Nonlinear Differential Equations and Applications (NoDEA)最新文献

In the context of Sobolev spaces with variable exponents, Poincaré–Wirtinger inequalities are possible as soon as Luxemburg norms are considered. On the other hand, modular versions of the inequalities in the expected form

are known to be false. As a result, all available modular versions of the Poincaré–Wirtinger inequality in the variable-exponent setting always contain extra terms that do not disappear in the constant exponent case, preventing such inequalities from reducing to the classical ones in the constant exponent setting. This obstruction is commonly believed to be an unavoidable anomaly of the variable exponent setting. The main aim of this paper is to show that this is not the case, i.e., that a consistent generalization of the Poincaré–Wirtinger inequality to the variable exponent setting is indeed possible. Our contribution is threefold. First, we establish that a modular Poincaré–Wirtinger inequality particularizing to the classical one in the constant exponent case is indeed conceivable. We show that if (Omega subset mathbb {R}^n ) is a bounded Lipschitz domain, and if (pin L^infty (Omega )), (p geqslant 1), then for every (fin C^infty (bar{Omega })) the following generalized Poincaré–Wirtinger inequality holds

where (langle frangle _{Omega }) denotes the mean of (f) over (Omega ), and (C>0) is a positive constant depending only on (Omega ) and (Vert pVert _{L^infty (Omega )}). Second, our argument is concise and constructive and does not rely on compactness results. Third, we additionally provide geometric information on the best Poincaré–Wirtinger constant on Lipschitz domains.

We study several discretizations of a second order gradient-like system with damping. We first consider an explicit scheme with a linear damping in finite dimension. We prove that every solution converges if the nonlinearity satisfies a global Lojasiewicz inequality. Convergence rates are also established. In the case of a strong nonlinear damping, we prove convergence of every solution for a fully implicit scheme in the one-dimensional case, even if the nonlinearity does not satisfy a Lojasiewicz inequality. The optimality of the damping is also established. Numerical simulations illustrate the theoretical results.

This work is devoted to study the relation between regularity and decay of solutions of some dissipative perturbations of the Korteweg-de Vries equation in weighted and asymmetrically weighted Sobolev spaces.

This paper is concerned with the following minimization problem

where energy functional (E_p(u)) is defined by

and V is a bounded potential. For (0<p< p^*:=frac{4-2,h}{N}(0<h<min {2,N})), it is shown that there exists a constant (M_0ge 0), such that the minimization problem exists at least one minimizer if (M> M_0). When (p=p^*,) the minimization problem exists at least one minimizer if (Min (M_{*},Vert Q_{p^*}Vert _{L^2}),) where constant (M_{*}ge 0) and (Q_{p^*}) is the unique positive radial solution of (-Delta u+u -| x|^{-h}|u |^{p^*} u=0,) and under some assumptions on V, there is no minimizer if (Mge Vert Q_{p^*}Vert _{L^2}). Moreover, when (0<p<p^*,) for fixed (M> Vert Q_{p^*}Vert _{L^2}), we analyze the concentration behavior of minimizers as (p nearrow p^* ).

This paper is concerned with a class of parabolic-elliptic chemotaxis models with density-suppressed motility and general logistic source in an n-dimensional smooth bounded domain. With some conditions on the density-suppressed motility function, we show the convergence rate of solutions is exponential as time tends to infinity for such kind of models.

We study the existence of ground states (minimizers of an energy under a fixed masses constraint) for a system of coupled NLS equations with double power nonlinearities (classical and point). We prove that the presence of at least one subcritical power parameter guarantees existence of a ground state for masses below specific values. Moreover, we show that each ground state is given by a pair of strictly positive functions (up to rotation). Using the concentration-compactness approach, under certain restrictions we prove the compactness of each minimizing sequence. One of the principal peculiarities of the model is that in presence of critical power parameters existence of ground states depends on concrete mass parameters related with the optimal function for the Gagliardo–Nirenberg inequality.

The present paper presents an abstract theory for proving (local-in-time) existence of energy solutions to some doubly-nonlinear evolution equations of gradient flow type involving time-dependent subdifferential operators with a quantitative estimate for the local-existence time. Furthermore, the abstract theory is employed to obtain an optimal existence result for some doubly-nonlinear parabolic equations involving space-time variable exponents, which are (possibly) non-monotone in time. More precisely, global-in-time existence of solutions is proved for the parabolic equations.

In this note, we prove the local well-posedness in the energy space of the k-generalized Zakharov–Kuznetsov equation posed on ( mathbb {R}times mathbb {T}) for any power non-linearity ( kge 2). Moreover, we obtain global solutions under a precise smallness assumption on the initial data by proving a sharp Gagliardo Nirenberg type inequality.

In this paper we show how to construct Morse homology for an explicit class of functionals involving the 2p-area functional. The natural domain of definition of such functionals is the Banach space (W^{1,2p}_0(Omega )), where (p>n/2) and (Omega subset mathbb {R}^n) is a bounded domain with sufficiently smooth boundary. As (W^{1,2p}_0(Omega )) is not isomorphic to its dual space,critical points of such functionals cannot be non-degenerate in the usual sense, and hence in the construction of Morse homology we only require that the second differential at each critical point be injective. Our result upgrades, in the case (p>n/2), the results in Cingolani and Vannella (Ann Inst H Poincaré Anal Non Linéaire 2:271–292, 2003; Ann Mat Pura Appl 186:155–183, 2007), where critical groups for an analogous class of functionals are computed, and provides in this special case a positive answer to Smale’s suggestion that injectivity of the second differential should be enough for Morse theory

We prove partial regularity of solutions u of the nonlinear quasimonotone system ({text {div}}Aleft( x,u,Duright) +Bleft( x,u,Duright) =0) under natural polynomial growth of its coefficient functions A and B. We propose a new direct method based on an (L^{p}) estimate with low exponent (p>1) for a linear elliptic system with constant coefficient.