Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem

We consider the stability in an inverse problem of determining the potential q entering the wave equation in a bounded smooth domain of Rd from boundary observations. The observation is given by a hyperbolic (dynamic) Dirichlet to Neumann map associated to a wave equation. We prove a log-type stability estimate in determining q from a partial Dirichlet to Neumann map provided that q is a priori known in a neighbourhood of the boundary of the spatial domain and satisfies an additional condition. Next, we use this result to establish a stability estimate related to the multidimensional Borg-Levinson theorem.     

A new transformation of Burger’s equation for an exact solution in a bounded region necessary for certain boundary conditions

In this work, the transient analytic solution is found for the initial-boundary-value Burgers equation in 0?x?L. The boundary conditions are a homogeneous Dirichlet condition at x=0 and a constant total flux at x=L. The technique used consists of applying the transformation that reduces Burgers equation to a linear diffusion-advection equation. Previous work on this equation in a bounded region has only applied the Cole-Hopf transformation , which transforms Burgers equation to the linear diffusion equation. The Cole-Hopf transformation can only solve Burgers equation with constant Dirichlet boundary conditions, or time-dependent Dirichlet boundary conditions of the form u(0,t)=F₁(t) and u(L,t)=F₂(t),0?x?L. In this work, it is shown that the Cole-Hopf transformation will not solve Burgers equation in a bounded region with the boundary conditions dealt with in this work.     

Random attractors for stochastic lattice dynamical systems in weighted spaces

In this paper, we first provide some sufficient conditions for the existence of global compact random attractors for general random dynamical systems in weighted space (p?1) of infinite sequences. Then we consider the existence of global compact random attractors in weighted space for stochastic lattice dynamical systems with random coupled coefficients and multiplicative/additive white noises. Our results recover many existing ones on the existence of global random attractors for stochastic lattice dynamical systems with multiplicative/additive white noises in regular l² space of infinite sequences.     

Localization for discrete one-dimensional random word models

We consider Schrödinger operators in whose potentials are obtained by randomly concatenating words from an underlying set according to some probability measure ν on . Our assumptions allow us to consider models with local correlations, such as the random dimer model or, more generally, random polymer models. We prove spectral localization and, away from a finite set of exceptional energies, dynamical localization for such models. These results are obtained by employing scattering theoretic methods together with Furstenberg's theorem to verify the necessary input to perform a multiscale analysis.     

Dynamics of the viscous Cahn-Hilliard equation

We study generalized viscous Cahn-Hilliard problems with nonlinearities satisfying critical growth conditions in , where Ω is a bounded smooth domain in Rn, n?3. In the critical growth case, we prove that the problems are locally well posed and obtain a bootstrapping procedure showing that the solutions are classical. For p=2 and almost critical dissipative nonlinearities we prove global well posedness, existence of global attractors in and, uniformly with respect to the viscosity parameter, L^∞(Ω) bounds for the attractors. Finally, we obtain a result on continuity of regular attractors which shows that, if n=3,4, the attractor of the Cahn-Hilliard problem coincides (in a sense to be specified) with the attractor for the corresponding semilinear heat equation.     

Superlinear elliptic equations with singular coefficients on the boundary

In this paper, a superlinear elliptic equation whose coefficient diverges on the boundary is studied in any bounded domain Ω under the zero Dirichlet boundary condition. Although the equation has a singularity on the boundary, a solution is smooth on the closure of the domain. Indeed, it is proved that the problem has a positive solution and infinitely many solutions without positivity, which belong to or . Moreover, it is proved that a positive solution has a higher order regularity up to .     

U(1)-invariant special Lagrangian 3-folds. III. Properties of singular solutions

This is the last of three papers studying special Lagrangian 3-submanifolds (SLV 3-folds) N in invariant under the U(1)-action e^iθ:(z₁,z₂,z₃)?(e^iθz₁,e^iθz₂,z₃), using analytic methods. If N is such a 3-fold then |z₁|²−|z₂|²=2a on N for some . Locally, N can be written as a kind of graph of functions satisfying a nonlinear Cauchy-Riemann equation depending on a, so that u+iv is like a holomorphic function of x+iy.The first paper studied the case a nonzero, and proved existence and uniqueness for solutions of two Dirichlet problems derived from the nonlinear Cauchy-Riemann equation. This yields existence and uniqueness of a large class of nonsingular U(1)-invariant SL 3-folds in , with boundary conditions. The second paper extended these results to weak solutions of the Dirichlet problems when a=0, giving existence and uniqueness of many singular U(1)-invariant SL 3-folds in , with boundary conditions.This third paper studies the singularities of these SL 3-folds. We show that under mild conditions the singularities are isolated, and have a multiplicityn>0, and one of two types. Examples are constructed with every multiplicity and type. We also prove the existence of large families of U(1)-invariant special Lagrangian fibrations of open sets in , including singular fibres.     

The Klein–Gordon equation in the Anti-de Sitter cosmology

This paper deals with the Klein–Gordon equation on the Poincaré chart of the 5-dimensional Anti-de Sitter universe. When the mass μ is larger than , the Cauchy problem is well-posed despite the loss of global hyperbolicity due to the time-like horizon. We express the finite energy solutions in the form of a continuous Kaluza–Klein tower and we deduce a uniform decay as . We investigate the case , ν∈N^?, which encompasses the gravitational fluctuations, ν=4, and the electromagnetic waves, ν=2. The propagation of the wave front set shows that the horizon acts like a perfect mirror. We establish that the smooth solutions decay as , and we get global Lp estimates of Strichartz type. When ν is even, there appears a lacuna and the equipartition of the energy occurs at finite time for the compactly supported initial data, although the Huygens principle fails. We address the cosmological model of the negative-tension Minkowski brane, on which a Robin boundary condition is imposed. We prove the hyperbolic mixed problem is well-posed and the normalizable solutions can be expanded into a discrete Kaluza–Klein tower. We establish some L²−L^∞ estimates in suitable weighted Sobolev spaces.     

U(1)-invariant special Lagrangian 3-folds. II. Existence of singular solutions

This is the second of three papers studying special Lagrangian 3-submanifolds (SLV 3-folds) N in invariant under the U(1)-action e^iθ:(z₁,z₂,z₃)?(e^iθz₁,e^−iθz₂,z₃), using analytic methods. If N is such a 3-fold then |z₁|²−|z₂|²=2a on N for some . Locally, N can be written as a kind of graph of functions satisfying a nonlinear Cauchy-Riemann equation depending on a, so that u+iv is like a holomorphic function of x+iy.The first paper studied the case when a is nonzero. Then u,v are smooth and N is nonsingular. It proved existence and uniqueness for solutions of two Dirichlet problems derived from the equations on u,v. This implied existence and uniqueness for a large class of nonsingular U(1)-invariant SL 3-folds in , with boundary conditions.In this paper and its sequel we focus on the case a=0. Then the nonlinear Cauchy-Riemann equation is not always elliptic. Because of this there may be points (x,0) where u,v are not differentiable, corresponding to singular points of N. This paper is concerned largely with technical analytic issues, and the sequel with the geometry of the singularities of N. We prove a priori estimates for derivatives of solutions of the nonlinear Cauchy-Riemann equation, and use them to show existence and uniqueness of weak solutions u,v to the two Dirichlet problems when a=0, which are continuous and weakly differentiable. This gives existence and uniqueness for a large class of singular U(1)-invariant SL 3-folds in , with boundary conditions.     

The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations

In this paper, first, we introduce a new concept, called the norm-to-weak continuous semigroup in a Banach space, and give a technical theorem to verify this notion of continuity. Then we establish a general method which is necessary and sufficient to obtain the existence of the global attractor for this kind of semigroup. As an application, we obtain the existence of the global attractor for a nonlinear reaction-diffusion equation with a polynomial growth nonlinearity of arbitrary order and with some weak derivatives in the inhomogeneous term, the global attractors are obtained in L^p(Ω), and , respectively. A new a priori estimate method, called asymptotic a priori estimate, has been introduced. Since the solutions of the equation has no higher regularity and the semigroup associated the solutions is not continuous in L^p(Ω), and , the results in this part are new and appear to be optimal.     

Hardy-type theorem for orthogonal functions with respect to their zeros. The Jacobi weight case

Motivated by the G.H. Hardy's 1939 results [G.H. Hardy, Notes on special systems of orthogonal functions II: On functions orthogonal with respect to their own zeros, J. London Math. Soc. 14 (1939) 37-44] on functions orthogonal with respect to their real zeros λn, , we will consider, under the same general conditions imposed by Hardy, functions satisfying an orthogonality with respect to their zeros with Jacobi weights on the interval (0,1), that is, the functions f(z)=zνF(z), ν∈R, where F is entire and

     

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⁺.     

Exact boundary controllability of the nonlinear Schrödinger equation

This paper studies the exact boundary controllability of the semi-linear Schrödinger equation posed on a bounded domain Ω⊂Rn with either the Dirichlet boundary conditions or the Neumann boundary conditions. It is shown that if

     

Existence and regularity of the global attractor for a weakly damped forced shallow water equation in H

In this paper we prove the existence of the global attractor for a weakly damped forced shallow water equation in . Moreover, we show that the global attractor is actually compact in . The main method is the combination of the energy equation, a suitable splitting of the solutions, and bilinear estimates in Bourgain spaces.     

Inhomogeneous infinity Laplace equation

We present the theory of the viscosity solutions of the inhomogeneous infinity Laplace equation in domains in Rn. We show existence and uniqueness of a viscosity solution of the Dirichlet problem under the intrinsic condition f does not change its sign. We also discover a characteristic property, which we call the comparison with standard functions property, of the viscosity sub- and super-solutions of the equation with constant right-hand side. Applying these results and properties, we prove the stability of the inhomogeneous infinity Laplace equation with nonvanishing right-hand side, which states the uniform convergence of the viscosity solutions of the perturbed equations to that of the original inhomogeneous equation when both the right-hand side and boundary data are perturbed. In the end, we prove the stability of the well-known homogeneous infinity Laplace equation , which states the viscosity solutions of the perturbed equations converge uniformly to the unique viscosity solution of the homogeneous equation when its right-hand side and boundary data are perturbed simultaneously.