Persistency of Analyticity for Nonlinear Wave Equations: An Energy-like Approach

We study the persistence of the Gevrey class regularity of solutions to nonlinear wave equations with real analytic nonlinearity. Specifically, it is proven that the solution remains in a Gevrey class, with respect to some of its spatial variables, during its whole life-span, provided the initial data is from the same Gevrey class with respect to these spatial variables. In addition, for the special Gevrey class of analytic functions, we find a lower bound for the radius of the spatial analyticity of the solution that might shrink either algebraically or exponentially, in time, depending on the structure of the nonlinearity. The standard $L^2$ theory for the Gevrey class regularity is employed; we also employ energy-like methods for a generalized version of Gevrey classes based on the $\ell^1$ norm of Fourier transforms (Wiener algebra). After careful comparisons, we observe an indication that the $\ell^1$ approach provides a better lower bound for the radius of analyticity of the solutions than the $L^2$ approach. We present our results in the case of period boundary conditions, however, by employing exactly the same tools and proofs one can obtain similar results for the nonlinear wave equations and the nonlinear Schr\"odinger equation, with real analytic nonlinearity, in certain domains and manifolds without physical boundaries, such as the whole space $\mathbb{R}^n$, or on the sphere $\mathbb{S}^{n-1}$

On the backward behavior of some dissipative evolution equations

We prove that every solution of a KdV-Burgers-Sivashinsky type equation blows up in the energy space, backward in time, provided the solution does not belong to the global attractor. This is a phenomenon contrast to the backward behavior of the periodic 2D Navier-Stokes equations studied by Constantin-Foias-Kukavica-Majda [18], but analogous to the backward behavior of the Kuramoto-Sivashinsky equation discovered by Kukavica-Malcok [50]. Also we study the backward behavior of solutions to the damped driven nonlinear Schrodinger equation, the complex Ginzburg-Landau equation, and the hyperviscous Navier-Stokes equations. In addition, we provide some physical interpretation of various backward behaviors of several perturbations of the KdV equation by studying explicit cnoidal wave solutions. Furthermore, we discuss the connection between the backward behavior and the energy spectra of the solutions. The study of backward behavior of dissipative evolution equations is motivated by the investigation of the Bardos-Tartar conjecture stated in [5].Comment: 34 page

SYSTEMS OF NONLINEAR WAVE EQUATIONS WITH DAMPING AND SUPERCRITICAL SOURCES

We consider the local and global well-posedness of the coupled nonlinear wave equations utt – Δu + g1(ut) = f1(u, v) vtt – Δv + g2(vt) = f2(u, v); in a bounded domain Ω subset of the real numbers (Rn) with a nonlinear Robin boundary condition on u and a zero boundary conditions on v. The nonlinearities f1(u, v) and f2(u, v) are with supercritical exponents representing strong sources, while g1(ut) and g2(vt) act as damping. It is well-known that the presence of a nonlinear boundary source causes significant difficulties since the linear Neumann problem for the single wave equation is not, in general, well-posed in the finite-energy space H1(Ω) × L2(∂Ω) with boundary data from L2(∂Ω) (due to the failure of the uniform Lopatinskii condition). Additional challenges stem from the fact that the sources considered in this dissertation are non-dissipative and are not locally Lipschitz from H1(Ω) into L2(Ω) or L2(∂Ω). By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data. Under some restrictions on the parameters, we also prove that every weak solution to our system blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system. Additional results are obtained via careful analysis involving the Nehari Manifold. Specifically, we prove the existence of a unique global weak solution with initial data coming from the “good part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin. Moreover, we prove a blow up result for weak solutions with nonnegative initial energy. Finally, we establish important generalization of classical results by H. Brézis in 1972 on convex integrals on Sobolev spaces. These results allowed us to overcome a major technical difficulty that faced us in the proof of the local existence of weak solutions

Non-viscous Regularization of the Davey-Stewartson Equations: Analysis and Modulation Theory

In the present study we are interested in the Davey-Stewartson equations (DSE) that model packets of surface and capillary-gravity waves. We focus on the elliptic-elliptic case, for which it is known that DSE may develop a finite-time singularity. We propose three systems of non-viscous regularization to the DSE in variety of parameter regimes under which the finite blow-up of solutions to the DSE occurs. We establish the global well-posedness of the regularized systems for all initial data. The regularized systems, which are inspired by the $\alpha$-models of turbulence and therefore are called the $\alpha$-regularized DSE, are also viewed as unbounded, singularly perturbed DSE. Therefore, we also derive reduced systems of ordinary differential equations for the $\alpha$-regularized DSE by using the modulation theory to investigate the mechanism with which the proposed non-viscous regularization prevents the formation of the singularities in the regularized DSE. This is a follow-up of the work of Cao, Musslimani and Titi on the non-viscous $\alpha$-regularization of the nonlinear Schr\"odinger equation

Sparse distribution of lattice points in annular regions

This paper is inspired by Richards' work on large gaps between sums of two squares [10]. It is demonstrated in [10] that there exist arbitrarily large values of $\lambda$ and $\mu$, where $\mu \geq C \log \lambda$, such that intervals $[\lambda, \,\lambda + \mu ]$ do not contain any sums of two squares. Geometrically, these gaps between sums of two squares correspond to annuli in $\mathbb R^2$ that do not contain any integer lattice points. The primary objective of this paper is to investigate the sparse distribution of integer lattice points within annular regions in $\mathbb R^2$. Specifically, we establish the existence of annuli $\{x\in \mathbb R^2: \lambda \leq |x|^2 \leq \lambda + \kappa\}$ with arbitrarily large values of $\lambda$ and $\kappa$, where $\kappa \geq C \lambda^s$ with $0<s<\frac{1}{4}$, satisfying that any two integer lattice points within any one of these annuli must be sufficiently far apart. Furthermore, we extend our analysis to include the sparse distribution of lattice points in spherical shells in $\mathbb R^3$.Comment: 13 page

Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping

Presented here is a study of a viscoelastic wave equation with supercritical source and damping terms. We employ the theory of monotone operators and nonlinear semigroups, combined with energy methods to establish the existence of a unique local weak solution. In addition, it is shown that the solution depends continuously on the initial data and is global provided the damping dominates the source in an appropriate sense.Comment: The 2nd version includes a new proof of the energy identit