On well-posedness for some dispersive perturbations of Burgers' equation

We show that the Cauchy problem for a class of dispersive perturbations of Burgers' equations containing the low dispersion Benjamin–Ono equation

${?}_{t}u?D_x^{\alpha }{?}_{x}u={?}_x(u^2),0<\alpha \le 1,$

is locally well-posed in $H^s(\mathbb{R})$ when $s>s_{\alpha }:=\frac{3}{2}?\frac{5\alpha }{4}$. As a consequence, we obtain global well-posedness in the energy space $H^{\frac{\alpha }{2}}(\mathbb{R})$ as soon as $\frac{\alpha }{2}>s_{\alpha }$, i.e. $\alpha >\frac{6}{7}$.     

Global attractor for the damped Benjamin–Bona–Mahony equations on R1

We utilize a new necessary and sufficient condition to verity the asymptotic compactness of an evolution equation defined in an unbounded domain, which involves the Littlewood–Paley projection operators. We then use this condition to prove the existence of an attractor for the damped Benjamin–Bona–Mahony equation in the phase space H ¹(R ¹) by showing the solutions are point dissipative and asymptotically compact. Moreover the attractor is in fact smoother and it belongs to H ^3/2?? for every ?>0.     

Global Well-Posedness for a Nonlocal Gross–Pitaevskii Equation with Non-Zero Condition at Infinity

We study the Gross–Pitaevskii equation involving a nonlocal interaction potential. Our aim is to give sufficient conditions that cover a variety of nonlocal interactions such that the associated Cauchy problem is globally well-posed with non-zero boundary condition at infinity, in any dimension. We focus on even potentials that are positive definite or positive tempered distributions.     

Global solutions for the generalized Camassa–Holm equation

In this paper, we study the Cauchy problem of the generalized Camassa–Holm equation. Firstly, we prove the existence of the global strong solutions provide the initial data satisfying a certain sign condition. Then, we obtain the existence and the uniqueness of the global weak solutions under the same sign condition of the initial data.     

Global Existence for the n-Dimensional Semilinear Tricomi-Type Equations

In this article we investigate the issue of global existence of the solutions of the Cauchy problem for semilinear Tricomi-type equations in ? ⁿ⁺¹, n > 1. We give some sufficient conditions for existence of the global weak solutions. These conditions tie together nonlinearity with the speed of propagation and with the dimension n. We also prove necessity of these (or close) conditions. In fact, we extend these necessity results to the nonlocal semilinear equations.     

Global spherically symmetric flows for a viscous radiative and reactive gas in an exterior domain

We consider the equations for a viscous, compressible, radiative and reactive gas (pressure $P=R\rho \theta +a{\theta }^4/3$, internal energy $e=c_v\theta +a{\theta }^4/\rho$) over an unbounded exterior domain in ${\mathbb{R}}^n$, where $n\ge 2$ is the space dimension. The existence, uniqueness, and large-time behavior of global spherically symmetric solutions are established for large initial data. The key point in the analysis is to deduce certain uniform a priori estimates on the solutions, especially on lower and upper bounds of the specific volume and temperature.     

The Cauchy problem for a generalized Camassa–Holm equation with the velocity potential

In this paper, we discuss a generalized Camassa–Holm equation whose solutions are velocity potentials of the classical Camassa–Holm equation. By exploiting the connection between these two equations, we first establish the local well-posedness of the new equation in the Besov spaces and deduce several blow-up criteria and blow-up results. Then, we investigate the existence of global strong solutions and present a class of cuspon weak solutions for the new equation.     

Nonlinear Anisotropic Elliptic and Parabolic Equations in R<Superscript>N</Superscript> with Advection and Lower Order Terms and Locally Integrable Data

We prove existence and regularity results for distributional solutions in R^N for nonlinear elliptic and parabolic equations with general anisotropic diffusivities as well as advection and lower-order terms that satisfy appropriate growth conditions. The data are assumed to be merely locally integrable. Mathematics Subject Classifications (2000) 35J60, 35K55.This work was supported by the BeMatA program of the Research Council of Norway and the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282. This work was done while M. Bendahmane visited the Centre of Mathematics for Applications (CMA) at the University of Oslo, Norway.     

Meshless formulation to two-dimensional nonlinear problem of generalized Benjamin–Bona–Mahony–Burgers through singular boundary method: Analysis of stability and convergence

In this study, the singular boundary method (SBM) is employed for the simulation of nonlinear generalized Benjamin–Bona–Mahony–Burgers problem with initial and Dirichlet-type boundary conditions. The θ-weighted finite difference method is used to discretize the time derivatives. Then the original equations are split into a system of partial differential equations. A splitting scheme is applied to split the solution of the inhomogeneous governing equation into homogeneous solution and particular solution. To solve this system, the method of particular solution (MPS) in combination with the SBM is used where the SBM is used for homogeneous solution and MPS is used for particular solution. Furthermore, the stability and convergence of the proposed method is conducted. Finally, several numerical examples with different domains are provided and compared with the exact analytical solutions to show the accuracy and efficiency in comparison with existing other methods.     

On a generalized Camassa–Holm equation with the flow generated by velocity and its gradient

In this paper, we consider a generalized Camassa–Holm equation with the flow generated by the vector field and its gradient. We first establish the local well-posedness of equation in the sense of Hadamard in both critical Besov spaces and supercritical Besov spaces. Then we gain a blow-up criterion. Under a sign condition we reach the sign-preserved property and a precise blow-up criterion. Applying this precise criterion we finally present two blow-up results and the precise blow-up rate for strong solutions to equation.