Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms

In this paper we consider the Cauchy problem of semilinear parabolic equations with nonlinear gradient terms a(x)|u|^q−1u|∇u|^p. We prove the existence of global solutions and self-similar solutions for small initial data. Moreover, for a class of initial data we show that the global solutions behave asymptotically like self-similar solutions as t→∞.     

Modified Wave Operators Without Loss of Regularity for Some Long-Range Hartree Equations: I

We reconsider the theory of scattering for some long-range Hartree equations with potential |x|^?γ with 1/2 <  γ <  1. More precisely we study the local Cauchy problem with infinite initial time, which is the main step in the construction of the modified wave operators. We solve that problem in the whole subcritical range without loss of regularity between the asymptotic state and the solution, thereby recovering a result of Nakanishi. Our method starts from a different parametrization of the solutions, already used in our previous papers. This reduces the proofs to energy estimates and avoids delicate phase estimates.     

L2 ASYMPTOTIC FOR THE EQUILIBRIUM SOLUTION OF THE F-M EQUATIONS IN THE SPACE-PERIODIC CASE

The L² exponetial asymptotical stability for the equilibrium solution of the F-M equations in the space-periodic case (n = 2) is considered. Under some assumptions on the external force, it can be shown that the weak solution of F-M equations with initial and boundary conditions in space-periodic case approaches the stationary solution of the system exponetially when time t goes to infinite.     

Existence and optimal decay rates of the compressible non-isentropic Navier–Stokes–Poisson models with external forces

In this paper, we consider the three dimensional compressible non-isentropic Navier–Stokes–Poisson equations with the potential external force. Under the smallness assumption of the external force in some Sobolev space, the existence of the stationary solution is established by solving a nonlinear elliptic system. Next, we show global well-posedness of the initial value problem for the three dimensional compressible non-isentropic Navier–Stokes–Poisson equations, provided the prescribed initial data is close to the stationary solution. Finally, based on the elaborate energy estimates for the nonlinear system and L²$L^2$-decay estimates for the semigroup generated by the linearized equation, we give the optimal L²$L^2$-convergence rates of the solutions toward the stationary solution.     

A note on the existence and uniqueness of the solution to neutral stochastic functional differential equations with infinite delay

In this note, we prove the existence and uniqueness of the solution to neutral stochastic functional differential equations with infinite delay (INSFDEs in short) in which the initial value belongs to the phase space BC((-∞,0]R^d), which denotes the family of bounded continuous R^d-value functions φ defined on (-∞,0] with norm ||φ||=sup_-∞<θ?0|φ(θ)|, under some Carathéodory-type conditions on the coefficients by means of the successive approximation. Especially, we extend the results appeared in Ren et al. [Y. Ren, S. Lu, N. Xia, Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay, J. Comput. Appl. Math. 220 (2008) 364-372], Ren and Xia [Y. Ren, N. Xia, Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay, Appl. Math. Comput. 210 (2009) 72-79] and Zhou and Xue [S. Zhou, M. Xue, The existence and uniqueness of the solutions for neutral stochastic functional differential equations with infinite delay, Math. Appl. 21 (2008) 75-83].     

On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain

We study the global existence of weak solutions to a multi-dimensional simplified Ericksen–Leslie system for compressible flows of nematic liquid crystals with large initial energy in a bounded domain Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, where N=2 or 3$N=2\text{or}3$. By exploiting a maximum principle, Nirenberg?s interpolation inequality and a smallness condition imposed on the N  -th component of initial direction field _d0${\mathbf{d}}_0$ to overcome the difficulties induced by the supercritical nonlinearity |∇d|²d${|\mathrm{\nabla }\mathbf{d}|}^2\mathbf{d}$ in the equations of angular momentum, and then adapting a modified three-dimensional approximation scheme and the weak convergence arguments for the compressible Navier–Stokes equations, we establish the global existence of weak solutions to the initial-boundary problem with large initial energy and without any smallness condition on the initial density and velocity.     

Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity

We study the asymptotic behavior in time of solutions to the initial value problem of the nonlinear Schrödinger equation with a subcritical dissipative nonlinearity λ|u|^p−1u, where 1<p<1+2/n, n is the space dimension and λ is a complex constant satisfying Imλ<0. We show the time decay estimates and the large-time asymptotics of the solution, when the space dimension n?3, p is sufficiently close to 1+2/n and the initial data is sufficiently small.     

Classification of radial solutions to the Emden–Fowler equation on the hyperbolic space

We study the Emden–Fowler equation ?Δu = |u| ^p?1 u on the hyperbolic space ${{\mathbb H}^n}$ . We are interested in radial solutions, namely solutions depending only on the geodesic distance from a given point. The critical exponent for such equation is p = (n + 2)/(n ? 2) as in the Euclidean setting, but the properties of the solutions show striking differences with the Euclidean case. While the papers (Bhakta and Sandeep, Poincaré Sobolev equations in the hyperbolic space, 2011; Mancini and Sandeep, Ann Sci Norm Sup Pisa Cl Sci 7(5):635–671, 2008) consider finite energy solutions, we shall deal here with infinite energy solutions and we determine the exact asymptotic behavior of wide classes of finite and infinite energy solutions.     

The initial boundary value problem for Navier-Stokes equations

By making full use of the estimates of solutions to nonstationary Stokes equations and the method discussing global stability, we establish the global existence theorem of strong solutions for Navier-Stokes equatios in arbitrary three dimensional domain with uniformlyC ³ boundary, under the assumption that |a| _L ²(Θ) + |f| _L ¹(0,∞;L ²(Θ)) or |∇a| _L ²(Θ) + |f| _L ²(0,∞;L ²(Θ)) small or viscosityv large. Herea is a given initial velocity andf is the external force. This improves on the previous results. Moreover, the solvability of the case with nonhomogeneous boundary conditions is also discussed. This work is supported by foundation of Institute of Mathematics, Academia Sinica     

On the classification of minimal mass blowup solutions of the focusing mass-critical Hartree equation

Consider the focusing mass-critical nonlinear Hartree equation iut+Δu=−(⁻²|⋅|∗²|u|)u for spherically symmetric initial data with ground state mass M(Q) in dimension d?5. We show that any global solution u which does not scatter must be the solitary wave eitQ up to phase rotation and scaling.     

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).     

Solutions of the Robin Problem for the System of Elastic Theory in External Domains

One considers the Robin problem for the linear system of elastic theory. Properties of its solutions are examined in the classes of functions with bounded energy integral with the power weight |x| ^a . For different values of the weight parameter a, uniqueness theorems are proved and exact formulas are obtained for the dimension of the space of solutions of the Robin problem in external domains.     

Non-commutative operator Bohr inequality

It is shown that if A, B, X are Hilbert space operators such that X?γI, for the positive real number γ, and p,q>1 with 1/p+1/q=1, then |A−B|²?p|A|²+q|B|² with equality if and only if (1−p)A=B and γ||||A−B|²|||?|||p|A|²X+qX|B|²||| for every unitarily invariant norm. Moreover, if in addition A, B are normal and X is any Hilbert-Schmidt operator, then ‖δ_A,B²(X)‖₂?‖p|A|²X+qX|B|²‖₂ with equality if and only if (1−p)AX=XB.     

Hypergroupes infinis ayant un nombre fini d’hyperproduits propres

A non-commutative version for a theorem of P. Corsini [1] is given. It is shown that if (H, o) is an infinite hypergroup such that the set {(h, k) ?H ²|Card(hok)>1} is finite then there exists (a,b) ?H ² with Card (aob)=CardH. All the hypergroups having at most two proper hyperproducts are determined.     

Quantum scattering at low energies

For a class of negative slowly decaying potentials, including V(x):=−γ|x|^−μ with 0<μ<2, we study the quantum mechanical scattering theory in the low-energy regime. Using appropriate modifiers of the Isozaki-Kitada type we show that scattering theory is well behaved on the whole continuous spectrum of the Hamiltonian, including the energy 0. We show that the modified scattering matrices S(λ) are well-defined and strongly continuous down to the zero energy threshold. Similarly, we prove that the modified wave matrices and generalized eigenfunctions are norm continuous down to the zero energy if we use appropriate weighted spaces. These results are used to derive (oscillatory) asymptotics of the standard short-range and Dollard type S-matrices for the subclasses of potentials where both kinds of S-matrices are defined. For potentials whose leading part is −γ|x|^−μ we show that the location of singularities of the kernel of S(λ) experiences an abrupt change from passing from positive energies λ to the limiting energy λ=0. This change corresponds to the behaviour of the classical orbits. Under stronger conditions one can extract the leading term of the asymptotics of the kernel of S(λ) at its singularities.     

On the standing wave for a class of nonlinear Schrödinger equations

This paper is concerned with the standing wave for a class of nonlinear Schrödinger equations

iφt+Δφ−²|x|φ+μ|φ|^p−1φ+γ|φ|^q−1φ=0,     

On the integral equation formulations of some 2D contact problems

Two integral equations, representing the mechanical response of a 2D infinite plate supported along a line and subject to a transverse concentrated force, are examined. The kernels of the integral operators are of the type (x−y)ln|x−y| and (x−y)²ln|x−y|. In spite of the fact that these are only weakly singular, the two equations are studied in a more general framework, which allows us to consider also solutions having non-integrable endpoint singularities. The existence and uniqueness of solutions of the equations are discussed and their endpoint singularities detected.Since the two equations are of interest in their own right, some properties of the associated integral operators are examined in a scale of weighted Sobolev type spaces. Then, new results on the existence and uniqueness of integrable solutions of the equations that in some sense are complementary to those previously obtained are derived.     

Two dimensional exterior mixed problem for semilinear damped wave equations

We consider two dimensional exterior mixed problems for a semilinear damped wave equation with a power type nonlinearity p|u|. For compactly supported initial data, which have a small energy we shall derive global in time existence results in the case when the power of the nonlinearity satisfies 2<p<+∞. This generalizes a previous result of [J. Differential Equations 200 (2004) 53-68], which dealt with a radially symmetric solution.     

On a family of relative quartic Thue inequalities

We consider the relative Thue inequalities

|X⁴−t²X²Y²+s²Y⁴|?²|t|−²|s|−2,