On the global existence of solution to one-dimensional fourth order nonlinear Sobolev type equations

We prove the global existence and uniqueness of a classical solution to initial boundary value problem for a class of Sobolev type equations under the Dirichlet boundary conditions. This class of evolution equations covers the well-known viscous Cahn-Hilliard equation and the viscous Camassa-Holm equation.     

Strichartz Estimates for the Schrödinger Equation on Polygonal Domains

We prove Strichartz estimates with a loss of derivatives for the Schrödinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates on the polygon follow from those on Euclidean surfaces with conical singularities. We develop a Littlewood-Paley squarefunction estimate with respect to the spectrum of the Laplacian on these spaces. This allows us to reduce matters to proving estimates at each frequency scale. The problem can be localized in space provided the time intervals are sufficiently small. Strichartz estimates then follow from a recent result of the second author regarding the Schrödinger equation on the Euclidean cone.     

Transmission of progressing waves across a moving interface

We describe global time existence nd uniqueness results for the wave equations with boundary conditions of Dirichlet type on a characteresitc cone and either Dirichlet or Neumann type on a timelike tube. We find that the solution is in general only half as regular as the data and we provide estimates which describe the differing differentiabilities of the solution in directions which are either tangent or transvers to the characteristic cone.     

Boundary Observability and Stabilization for Westervelt Type Wave Equations without Interior Damping

In this paper we show boundary observability and boundary stabilizability by linear feedbacks for a class of nonlinear wave equations including the undamped Westervelt model used in nonlinear acoustics. We prove local existence for undamped generalized Westervelt equations with homogeneous Dirichlet boundary conditions as well as global existence and exponential decay with absorbing type boundary conditions.     

Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations

We prove some new Strichartz estimates for a class of dispersive equations with radial initial data. In particular, we obtain the full radial Strichartz estimates up to some endpoints for the Schrödinger equation. Using these estimates, we obtain some new results related to nonlinear problems, including small data scattering and large data LWP for the nonlinear Schrödinger and wave equations with radial critical initial data and the well-posedness theory for the fractional order Schrödinger equation in the radial case.     

Initial boundary value problems for nonlinear dispersive wave equations

In this paper we study initial value boundary problems of two types of nonlinear dispersive wave equations on the half-line and on a finite interval subject to homogeneous Dirichlet boundary conditions. We first prove local well-posedness of the rod equation and of the b-equation for general initial data. Furthermore, we are able to specify conditions on the initial data which on the one hand guarantee global existence and on the other hand produce solutions with a finite life span. In the case of finite time singularities we are able to describe the precise blow-up scenario of breaking waves. Our approach is based on sharp extension results for functions on the half-line or on a finite interval and several symmetry preserving properties of the equations under discussion.     

Strichartz type estimates and the well-posedness of an energy critical 2D wave equation in a bounded domain

We study the well-posedness of the Cauchy problem with Dirichlet or Neumann boundary conditions associated to an H¹-critical semilinear wave equation on a smooth bounded domain Ω⊂R². First, we prove an appropriate Strichartz type estimate using the Lq spectral projector estimates of the Laplace operator. Our proof follows Burq, Lebeau and Planchon (2008) [4]. Then, we show the global well-posedness when the energy is below or at the threshold given by the sharp Moser-Trudinger inequality. Finally, in the supercritical case, we prove an instability result using the finite speed of propagation and a quantitative study of the associated ODE with oscillatory data.     

Attractor and dimension for strongly damped nonlinear wave equation

1. IntroductionConsider the strongly damped nonlinear wave equationwith the Dirichlet boundary conditionand the initial value conditionswhere u = u(x, t) is a real--valued function on fi x [0, co), fi is an open bounded set of R"with smooth boundary off, or > 0, g e L'(fl), D(--Q) ~ Ha(~~) n H'(fl).We assume for the function f(u) as follows'f(u) E CI (R, R) satisfiesfor any ig al, uZ E R, where k, hi > 0, i ~ 0, 1, 2, 61 > 0 and 0 5 6o < 1'The type of equation (1) can be regarded as the…     

Global Strichartz estimates for solutions to the wave equation exterior to a convex obstacle

In this paper, we show that certain local Strichartz estimates for solutions of the wave equation exterior to a convex obstacle can be extended to estimates that are global in both space and time. This extends the work that was done previously by H. Smith and C. Sogge in odd spatial dimensions. In order to prove the global estimates, we explore weighted Strichartz estimates for solutions of the wave equation when the Cauchy data and forcing term are compactly supported.

     

Kernel sections for non-autonomous strongly damped wave equations

We prove the existence of compact kernel sections for the process generated by a non-autonomous strongly damped wave equation with homogeneous Dirichlet boundary condition. We show that the upper bound of the Hausdorff dimension of sections decreases as the damping grows for large strong damping.     

Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential

We prove spacetime weighted-L² estimates for the Schrödinger and wave equation with an inverse-square potential. We then deduce Strichartz estimates for these equations.     

Periodic solutions of a quasilinear wave equation

We study the properties of wave operators satisfying the periodicity condition with respect to time and homogeneous boundary conditions of the third kind and of Dirichlet type. We prove the existence of a nontrivial periodic (in time) sine-Gordon solution with homogeneous boundary conditions of the third kind and of Dirichlet type. We obtain theorems on the existence of periodic solutions of a quasilinear wave equation with variable (in x) coefficients and a boundary condition of the third kind.     

Periodic problem for the nonlinear damped wave equation with convective nonlinearity

We study the nonlinear damped wave equation with a linear pumping and a convective nonlinearity. We consider the solutions, which satisfy the periodic boundary conditions. Our aim is to prove global existence of solutions to the periodic problem for the nonlinear damped wave equation by applying the energy-type estimates and estimates for the Green operator. Moreover, we study the asymptotic profile of global solutions.     

Strichartz and Smoothing Estimates for Dispersive Equations with Magnetic Potentials

We prove global smoothing and Strichartz estimates for the Schrödinger, wave, Klein–Gordon equations and for the massless and massive Dirac systems, perturbed with singular electromagnetic potentials. We impose a smallness condition on the magnetic part, while the electric part can be large. The decay and regularity assumptions on the coefficients are close to critical.     

Dispersion and Strichartz estimates for the Liouville equation

We consider the Liouville equation associated with a metric g of class C² and we prove dispersion and Strichartz estimates for the solution of this equation in terms of geodesics associated with g. We introduce the notion of focusing and dispersive metric to characterize metrics such that the same dispersion estimate as in the Euclidean case holds. To deal with the case of non-trapped long range perturbation of the Euclidean metric, we prove a global velocity moments effect on the solution. In particular, we obtain global in time Strichartz estimates for metrics such that the dispersion estimate is not satisfied.     

Attractors for Damped Quintic Wave Equations in Bounded Domains

The dissipative wave equation with a critical quintic non-linearity in smooth bounded three-dimensional domain is considered. Based on the recent extension of the Strichartz estimates to the case of bounded domains, the existence of a compact global attractor for the solution semigroup of this equation is established. Moreover, the smoothness of the obtained attractor is also shown.     

The Radial Defocusing Nonlinear Schrödinger Equation in Three Space Dimensions

We study the defocusing nonlinear Schrödinger equation in three space dimensions. We prove that any radial solution that remains bounded in the critical Sobolev space must be global and scatter. In the energy-supercritical setting we employ a space-localized Lin–Strauss Morawetz inequality of Bourgain. In the intercritical regime we prove long-time Strichartz estimates and frequency-localized Lin–Strauss Morawetz inequalities.     

Scattering Problem for Klein–Gordon Equation with Cubic Convolution Nonlinearity

The scattering problem for the Klein–Gordon equation with cubic convolution nonlinearity is considered. Based on the Strichartz estimates for the inhomogeneous Klein–Gordon equation, we prove the existence of the scattering operator, which improves the known results in some sense.     

Strichartz estimates for the Dunkl wave equation and application

In this paper, we prove dispersive and Strichartz estimates associated for the Dunkl wave equation.     

Strichartz estimates and nonlinear wave equation on nontrapping asymptotically conic manifolds

We prove the global-in-time Strichartz estimates for wave equations on the nontrapping asymptotically conic manifolds. We obtain estimates for the full set of wave admissible indices, including the endpoint. The key points are the properties of the microlocalized spectral measure of Laplacian on this setting showed in [18] and a Littlewood–Paley squarefunction estimate. As applications, we prove the global existence and scattering for a family of nonlinear wave equations on this setting.