Search

Abstract We are concerned with positive solutions of the fractional elliptic system (E) { (− Δ) s u = f (v) in Ω , (− Δ) s v = g (u) in Ω , where Ω is an arbitrary domain in R N (N > 2 s), s ∈ (1 2 , 1) and f , g ∈ C l o c β (R) , for some β ∈ (0 , 1). We establish universal a priori estimate for positive solutions of (E). Then for C 2 bounded domain Ω, we prove the existence of positive solutions of (E) with prescribed boundary value u = μ and v = ν where μ , ν are positive bounded measure on ∂Ω and discuss regularity property of the solutions. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we are concerned with an initial–boundary value problem for a class of non-Newtonian fluids. The viscosity term of momentum equation possesses singularity and nonlinearity, and the initial vacuum is allowed. The main feature which distinguishes this paper from other related works lies in the fact that we study the global strong solution of the compressible non-Newtonian fluids with singularity and vacuum in one-dimensional bounded intervals, and show that there exists a unique global-in-time strong solution to the problem while the initial energy is small. [ABSTRACT FROM AUTHOR]

Abstract We study the linear finite element approximation of the elasticity equations with and without unilateral friction contact (of Tresca type) conditions in a polygonal or polyhedral domain. The unilateral contact condition is weakly imposed by the penalty method. We derive error estimates which depend on the penalty parameter ε and the mesh size h. In fact, under the H 3 2 + ν (Ω) , 0 < ν ≤ 1 2 , regularity of the solution of the contact problems (with and without friction) and with the requirement ε > h , we prove a convergence rate of O (h 1 2 + ν + ε 1 2 + ν) in the energy norm. Therefore, if the penalty parameter is taken as ε : = c h θ where 0 < θ ≤ 1 , the convergence rate of O (h θ (1 2 + ν) ) is obtained. In particular, we obtain an optimal linear convergence when ε behaves like h (i.e. θ = 1) and ν = 1 2. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for an attraction–repulsion chemotaxis system in R 2 with the chemotactic coefficient of the attractant β 1 and that of the repellent β 2 . It is known that in the repulsive dominant case β 1 < β 2 or the balance case β 1 = β 2 , the nonnegative solutions to the Cauchy problem exist globally in time, whereas in the attractive dominant case β 1 > β 2 , there are blowing-up solutions in finite time under the assumption ( β 1 − β 2 ) ∫ R 2 u 0 d x > 8 π on the nonnegative initial data u 0 . In this paper, we show the global existence of nonnegative solutions to the Cauchy problem under the assumption ( β 1 − β 2 ) ∫ R 2 u 0 d x < 8 π in the attractive dominant case. [ABSTRACT FROM AUTHOR]

Asymptotic stability of monotone increasing traveling wave solutions is investigated for a class of viscous compressible fluid equations with capillarity term. By using the theory of planar dynamical systems, the existence has been proved for the monotone increasing traveling wave solution. Moreover, through building uniformly a priori energy estimate for the perturbation of the traveling wave solution, we prove that the monotone increasing traveling wave solution is asymptotically stable in H 2 × H 1 . [ABSTRACT FROM AUTHOR]

Abstract: We consider the solutions to the prescribed scalar curvature equation on a four-dimensional Riemannian manifold M. We prove an upper bound for the supremum of all the solutions on every compact subset K of M, provided that all the solutions on M are bounded from below by a positive number. [Copyright &y& Elsevier]

Abstract: This paper deals with a semilinear Schrödinger equation whose nonlinear term involves a positive parameter λ and a real function which satisfies a superlinear growth condition just in a neighborhood of zero. By proving an a priori estimate (for a suitable class of solutions) we are able to avoid further restrictions on the behavior of at infinity in order to prove, for λ sufficiently large, the existence of one-sign and sign-changing solutions. Minimax methods are employed to establish this result. [Copyright &y& Elsevier]

Abstract: This paper discusses a prey–predator system with strongly coupled nonlinear diffusion terms. We give a sufficient condition for the existence of positive steady state solutions. Our proof is based on the bifurcation theory. Some a priori estimates for steady state solutions will play an important role in the proof. [Copyright &y& Elsevier]

Abstract: We study a priori estimates of positive solutions of the equation , , , satisfying the homogeneous Dirichlet boundary conditions. Here Ω is a bounded domain in , , is subcritical, changes sign and satisfy some additional technical hypotheses. Assume that the solution u blows up in a finite time T and the set is connected. Using our a priori bounds, we show that u blows up completely in at and the blow-up time T depends continuously on the initial data. [Copyright &y& Elsevier]

In a framework of a theory of weighted anisotropic Sobolev spaces an a priori estimate for a model semi-elliptic differential operatorAu = −∂2u/∂x21+∂4u/∂x42 + ∂4u/∂x42is obtained. [Copyright &y& Elsevier]