A class of quasilinear degenerate elliptic problems

We establish the existence of solutions for a class of quasilinear degenerate elliptic equations. The equations in this class satisfy a structure condition which provides ellipticity in the interior of the domain, and degeneracy only on the boundary. Equations of transonic gas dynamics, for example, satisfy this property in the region of subsonic flow and are degenerate across the sonic surface. We prove that the solution is smooth in the interior of the domain but may exhibit singular behavior at the degenerate boundary. The maximal rate of blow-up at the degenerate boundary is bounded by the “degree of degeneracy” in the principal coefficients of the quasilinear elliptic operator. Our methods and results apply to the problems recently studied by several authors which include the unsteady transonic small disturbance equation, the pressure-gradient equations of the compressible Euler equations, and the singular quasilinear anisotropic elliptic problems, and extend to the class of equations which satisfy the structure condition, such as the shallow water equation, compressible isentropic two-dimensional Euler equations, and general two-dimensional nonlinear wave equations. Our study provides a general framework to analyze degenerate elliptic problems arising in the self-similar reduction of a broad class of two-dimensional Cauchy problems.     

Well-posedness of the IBVP for 2-D Euler equations with damping

In this paper we focus on the initial-boundary value problem of the 2-D isentropic Euler equations with damping. We prove the global-in-time existence of classical solution to the initial-boundary value problem for small smooth initial data by the method of local existence of solution combined with a priori energy estimates, where the appropriate boundary condition plays an important role.     

Almost global existence for nonlinear wave equations in an exterior domain in two space dimensions

In this paper we deal with the exterior problem for a system of nonlinear wave equations in two space dimensions, assuming that the initial data is small and smooth. We establish the same type of lower bound of the lifespan for the problem as that for the Cauchy problem, despite of the weak decay property of the solution in two space dimensions.     

Partially overdetermined elliptic boundary value problems

We consider semilinear elliptic Dirichlet problems in bounded domains, overdetermined with a Neumann condition on a proper part of the boundary. Under different kinds of assumptions, we show that these problems admit a solution only if the domain is a ball. When these assumptions are not fulfilled, we discuss possible counterexamples to symmetry. We also consider Neumann problems overdetermined with a Dirichlet condition on a proper part of the boundary, and the case of partially overdetermined problems on exterior domains.     

On a quasilinear parabolic–elliptic chemotaxis system with logistic source

This paper deals with a quasilinear parabolic–elliptic chemotaxis system with logistic source, under homogeneous Neumann boundary conditions in a smooth bounded domain. For the case of positive diffusion function, it is shown that the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded. Moreover, if the diffusion function is zero at some point, or a positive diffusion function and the logistic damping effect is rather mild, we proved that the weak solutions are global existence. Finally, it is asserted that the solutions approach constant equilibria in the large time for a specific case of the logistic source.     

General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source

In this paper we consider a quasilinear viscoelastic wave equation in canonical form with the homogeneous Dirichlet boundary condition. We prove that, for certain class of relaxation functions and certain initial data in the stable set, the decay rate of the solution energy is similar to that of the relaxation function. This result improves earlier ones obtained by Messaoudi and Tatar [S.A. Messaoudi, N.-E. Tatar, Global existence and uniform stability of solutions for a quasilinear viscoelastic problem, Math. Methods Appl. Sci. 30 (2007) 665-680] in which only the exponential and polynomial decay rates are considered. Conversely, for certain initial data in the unstable set, there are solutions that blow up in finite time. The last result is new, since it allows a larger class of initial energy which may take positive values.     

On the existence of solutions for quasilinear elliptic problems with radial potentials on exterior ball

In this paper, we are concerned with a class of quasilinear elliptic problems with radial potentials and a mixed nonlinear boundary condition on exterior ball domain. Based on a compact embedding from a weighted Sobolev space to a weighted L^s space, the existence of nontrivial solutions is obtained via variational methods.     

Analytic solution of an exterior Neumann problem in a non‐convex domain

A new method, based on the Kelvin transformation and the Fokas integral method, is employed for solving analytically a potential problem in a non‐convex unbounded domain of ?², assuming the Neumann boundary condition. Taking advantage of the property of the Kelvin transformation to preserve harmonicity, we apply it to the present problem. In this way, the exterior potential problem is transformed to an equivalent one in the interior domain which is the Kelvin image of the original exterior one. An integral representation of the solution of the interior problem is obtained by employing the Kelvin inversion in ?² for the Neumann data and the ‘Neumann to Dirichlet’ map for the Dirichlet data. Applying next the ‘reverse’ Kelvin transformation, we finally obtain an integral representation of the solution of the original exterior Neumann problem. Copyright © 2010 John Wiley & Sons, Ltd.     

半线性波动方程的高维古沙问题

In this paper we study the Goursat problem for semilinear wave equations with zero boundary condition in which the boundary is the characteristic cone for wave operator. Our result states that the solution is Lipschitz and is smooth awayfrom the characteristic cone.     

Blow-up of periodic solutions to reducible quasilinear hyperbolic systems

Under general assumption, we prove that the smooth solution to reducible quasilinear hyperbolic system with periodic initial data in general develop singularities in finite time, provided that the total variation on one period of the initial data is small.     

On the decay of solutions to the 2D Neumann exterior problem for the wave equation

We consider the exterior problem in the plane for the wave equation with a Neumann boundary condition and study the asymptotic behavior of the solution for large times. For possible application we are interested to show a decay estimate which does not involve weighted norms of the initial data. In the paper we prove such an estimate, by a combination of the estimate of the local energy decay and decay estimates for the free space solution.     

Neumann and Robin problems in a cracked domain with jump conditions on cracks

The boundary value problem for the Laplace equation is studied on a domain with smooth compact boundary and with smooth internal cracks. The Neumann or the Robin condition is given on the boundary of the domain. The jump of the function and the jump of its normal derivative is prescribed on the cracks. The solution is looked for in the form of the sum of a single layer potential and a double layer potential. The solvability of the corresponding integral equation is determined and the explicit solution of this equation is given in the form of the Neumann series. Estimates for the absolute value of the solution of the boundary value problem and for the absolute value of the gradient of the solution are presented.     

Bloch wave homogenization of a non-homogeneous Neumann problem

In this paper, we use the Bloch wave method to study the asymptotic behavior of the solution of the Laplace equation in a periodically perforated domain, under a non-homogeneous Neumann condition on the boundary of the holes, as the size of the holes goes to zero more rapidly than the domain period. This method allows to prove that, when the hole size exceeds a given threshold, the non-homogeneous boundary condition generates an additional term in the homogenized problem, commonly referred to as “the strange term” in the literature.     

Global smooth solutions for the quasilinear wave equation with boundary dissipation

We consider the existence of global solutions of the quasilinear wave equation with a boundary dissipation structure of an input-output in high dimensions when initial data and boundary inputs are near a given equilibrium of the system. Our main tool is the geometrical analysis. The main interest is to study the effect of the boundary dissipation structure on solutions of the quasilinear system. We show that the existence of global solutions depends not only on this dissipation structure but also on a Riemannian metric, given by the coefficients and the equilibrium of the system. Some geometrical conditions on this Riemannian metric are presented to guarantee the existence of global solutions. In particular, we prove that the norm of the state of the system decays exponentially if the input stops after a finite time, which implies the exponential stabilization of the system by boundary feedback.     

Formation of singularities for wave equations including the nonlinear vibrating string

Under very general assumptions, the authors prove that smooth solutions of quasilinear wave equations with small-amplitude periodic initial data always develop singularities in the second derivatives in finite time. One consequence of these results is the fact that all solutions of the classical nonlinear vibrating string equation satisfying either Dirichlet or Neumann boundary conditions and with sufficiently small nontriviai initial data necessarily develop singularities. In particular, there are no nontrivial smooth small-amplitude time-periodic solutions.     

The exterior Robin problem for the Helmholtz equation

A new method is given for solving the exterior Robin problem for the Helmholtz equation. The problem is reformulated as a new integral equation which is continuous as the field point approaches the boundary. It is shown that its solution can be represented as a convergent Neumann series for convex surfaces, for small values of the wave number. Examples are included which illustrate the method.     

Global solvability for the Kirchhoff equations in exterior domains of dimension three

We consider the initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole space of dimension three, and show that these problems admit time-global solutions, provided the norms of the initial data in the usual Sobolev spaces of appropriate order are sufficiently small. We obtain uniform estimates of the L¹(R) norms with respect to time variable at each point in the domain, of solutions of initial (boundary) value problem for the linear wave equations. We then show that the estimates above yield the unique global solvability for the Kirchhoff equations.     

Boundedness in a quasilinear parabolic-parabolic chemotaxis system with nonlinear logistic source

We study a quasilinear parabolic-parabolic chemotaxis system with nonlinear logistic source, under homogeneous Neumann boundary conditions in a smooth bounded domain. By establishing proper a priori estimates we prove that, with both the diffusion function and the chemotaxis sensitivity function being positive, the corresponding initial boundary value problem admits a unique global classical solution which is uniformly bounded. The result of this paper is a generalization of that of Cao (2014).     

Symmetry results for overdetermined problems on convex domains via Brunn–Minkowski inequalities

We consider some well-posed Dirichlet problems for elliptic equations set on the interior or the exterior of a convex domain (examples include the torsional rigidity, the first Dirichlet eigenvalue, and the electrostatic capacity), and we add an overdetermined Neumann condition which involves the Gauss curvature of the boundary. By using concavity inequalities of Brunn–Minkowski type satisfied by the corresponding variational energies, we prove that the existence of a solution implies the symmetry of the domain. This provides some new characterizations of spheres, in models going from solid mechanics to electrostatics.     

An energy-preserving nonlinear system modeling a string with input and output on the boundary

We study an energy conserving distributed parameter system described by a nonlinear string equation with the input and output at the boundary. We prove the existence of global smooth solutions to this quasilinear hyperbolic system if the initial data and the boundary input are small. If, moreover, the input function becomes zero after some finite time, then the state trajectories decay exponentially.