It is shown that a limit cycle is generated in a supercritical Hopf bifurcation and terminated in a homoclinic bifurcation, as the parameters vary. Analytic proofs are provided for all results, which generalize the results presented in [11] . [ABSTRACT FROM AUTHOR]

This paper is concerned with the inflow problem for the one-dimensional compressible Navier–Stokes equations. For such a problem, Matsumura and Nishihara showed in [10] that there exists boundary layer solution to the inflow problem, and that both the boundary layer solution, the rarefaction wave, and the superposition of boundary layer solution and rarefaction wave are nonlinear stable under small initial perturbation. The main purpose of this paper is to show that similar stability results for the boundary layer solution and the supersonic rarefaction wave still hold for a class of large initial perturbation which can allow the initial density to have large oscillation. The proofs are given by an elementary energy method and the key point is to deduce the desired lower and upper bounds on the density function. [ABSTRACT FROM AUTHOR]

In this paper the centroaffine Minkowski problem, a critical case of the L p -Minkowski problem in the n + 1 dimensional Euclidean space, is studied. By its variational structure and the method of blow-up analyses, we obtain two sufficient conditions for the existence of solutions, for a generalized rotationally symmetric case of the problem. [ABSTRACT FROM AUTHOR]

In this paper we consider a diffusion system with the Belousov–Zhabotinskii (BZ for short) chemical reaction. Following Brazhnik and Tyson [4] and Pérez-Muñuzuri et al. [45] , who predicted V-shaped fronts theoretically and discovered V-shaped fronts by experiments respectively, we give a rigorous mathematical proof of their results. We establish the existence of V-shaped traveling fronts in R 2 by constructing a proper supersolution and a subsolution. Furthermore, we establish the stability of the V-shaped front in R 2 . [ABSTRACT FROM AUTHOR]

We consider a nonlinear dispersive equation with a quasilinear quadratic term. We establish two results. First, we show that solutions to this equation with initial data of order O ( ε ) in Sobolev norms exist for a time span of order O ( ε − 2 ) for sufficiently small ε . Secondly, we derive the Nonlinear Schrödinger (NLS) equation as a formal approximation equation describing slow spatial and temporal modulations of the envelope of an underlying carrier wave, and justify this approximation with the help of error estimates in Sobolev norms between exact solutions of the quasilinear equation and the formal approximation obtained via the NLS equation. The proofs of both results rely on estimates of appropriate energies whose constructions are inspired by the method of normal-form transforms. To justify the NLS approximation, we have to overcome additional difficulties caused by the occurrence of resonances. We expect that the method developed in the present paper will also allow to prove the validity of the NLS approximation for a larger class of quasilinear dispersive systems with resonances. [ABSTRACT FROM AUTHOR]

This paper concerns the equation (0.1) u t = u x x + f ( x − c t , u ) , x ∈ R , where c ≥ 0 is a forcing speed and f : ( s , u ) ∈ R × R + → R is asymptotically of KPP type as s → − ∞ . We are interested in the questions of whether such a forced moving KPP nonlinearity from behind can give rise to traveling waves with the same speed and how they attract solutions of initial value problems when they exist. Under a sublinearity condition on f ( s , u ) , we obtain the complete existence and multiplicity of forced traveling waves as well as their attractivity except for some critical cases. In these cases, we provide examples to show that there is no definite answer unless one imposes further conditions depending on the heterogeneity of f in s ∈ R . [ABSTRACT FROM AUTHOR]

This paper presents our study of regularity for p -harmonic map heat flows. We devise a monotonicity-type formula of scaled energy and establish a criterion for a uniform regularity estimate for regular p -harmonic map heat flows. As application we show the small data global in the time existence of regular p -harmonic map heat flow. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for a two-phase model with magnetic field in three dimensions. The global existence and uniqueness of strong solution as well as the time decay estimates in H 2 ( R 3 ) are obtained by introducing a new linearized system with respect to ( n γ − n ˜ γ , n − n ˜ , P − P ˜ , u , H ) for constants n ˜ ≥ 0 and P ˜ > 0 , and doing some new a priori estimates in Sobolev Spaces to get the uniform upper bound of ( n − n ˜ , n γ − n ˜ γ ) in H 2 ( R 3 ) norm. [ABSTRACT FROM AUTHOR]

The chemotaxis system { u t = Δ u − ∇ ⋅ ( u v ∇ v ) , v t = Δ v − u v , ( ⋆ ) is considered under homogeneous Neumann boundary conditions in the ball Ω = B R ( 0 ) ⊂ R n , where R > 0 and n ≥ 2 . Despite its great relevance as a model for the spontaneous emergence of spatial structures in populations of primitive bacteria, since its introduction by Keller and Segel in 1971 this system has been lacking a satisfactory theory even at the level of the basic questions from the context of well-posedness; global existence results in the literature are restricted to spatially one- or two-dimensional cases so far, or alternatively require certain smallness hypotheses on the initial data. For all suitably regular and radially symmetric initial data ( u 0 , v 0 ) satisfying u 0 ≥ 0 and v 0 > 0 , the present paper establishes the existence of a globally defined pair ( u , v ) of radially symmetric functions which are continuous in ( Ω ¯ ∖ { 0 } ) × [ 0 , ∞ ) and smooth in ( Ω ¯ ∖ { 0 } ) × ( 0 , ∞ ) , and which solve the corresponding initial-boundary value problem for (⋆) with ( u ( ⋅ , 0 ) , v ( ⋅ , 0 ) ) = ( u 0 , v 0 ) in an appropriate generalized sense. To the best of our knowledge, this in particular provides the first result on global existence for the three-dimensional version of (⋆) involving arbitrarily large initial data. [ABSTRACT FROM AUTHOR]

We consider kinetic models for a multi component gas mixture without chemical reactions. In the literature, one can find two types of BGK models in order to describe gas mixtures. One type has a sum of BGK type interaction terms in the relaxation operator, for example the model described by Klingenberg, Pirner and Puppo [20] which contains well-known models of physicists and engineers for example Hamel [16] and Gross and Krook [15] as special cases. The other type contains only one collision term on the right-hand side, for example the well-known model of Andries, Aoki and Perthame [1] . For each of these two models [20] and [1] , we prove existence, uniqueness and positivity of solutions in the first part of the paper. In the second part, we use the first model [20] in order to determine an unknown function in the energy exchange of the macroscopic equations for gas mixtures described by Dellacherie [11] . [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of weak solutions for the equations describing the unsteady motion of an incompressible, viscous and homogeneous generalized Newtonian fluid in a non-cylindrical domain ⋃ t ∈ I { t } × Ω ( t ) . [ABSTRACT FROM AUTHOR]

In this paper, we discuss several results regarding existence, non-existence and asymptotic properties of solutions to u ⁗ + q u ″ + f ( u ) = 0 , under various hypotheses on the parameter q and on the potential F ( t ) = ∫ 0 t f ( s ) d s , generally assumed to be bounded from below. We prove a non-existence result in the case q ≤ 0 and an existence result of periodic solution for: 1) almost every suitably small (depending on F ), positive values of q ; 2) all suitably large (depending on F ) values of q . Finally, we describe some conditions on F which ensure that some (or all) solutions u q to the equation satisfy ‖ u q ‖ ∞ → 0 , as q ↓ 0 . [ABSTRACT FROM AUTHOR]

We consider the Brezis–Niremberg problem: ( P ε ) { − △ u = | u | p − 1 u + ε u in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain in R n , n = 4 , 5 , 6 , p + 1 = 2 n n − 2 is the critical Sobolev exponent and ε is a positive parameter. The main result of the paper generalizes the result of A. Iacopetti and F. Pacella [10] . Precisely we show that there are no low energy sign-changing solutions u ε with max ⁡ u ε / min ⁡ u ε → 0 or −∞ as ε goes to zero. [ABSTRACT FROM AUTHOR]

This paper is concerned with a system of quasilinear reaction–diffusion equations with density dependent diffusion coefficients and mixed quasimonotone reaction functions. The equations are allowed to be degenerate and the boundary conditions are of the nonlinear type. The main goals are to prove the existence and uniqueness of the weak solution between a pair of coupled upper and lower solutions; show that the weak solution evolves into the classical solution, and analyze the asymptotic behavior of the solution using quasi-solutions of the steady-state system. The general results are applied to a degenerate Lotka–Volterra competition model. Conditions are given for the solution to exist globally, to evolve into the classical solution, and to be attracted into a sector formed by quasi-solutions of the elliptic system. Especially for the Neumann problem we give a simple condition for the solution to converge to a unique constant steady-state solution which is a global attractor. [ABSTRACT FROM AUTHOR]

In this paper, we study systems of lattice differential equations of reaction–diffusion type. First, we establish some basic properties such as the local existence and global uniqueness of bounded solutions. Then we proceed to our main goal, which is the study of invariant regions. Our main result can be interpreted as an analogue of the weak maximum principle for systems of lattice differential equations. It is inspired by existing results for parabolic differential equations, but its proof is different and relies on the Euler approximations of solutions to lattice differential equations. As a corollary, we obtain a global existence theorem for nonlinear systems of lattice reaction–diffusion equations. The results are illustrated on examples from population dynamics. [ABSTRACT FROM AUTHOR]