The global Goursat problem and scattering for nonlinear wave equations

AbstractThe Goursat problem for nonlinear scalar equations on the Einstein Universe M̃, with finite-energy datum, has a unique global solution in the positive-energy, Sobolev-controllable case. Such equations include those of the form □ϑ + H′(ϑ) = 0, where H denotes a hamiltonian that is a fourth-order polynomial, bounded below, in components of the multicomponent scalar section ϑ. In particular, the conformally invariant equation (□ + 1)ϑ + λϑ3 = 0 (λ ⩾ 0) is included. In the higher-dimensional analog R × Sn to the Einstein Universe the same result holds under the stronger conditions on H required for Sobolev controllability. Irrespective of energy positivity, there is a unique local-in-time solution for arbitrary finite-energy Goursat datum, for all n ⩾ 3, establishing evolution from the given lightcone to any sufficiently close lightcone. These results show the existence of wave operators in the sense of scattering theory, and their continuity in the (Einstein) energy metric, for positive-energy equations of the indicated type. They also permit the comprehensive reduction of scattering theory for conformally invariant wave equations in Minkowski space M0 to the Goursat problem in M̃. In particular, any solution of the equation arising from a nonnegative conformally invariant biquadratic interaction Lagrangian on multicomponent scalar sections, having finite Einstein energy at any one time, is asymptotic to solutions of the corresponding multicomponent free wave equation as the Minkowski time x0 → ± ∞. Thus given a finite-Einstein-energy solution of the equation □ƒ + λƒ3 = 0 on M0 (λ ⩾ 0) there exist unique solutions ƒ± of the free wave equation which approach ƒ in the Minkowski energy norm as x0 → ± ∞, and every finite-Einstein-energy solution of the free wave equation is of the form ƒ+ (or ƒ−) for a unique solution ƒ of the nonlinear equation. This generalizes, in part in maximality sharp form, earlier results of Strauss for this equation

Functional Inequalities: New Perspectives and New Applications

This book is not meant to be another compendium of select inequalities, nor does it claim to contain the latest or the slickest ways of proving them. This project is rather an attempt at describing how most functional inequalities are not merely the byproduct of ingenious guess work by a few wizards among us, but are often manifestations of certain natural mathematical structures and physical phenomena. Our main goal here is to show how this point of view leads to "systematic" approaches for not just proving the most basic functional inequalities, but also for understanding and improving them, and for devising new ones - sometimes at will, and often on demand.Comment: 17 pages; contact Nassif Ghoussoub (nassif @ math.ubc.ca) for a pre-publication pdf cop

Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach

Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1, \quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For the semilinear heat equation $u_t= \Delta u+ u^p$, various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure

Strichartz Estimates for the Vibrating Plate Equation

We study the dispersive properties of the linear vibrating plate (LVP) equation. Splitting it into two Schr\"odinger-type equations we show its close relation with the Schr\"odinger equation. Then, the homogeneous Sobolev spaces appear to be the natural setting to show Strichartz-type estimates for the LVP equation. By showing a Kato-Ponce inequality for homogeneous Sobolev spaces we prove the well-posedness of the Cauchy problem for the LVP equation with time-dependent potentials. Finally, we exhibit the sharpness of our results. This is achieved by finding a suitable solution for the stationary homogeneous vibrating plate equation.Comment: 18 pages, 4 figures, some misprints correcte

On the Schr\"odinger equations with isotropic and anisotropic fourth-order dispersion

This paper deals with the Cauchy problem associated to the nonlinear fourth-order Schr\"odinger equation with isotropic and anisotropic mixed dispersion. This model is given by the equation $i\partial _{t}u+\epsilon \Delta u+\delta A u+\lambda|u|^\alpha u=0,$ $x\in \mathbb{R}^{n},$ $t\in \mathbb{R},$ where $A$ represents either the operator $\Delta^2$ (isotropic dispersion) or $\sum_{i=1}^d\partial_{x_ix_ix_ix_i},\ 1\leq d<n$ (anisotropic dispersion), and $\alpha, \epsilon, \lambda$ are given real parameters. We obtain local and global well-posedness results in spaces of initial data with low regularity, such as weak-$L^p$ spaces. Our analysis also includes the biharmonic and anisotropic biharmonic equation $(\epsilon=0)$ for which, the existence of self-similar solutions is obtained as consequence of his scaling invariance. In a second part, we investigate the vanishing second order dispersion limit in the framework of weak-$L^p$ spaces. We also analyze the convergence of the solutions for the nonlinear fourth-order Schr\"odinger equation $i\partial _{t}u+\epsilon \Delta u+\delta \Delta^2 u+\lambda|u|^\alpha u=0$, as $\epsilon$ goes to zero, in $H^2$-norm, to the solutions of the corresponding biharmonic equation $i\partial _{t}u+\delta \Delta^2 u+\lambda|u|^\alpha u=0$