On the low regularity of the modified Korteweg-de Vries equation with a dissipative term

We consider a dissipative version of the modified Korteweg-de Vries equation ut+uxxx−uxx+x(u³)=0. We prove global well-posedness results on the associated Cauchy problem in the Sobolev spaces Hs(R) for s>−1/4 while for s<−1/2 we prove some ill-posedness issues.     

On the well-posedness of the Schrödinger-Korteweg-de Vries system

We prove that the Cauchy problem for the Schrödinger-Korteweg-de Vries system is locally well-posed for the initial data belonging to the Sobolev spaces L²(R)×H−3/4(R), and Hs(R)×H−3/4(R) (s>−1/16) for the resonant case. The new ingredient is that we use the -type space, introduced by the first author in Guo (2009) [10], to deal with the KdV part of the system and the coupling terms. In order to overcome the difficulty caused by the lack of scaling invariance, we prove uniform estimates for the multiplier. This result improves the previous one by Corcho and Linares (2007) [6].     

Sharp well-posedness for the Benjamin equation

Having the ill-posedness in the range s<−3/4 of the Cauchy problem for the Benjamin equation with an initial H^s(R) data, we prove that the already-established local well-posedness in the range s>−3/4 of this initial value problem is extendable to s=−3/4 and also that such a well-posed property is globally valid for s∈[−3/4,∞).     

Blowup stability of solutions of the nonlinear heat equation with a large life span

We study the Cauchy problem for the nonlinear heat equation u_t-?u=|u|^p-1u in R^N. The initial data is of the form u₀=λ?, where ?∈C₀(R^N) is fixed and λ>0. We first take 1<p<p_f, where p_f is the Fujita critical exponent, and ?∈C₀(R^N)∩L¹(R^N) with nonzero mean. We show that u(t) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1<p<p_s, where p_s is the Sobolev critical exponent, and ?(x) decaying as |x|^-σ at infinity, where p<1+2/σ. We also prove that u(t) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ? is not radial.     

Global solutions and blow-up phenomena to a shallow water equation

A nonlinear shallow water equation, which includes the famous Camassa-Holm (CH) and Degasperis-Procesi (DP) equations as special cases, is investigated. The local well-posedness of solutions for the nonlinear equation in the Sobolev space Hs(R) with is developed. Provided that does not change sign, u₀∈Hs () and u₀∈L¹(R), the existence and uniqueness of the global solutions to the equation are shown to be true in u(t,x)∈C([0,∞);Hs(R))∩C¹([0,∞);Hs−1(R)). Conditions that lead to the development of singularities in finite time for the solutions are also acquired.     

Well-posedness and weak rotation limit for the Ostrovsky equation

We consider the Cauchy problem of the Ostrovsky equation. We first prove the time local well-posedness in the anisotropic Sobolev space Hs,a with s>−a/2−3/4 and 0?a?−1 by the Fourier restriction norm method. This result include the time local well-posedness in Hs with s>−3/4 for both positive and negative dissipation, namely for both βγ>0 and βγ<0. We next consider the weak rotation limit. We prove that the solution of the Ostrovsky equation converges to the solution of the KdV equation when the rotation parameter γ goes to 0 and the initial data of the KdV equation is in L². To show this result, we prove a bilinear estimate which is uniform with respect to γ.     

On global solution to the Klein-Gordon-Hartree equation below energy space

In this paper, we consider the Cauchy problem for Klein-Gordon equation with a cubic convolution nonlinearity in R³. By making use of Bourgain's method in conjunction with a precise Strichartz estimate of S. Klainerman and D. Tataru, we establish the Hs (s<1) global well-posedness of the Cauchy problem for the cubic convolution defocusing Klein-Gordon-Hartree equation. Before arriving at the previously discussed conclusion, we obtain global solution for this non-scaling equation with small initial data in Hs₀×Hs₀−1 where but not , for this equation that we consider is a subconformal equation in some sense. In doing so a number of nonlinear prior estimates are already established by using Bony's decomposition, flexibility of Klein-Gordon admissible pairs which are slightly different from that of wave equation and a commutator estimate. We establish this commutator estimate by exploiting cancellation property and utilizing Coifman and Meyer multilinear multiplier theorem. As far as we know, it seems that this is the first result on low regularity for this Klein-Gordon-Hartree equation.     

A splitting method for the nonlinear Schrödinger equation

We introduce a splitting method for the semilinear Schrödinger equation and prove its convergence for those nonlinearities which can be handled by the classical well-posedness L²(Rd)-theory. More precisely, we prove that the scheme is of first order in the L²(Rd)-norm for H²(Rd)-initial data.     

Small solutions to nonlinear Schrödinger equations in the Sobolev spaces

The local and global well-posedness for the Cauchy problem for a class of nonlinear Schrödinger equations is studied. The global well-posedness of the problem is proved in the Sobolev spaceH ^s=H^s(R ⁿ) of fractional orders>n/2 under the following assumptions. (1) Concerning the Cauchy data ?∈H ^s: ‖?;L ²‖ is relatively small with respect to ‖?;H ^σ‖ for any fixed σ withn/2<σ≤s. (2) Concerning the nonlinearityf: f(u) behaves as a conformal poweru ¹⁺⁴/n near zero and has an arbitrary growth rate at infinity.     

Positive solutions of reaction diffusion equations with super-linear absorption: Universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions

Consider classical solutions u∈C²(Rⁿ×(0,∞))∩C(Rⁿ×[0,∞)) to the parabolic reaction diffusion equation

     

Well-posedness for the nonlocal nonlinear Schrödinger equation

We establish local well-posedness for small initial data in the usual Sobolev spaces Hs(R), s?1, and global well-posedness in H¹(R), for the Cauchy problem associated to the nonlocal nonlinear Schrödinger equation

     

Blow-up results for a class of first-order nonlinear evolution inequalities

We investigate the non-existence of solutions to a class of evolution inequalities; in this case, as it happens in a relatively small number of blow-up studies, nonlinearities depend also on time-variable t and spatial derivatives of the unknown. The present results, which in great part do not require any assumption on the regularity of data, are completely new and shown with various applications. Some of these results referring to the problem u_t=Δu+a(x)|u|^p+λf(x) in R^N, t>0 include the non-existence results of positive global solutions obtained by Fujita and others when a≡1 and f≡0, Bandle-Levine and Levine-Meier when a≡|x|^m and f≡0, Pinsky when either f≡0 or f?0 and λ>0, Zhang and Bandle-Levine-Zhang when a≡1 and λ=1.     

Asymptotic equipartition and long time behavior of solutions of a thin-film equation

We investigate the large-time behavior of classical solutions to the thin-film type equation ut=−x(uuxxx). It was shown in previous work of Carrillo and Toscani that for non-negative initial data u₀ that belongs to H¹(R) and also has a finite mass and second moment, the strong solutions relax in the L¹(R) norm at an explicit rate to the unique self-similar source type solution with the same mass. The equation itself is gradient flow for an energy functional that controls the H¹(R) norm, and so it is natural to expect that one should also have convergence in this norm. Carrillo and Toscani raised this question, but their methods, using a different Lyapunov functions that arises in the theory of the porous medium equation, do not directly address this since their Lyapunov functional does not involve derivatives of u. Here we show that the solutions do indeed converge in the H¹(R) norm at an explicit, but slow, rate. The key to establishing this convergence is an asymptotic equipartition of the excess energy. Roughly speaking, the energy functional whose dissipation drives the evolution through gradient flow consists of two parts: one involving derivatives of u, and one that does not. We show that these must decay at related rates—due to the asymptotic equipartition—and then use the results of Carrillo and Toscani to control the rate for the part that does not depend on derivatives. From this, one gets a rate on the dissipation for all of the excess energy.     

Well-posedness of a higher order modified Camassa–Holm equation in spaces of low regularity

In this paper we consider the Cauchy problem for a higher order modified Camassa–Holm equation. By using the Fourier restriction norm method introduced by Bourgain, we establish the local well-posedness for the initial data in the H ^s (R) with ${s > -n+\frac{5}{4},\,n\in {\bf N}^{+}.}${s > -n+\frac{5}{4},\,n\in {\bf N}^{+}.} As a consequence of the conservation of the energy ||u||_H¹(R),{{||u||_{H^{1}(R)},}} we have the global well-posedness for the initial data in H ¹(R).     

On the stochastic Benjamin-Ono equation

We discuss the Cauchy problem for the stochastic Benjamin-Ono equation in the function class Hs(R), s>3/2. When there is a zero-order dissipation, we also establish the existence of an invariant measure with support in H²(R). Many authors have discussed the Cauchy problem for the deterministic Benjamin-Ono equation. But our results are new for the stochastic Benjamin-Ono equation. Our goal is to extend known results for the deterministic equation to the stochastic equation.     

Dirichlet inhomogeneous boundary value problem for the n+1 complex Ginzburg-Landau equation

We study the following complex Ginzburg-Landau equation with cubic nonlinearity on for under initial and Dirichlet boundary conditions u(x,0)=h(x) for x∈Ω, u(x,t)=Q(x,t) on ∂Ω where h,Q are given smooth functions. Under suitable conditions, we prove the existence of a global solution in H¹. Further, this solution approaches to the solution of the NLS limit under identical initial and boundary data as a,b→0⁺.     

Well posedness for a higher order modified Camassa-Holm equation

We show that the Cauchy problem for a higher order modification of the Camassa-Holm equation is locally well posed for initial data in the Sobolev space Hs(R) for s>s^′, where 1/4?s^′<1/2 and the value of s^′ depends on the order of equation. Employing harmonic analysis methods we derive the corresponding bilinear estimate and then use a contraction mapping argument to prove existence and uniqueness of solutions.     

Global existence and blowup for sign-changing solutions of the nonlinear heat equation

In this paper, given 0<α<2/N, we prove the existence of a function ψ with the following properties. The solution of the equation ut−Δu=α|u|u on RN with the initial condition u(0)=ψ is global. On the other hand, the solution with the initial condition u(0)=λψ blows up in finite time if λ>0 is either sufficiently small or sufficiently large.     

Well-posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces

In this paper we prove the local and global well-posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces under certain conditions of s, q, and σ.