Nonlinear Differential Equations and Applications (NoDEA)最新文献

The purpose of this paper consists in using variational methods to establish the existence of heteroclinic solutions for some classes of prescribed mean curvature equations of the type

where (epsilon >0) and V is a double-well potential with minima at (t=alpha ) and (t=beta ) with (alpha <beta ). Here, we consider some class of functions A(x, y) that are oscillatory in the variable y and satisfy different geometric conditions such as periodicity in all variables or asymptotically periodic at infinity.

We study the decay properties of Wigner kernels for Fourier integral operators of types I and II. The symbol spaces that allow a nice decay of these kernels are the Shubin classes (Gamma ^m({mathbb {R}^{2d}})), with negative order m. The phases considered are the so-called tame ones, which appear in the Schrödinger propagators. The related canonical transformations are allowed to be nonlinear. It is the nonlinearity of these transformations that are the main obstacles for nice kernel localizations when symbols are taken in the Hörmander’s class (S^{0}_{0,0}({mathbb {R}^{2d}})). Here we prove that Shubin classes overcome this problem and allow a nice kernel localization, which improves with the decreasing of the order m.

The present studies concern properties of set-valued Young integrals generated by families of (beta )-Hölder functions and differential inclusions governed by such a type of integrals. These integrals differ from classical set-valued integrals of set-valued functions constructed in an Aumann’s sense. Integrals and inclusions considered in the manuscript contain as a particular case set-valued integrals and inclusions driven by a fractional Brownian motion. Our study is focused on topological properties of solutions to Young differential inclusions. In particular, we show that the set of all solutions is compact in the space of continuous functions. We also study its dependence on initial conditions as well as properties of reachable sets of solutions. The results obtained in the paper are finally applied to some optimality problems driven by Young differential inclusions. The properties of optimal solutions and their reachable sets are discussed.

We consider the initial value problems (IVPs) for the modified Korteweg–de Vries (mKdV) equation

where u is a real valued function and (mu =pm 1), and the cubic nonlinear Schrödinger equation with third order dispersion (tNLS equation in short)

where (alpha , beta ) and (gamma ) are real constants and v is a complex valued function. In both problems, the initial data (u_0) and (v_0) are analytic on (mathbb {R}) and have uniform radius of analyticity (sigma _0) in the space variable. We prove that the both IVPs are locally well-posed for such data by establishing an analytic version of the trilinear estimates, and showed that the radius of spatial analyticity of the solution remains the same (sigma _0) till some lifespan (0<T_0le 1). We also consider the evolution of the radius of spatial analyticity (sigma (t)) when the local solution extends globally in time and prove that for any time (Tge T_0) it is bounded from below by (c T^{-frac{4}{3}}), for the mKdV equation in the defocusing case ((mu = -1)) and by (c T^{-(4+varepsilon )}), (varepsilon >0), for the tNLS equation. The result for the mKdV equation improves the one obtained in Bona et al. (Ann Inst Henri Poincaré 22:783–797, 2005) and, as far as we know, the result for the tNLS equation is the new one.

In this paper we establish new and optimal estimates for the existence time of the maximal solutions to the nonlinear parabolic system (partial _t u=Delta u+|v|^{p-1} v,; partial _t v=Delta v+|u|^{q-1} u,) (qge pge 1,; q>1) with initial values in Lebesgue or weighted Lebesgue spaces. The lower-bound estimates hold without any restriction on the sign or the size of the components of the initial data. To prove the upper-bound estimates, necessary conditions for the existence of nonnegative solutions are established. These necessary conditions allow us to give new sufficient conditions for finite time blow-up with initial values having critical decay at infinity.

We consider a scalar conservation law with source in a bounded open interval (Omega subseteq mathbb R). The equation arises from the macroscopic evolution of an interacting particle system. The source term models an external effort driving the solution to a given function (varrho ) with an intensity function (V:Omega rightarrow mathbb R_+) that grows to infinity at (partial Omega ). We define the entropy solution (u in L^infty ) and prove the uniqueness. When V is integrable, u satisfies the boundary conditions introduced by F. Otto (C. R. Acad. Sci. Paris, 322(1):729–734, 1996), which allows the solution to attain values at (partial Omega ) different from the given boundary data. When the integral of V blows up, u satisfies an energy estimate and presents essential continuity at (partial Omega ) in a weak sense.

We consider a stochastic electroconvection model describing the nonlinear evolution of a surface charge density in a two-dimensional fluid with additive stochastic forcing. We prove the existence and uniqueness of solutions, we define the corresponding Markov semigroup, and we study its Feller properties. When the noise forces enough modes in phase space, we obtain the uniqueness of the smooth invariant measure for the Markov transition kernels associated with the model.

This paper studies the upscaling of an elliptic problem with a highly oscillating quasilinear matrix coefficient, a quasilinear term, and a semilinear term in domains periodically perforated with holes of critical size. A Signorini boundary condition is imposed on the boundary of the holes, while a Dirichlet boundary condition is prescribed on the exterior boundary. Using the periodic unfolding method, we obtain an obstacle problem with a nonnegativity spreading effect.

In this paper, we study the resonant prescribed T-curvature problem on a compact 4-dimensional Riemannian manifold with boundary. We derive sharp energy and gradient estimates of the associated Euler-Lagrange functional to characterize the critical points at infinity of the associated variational problem under a non-degeneracy on a naturally associated Hamiltonian function. Using this, we derive a Morse type lemma at infinity around the critical points at infinity. Using the Morse lemma at infinity, we prove new existence results of Morse theoretical type. Combining the Morse lemma at infinity and the Liouville version of the Barycenter technique of Bahri–Coron (Commun Pure Appl Math 41–3:253–294, 1988) developed in Ndiaye (Adv Math 277(277):56–99, 2015), we prove new existence results under a topological hypothesis on the boundary of the underlying manifold, the selection map at infinity, and the entry and exit sets at infinity.

The almost everywhere pointwise and uniform convergences for the generalized KP-II equation are investigated when the initial data is in anisotropic Sobolev space (H^{s_{1},s_{2}}({textbf{R}}^{2})). Firstly, we show that the solution u(x, y, t) converges pointwisely to the initial data (f(x, y)in H^{s_{1},s_{2}}({{textbf{R}}}^{2}) ) for a.e. ((x, y) in {textbf{R}}^{2}) when (s_{1}ge frac{1}{4}), (s_{2}ge frac{1}{4}). The proof relies upon the Strichartz estimate and high-low frequency decomposition. Secondly, We prove that (s_{1}ge frac{1}{4}), (s_{2}ge frac{1}{4}) is a necessary condition for the maximal function estimate of the generalized KP-II equation to hold. Finally, by using the Fourier restriction norm method, we establish the nonlinear smoothing estimate to show the uniform convergence of the generalized KP-II equation in (H^{s_{1},s_{2}} ({{textbf{R}}}^{2}) ) with ( s_{1}ge frac{3}{2}-frac{alpha }{4}+epsilon , s_{2}>frac{1}{2}) and (alpha ge 4 ).