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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]

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]

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]

In this paper we give a simple proof of the existence of global-in-time smooth solutions for the convective Brinkman–Forchheimer equations (also called in the literature the tamed Navier–Stokes equations) ∂ t u − μ Δ u + ( u ⋅ ∇ ) u + ∇ p + α u + β | u | r − 1 u = 0 on a 3D periodic domain, for values of the absorption exponent r larger than 3. Furthermore, we prove that global, regular solutions exist also for the critical value of exponent r = 3 , provided that the coefficients satisfy the relation 4 μ β ≥ 1 . Additionally, we show that in the critical case every weak solution verifies the energy equality and hence is continuous into the phase space L 2 . As an application of this result we prove the existence of a strong global attractor, using the theory of evolutionary systems developed by Cheskidov. [ABSTRACT FROM AUTHOR]

We prove the existence of smallest positive and greatest negative solutions in D 1 , p ( R N ) of the p -Laplacian problem of logistic-type in R N with 1 < p < N : u ∈ D 1 , p ( R N ) : − Δ p u = a ( x ) ( λ | u | p − 2 u − g ( u ) ) , and consider its extended version with a harvesting term μh as well. Unlike papers on the topic, here we only assume a superlinear growth for the nonlinearity g : R → R at zero and at infinity, i.e., even supercritical growth is allowed, and, moreover, no monotonicity for s ↦ g ( s ) s p − 1 is required. We prove a Vázquez type maximum principle for nonnegative D 1 , p ( R N ) -solutions of some p -Laplacian problem and establish a sub–supersolution method in D 1 , p ( R N ) , which along with Wolff potential estimates for superharmonic functions will be used as our main tools. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the time periodic Lotka–Volterra competition–diffusion system where are T-periodic functions, , . Under certain conditions, the system admits two stable semi-trivial periodic solutions and and a unique coexistence periodic solution , which is unstable and satisfies and for . In this paper we prove that the system admits a time periodic traveling wave solution connecting two periodic solutions and as , where c is the wave speed. By using a dynamical method, we show that the time periodic traveling wave solution is asymptotically stable and unique modulo translation for front-like initial values. [Copyright &y& Elsevier]

Abstract: In this paper we are concerned with the following Neumann problem where ε is a small positive parameter, f is a superlinear and subcritical nonlinearity, Ω is a smooth and bounded domain in . Solutions with multiple boundary peaks have been established for this problem. It has also been proved that for any integer k there exists an interior k-peak solution which concentrates, as , at k sphere packing points in Ω. In this paper we prove the existence of a second interior k-peak solution provided that k is large enough, and we conjecture that its peaks are located along a straight line. Moreover, when Ω is a two-dimensional strictly convex domain, we also construct a third interior k-peak solution provided that k is large enough, whose peaks are aligned on a closed curve near ∂Ω. [Copyright &y& Elsevier]

In this paper, we are interested in standing waves with a vortex for the nonlinear Chern–Simons–Schrödinger equations (CSS for short). We study the existence and the nonexistence of standing waves when a constant λ > 0 , representing the strength of the interaction potential, varies. We prove every standing wave is trivial if λ ∈ ( 0 , 1 ) , every standing wave is gauge equivalent to a solution of the first order self-dual system of CSS if λ = 1 and for every positive integer N , there is a nontrivial standing wave with a vortex point of order N if λ > 1 . We also provide some classes of interaction potentials under which the nonexistence of standing waves and the existence of a standing wave with a vortex point of order N are proved. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for every Finsler n -dimensional sphere ( S n , F ) with reversibility λ and flag curvature K satisfying ( λ 1 + λ ) 2 < K ≤ 1 , either there exist infinitely many closed geodesics, or there exist at least two elliptic closed geodesics and each linearized Poincaré map has at least one eigenvalue of the form e − 1 θ with θ being an irrational multiple of π . [ABSTRACT FROM AUTHOR]

In this paper, we prove a global in time unique existence theorem for the compressible viscous fluids in a bounded domain with slip boundary condition in the maximal L p - L q regularity class with 2 < p < ∞ and N < q < ∞ under the assumption that initial data are small enough and orthogonal to rigid motions if domain is rotationally symmetric. To prove the global well-posedness, we use the prolongation argument based on the maximal L p - L q regularity estimate of exponentially decay type. The same problem was first treated by Kobayashi and Zaja̧czkowski [5] in the L 2 framework by using the energy method; our approach is completely different from Kobayashi and Zaja̧czkowski [5] . [ABSTRACT FROM AUTHOR]

In this paper we introduce a new homotopy invariant – the cohomological span of LS -Conley index. We prove the theorems on the existence of critical points for a class of strongly indefinite functionals with the gradient of the form L x + K ( x ) , where L is bounded linear and K is completely continuous. We give examples of Hamiltonian systems for which our methods give better results than the Morse inequalities. We also give a formula for the LS -index of an isolated critical point, which is an extension of the classical Dancer theorem for the case of LS -index. [ABSTRACT FROM AUTHOR]

In this paper we prove existence and uniqueness of local solutions to the three-dimensional (3D) Navier–Stokes (N–S) equation driven by space–time white noise using two methods: first, the theory of regularity structures introduced by Martin Hairer in [16] and second, the paracontrolled distribution proposed by Gubinelli, Imkeller, Perkowski in [12] . We also compare the two approaches. [ABSTRACT FROM AUTHOR]

In the present paper we prove existence results for solutions to nonlinear elliptic Neumann problems whose prototype is { − Δ p u − div ( c ( x ) | u | p − 2 u ) ) = f in Ω , ( | ∇ u | p − 2 ∇ u + c ( x ) | u | p − 2 u ) ⋅ n ̲ = 0 on ∂ Ω , when f is just a summable function. Our approach also allows us to prove a stability result for renormalized solutions and an existence result for operator with a zero order term. [ABSTRACT FROM AUTHOR]

In this paper, we prove the continuity of the flow of KdV on spaces of probability measures with respect to a combination of Wasserstein distances on H s , s > 0 and L 2 . We are motivated by the existence of an invariant measure belonging to the spaces onto which these distances are defined. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the wave equation with sub-linearity in a ball with periodic-Dirichlet conditions. Combining the minimax principle with an approximation argument, we prove the existence of infinitely many time-periodic solutions for the problem. The proof also relies on the structure of the spectrum of the linearized problem. [Copyright &y& Elsevier]

Abstract: This paper deals with a nonlocal problem arising in plasma physics. This problem involves the terms and , which are neither local, nor continuous and monotone. By using the Galerkin approximate method and the topological degree theory, the existence of solutions will be proved. [Copyright &y& Elsevier]

Abstract: The present paper is devoted to the study of the global well-posedness for the two-dimensional nonlinear Boussinesq equations with vertical dissipation. In the absence of horizontal dissipation, we establish a growth estimate on vertical component of velocity, that is, which is close to and is bounded via the low-high decomposition technique. This together with the smoothing effect in vertical direction enables us to obtain the -estimate for velocity. Based on this, we prove the existence and uniqueness of classical solution without smallness assumptions. In addition, we also discuss the global well-posedness result for the rough initial data. [Copyright &y& Elsevier]

Abstract: In this paper, we prove existence of small amplitude quasi-periodic solutions for a non-autonomous, quasi-periodically forced nonlinear wave equations with periodic spatial boundary conditions via KAM theory. [Copyright &y& Elsevier]

Abstract: In this paper, we study the Schrödinger–Poisson system and prove the existence of ground state solutions for system (SP) under certain assumptions on the linear and nonlinear terms. Some recent results from different authors are extended. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the initial value problem for the nonlinear Klein–Gordon–Schrödinger (KGS) equations in time–space. By using viscous approach, the existence of the global finite-energy solution is established for the nonlinear KGS equations by compactness argument. In addition, the uniqueness of the solution is proved by introducing a function with integral form. [Copyright &y& Elsevier]

Abstract: In this paper, we study the open loop stabilization as well as the existence and regularity of solutions of the weakly damped defocusing semilinear Schrödinger equation with an inhomogeneous Dirichlet boundary control. First of all, we prove the global existence of weak solutions at the -energy level together with the stabilization in the same sense. It is then deduced that the decay rate of the boundary data controls the decay rate of the solutions up to an exponential rate. Secondly, we prove some regularity and stabilization results for the strong solutions in -sense. The proof uses the direct multiplier method combined with monotonicity and compactness techniques. The result for weak solutions is strong in the sense that it is independent of the dimension of the domain, the power of the nonlinearity, and the smallness of the initial data. However, the regularity and stabilization of strong solutions are obtained only in low dimensions with small initial and boundary data. [Copyright &y& Elsevier]

Abstract: In this paper we use a concentration and compactness argument to prove the existence of a nontrivial non-radial solution to the nonlinear Schrödinger–Poisson equations in , assuming on the nonlinearity the general hypotheses introduced by Berestycki and Lions. [ABSTRACT FROM AUTHOR]

Abstract: This paper examines the properties of the homentropic Euler equations when the characteristics of the equations have been spatially averaged. The new equations are referred to as the characteristically averaged homentropic Euler (CAHE) equations. An existence and uniqueness proof for the modified equations is given. The speed of shocks for the CAHE equations are determined. The Riemann problem is examined and a general form of the solutions is presented. Finally, numerically simulations on the homentropic Euler and CAHE equations are conducted and the behaviors of the two sets of equations are compared. [Copyright &y& Elsevier]

Abstract: In this paper we study the generalized BO–ZK equation in two space dimensions We review the existence theory for solitary waves and prove that they are nonlinearly unstable if p belongs to the range . We also establish Strichartz-type estimates. [Copyright &y& Elsevier]