The semilinear second-order hyperbolic equation with data strongly singular at one point

In this paper we study the initial value problem for the scalar semilinear strictly hyperbolic equation in multidimensional space with data strongly singular at one point. Under the assumption of the initial data being conormal with respect to one point and bounded or regular with a certain low degree, the existence of the solution to this problem is obtained; meanwhile, it is proved that the singularity of the solution will spread on the forward characteristic cone of the hyperbolic operator issuing from this point, and the solution is bounded and conormal with respect to this cone.     

Convergence from an electromagnetic fluid system to the full compressible MHD equations

We are concerned with the zero dielectric constant limit for the full electro-magneto-fluid dynamics in this article. This singular limit is justified rigorously for global smooth solution for both well-prepared and ill-prepared initial data. The explicit convergence rate is also obtained by a elaborate energy estimate. Moreover, we show that for the well-prepared initial data, there is no initial layer, and the electric field always converges strongly to the limit function. While for the ill-prepared data case, there will be an initial layer near t=0. The strong convergence results only hold outside the initial layer.     

A nonlinear singular diffusion equation with source

In this paper, the existence, uniqueness and dependence on initial value of solution for a singular diffusion equation with nonlinear boundary condition are discussed. It is proved that there exists a unique global smooth solution which depends on initial data in L¹ continuously.     

Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians

We study propagation of the Gabor wave front set for a Schrödinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with nonnegative real part. We point out that the singular space associated with the quadratic form plays a crucial role for the understanding of this propagation. We show that the Gabor singularities of the solution to the equation for positive times are always contained in the singular space, and that they propagate in this set along the flow of the Hamilton vector field associated with the imaginary part of the quadratic form. As an application we obtain for the heat equation a sufficient condition on the Gabor wave front set of the initial datum tempered distribution that implies regularization to Schwartz regularity for positive times.     

Singular asymptotic expansions and delta waves for Burgers' equation

This paper is concerned with the characterization of the weak limits (delta waves) associated to the Cauchy problem for the Burgers' equation and the inviscid Burgers' equation with strongly singular initial data in the form of a regularization by smooth mollifiers of sums of derivatives of Dirac measures. By means of Laplace's method we give the precise asymptotic expansion of the solutionsu in powers of . Then we apply these asymptotics in order to classify completely all possible delta waves under a suitable nondegeneracy condition on some mollifiers regularizing the leading singular term of the initial data. We propose also certain stability results for the weak limits under suitable perturbations of the initial data.Partially supported by 60% of MURST, ItalyPartially supported by grant MM-410/94 with MES, Bulgaria and by 40% of MURST, Italy     

Convergence to equilibrium for the Cahn–Hilliard equation with a logarithmic free energy

In this paper we investigate the asymptotic behavior of the nonlinear Cahn–Hilliard equation with a logarithmic free energy and similar singular free energies. We prove an existence and uniqueness result with the help of monotone operator methods, which differs from the known proofs based on approximation by smooth potentials. Moreover, we apply the Lojasiewicz–Simon inequality to show that each solution converges to a steady state as time tends to infinity.     

ON EXISTENCE AND UNIQUENESS IN A FREE BOUNDARY PROBLEM FROM COMBUSTION

ABSTRACT

We study a free boundary problem for the heat equation describing the propagation of laminar flames under certain geometric assumptions on the initial data. The problem arises as the limit of a singular perturbation problem, and generally no uniqueness of limit solutions can be expected. However, if the initial data is starshaped, we show that the limit solution is unique and coincides with the minimal classical supersolution. Under certain convexity assumption on the data, we prove first that the limit solution is a classical solution of the free boundary problem for a short time interval, and then that the solution, in fact, stays classical as long as it does not vanish identically.     

Global well-posedness for the diffusion equation of population genetics

In this paper, we study the forward diffusion equation of population genetics. We prove the global existence of smooth solutions if the initial value is smooth. We also show that if the initial value is singular on the boundary, in a weighted Sobolev space, the diffusion equation exists a unique weak solution which is a probability density function. Moreover, we investigate the asymptotic behavior of the weak solution by the entropy method.     

Measure-valued solutions for a hierarchically size-structured population

We present a hierarchically size-structured population model with growth, mortality and reproduction rates which depend on a function of the population density (environment). We present an example to show that if the growth rate is not always a decreasing function of the environment (e.g., a growth which exhibits the Allee effect) the emergence of a singular solution which contains a Dirac delta mass component is possible, even if the vital rates of the individual and the initial data are smooth functions. Therefore, we study the existence of measure-valued solutions. Our approach is based on the vanishing viscosity method.     

Rates of convergence to zero for a semilinear parabolic equation with a critical exponent

We study nonnegative solutions to the Cauchy problem for a semilinear parabolic equation with a nonlinearity which is critical in the sense of Joseph and Lundgren. We establish the rate of convergence to zero of solutions that start from initial data which are near the singular steady state. In the critical case, this rate contains a logarithmic term which does not appear in the supercritical case and makes the calculations more delicate.     

奇异扩散方程的非线性边值问题

本文讨论具有非线性边界条件的奇异扩散方程混合边值问题整体光滑解的存在性,唯一性和关于初值的连续依赖性。     

具非线性边界条件的奇异扩散方程

本文讨论具非线性第二边界条件的一端无界的奇异扩散方程的初边值间题,利用先验估计方法得到的主要结果是:存在唯一的整体光滑解,且解连续依赖于初值．     

Cauchy problem with general discontinuous initial data along a smooth curve for 2-d Euler system

We study the Cauchy problems for the isentropic 2-d Euler system with discontinuous initial data along a smooth curve. All three singularities are present in the solution: shock wave, rarefaction wave and contact discontinuity. We show that the usual restrictive high order compatibility conditions for the initial data are automatically satisfied. The local existence of piecewise smooth solution containing all three waves is established.     

Global smooth solutions of 3-D null-form wave equations in exterior domains with Neumann boundary conditions

The paper is devoted to investigating long time behavior of smooth small data solutions to 3-D quasilinear wave equations outside of compact convex obstacles with Neumann boundary conditions. Concretely speaking, when the surface of a 3-D compact convex obstacle is smooth and the quasilinear wave equation fulfills the null condition, we prove that the smooth small data solution exists globally provided that the Neumann boundary condition on the exterior domain is given. One of the main ingredients in the current paper is the establishment of local energy decay estimates of the solution itself. As an application of the main result, the global stability to 3-D static compressible Chaplygin gases in exterior domain is shown under the initial irrotational perturbation with small amplitude.     

On maps with given Jacobians involving the heat equation

In this paper we present and analyze two new algorithms to construct a smooth diffeomorphism of a domain with prescribed jacobian function. The first one is free from any restriction on the boundary, while the second one produces a diffeomorphism that coincides with the identity map on the boundary of the domain. Both are based on the solution of an initial value problem for the linear heat equation, and the second also uses solutions of the Stokes system of Fluid Mechanics.     

The Riemann problem for the Leray-Burgers equation

For Riemann data consisting of a single decreasing jump, we find that the Leray regularization captures the correct shock solution of the inviscid Burgers equation. However, for Riemann data consisting of a single increasing jump, the Leray regularization captures an unphysical shock. This behavior can be remedied by considering the behavior of the Leray regularization with initial data consisting of an arbitrary mollification of the Riemann data. As we show, for this case, the Leray regularization captures the correct rarefaction solution of the inviscid Burgers equation. Additionally, we prove the existence and uniqueness of solutions of the Leray-regularized equation for a large class of discontinuous initial data. All of our results make extensive use of a reformulation of the Leray-regularized equation in the Lagrangian reference frame. The results indicate that the regularization works by bending the characteristics of the inviscid Burgers equation and thereby preventing their finite-time crossing.     

The initial value problem for cohomogeneity one Einstein metrics

The PDE Ric(g) = λ · g for a Riemannian Einstein metric g on a smooth manifold M becomes an ODE if we require g to be invariant under a Lie group G acting properly on M with principal orbits of codimension one. A singular orbit of the G-action gives a singularity of this ODE. Generically, an equation with such type of singularity has no smooth solution at the singularity. However, in our case, the very geometric nature of the equation makes it solvable. More precisely, we obtain a smooth G-invariant Einstein metric (with any Einstein constant λ) in a tubular neighbourhood around a singular orbit Q ⊂ M for any prescribed G-invariant metric g_Q and second fundamental form L_Q on Q, provided that the following technical condition is satisfied (which is very often the case): the representations of the principal isotropy group on the tangent and the normal space of the singular orbit Q have no common sub-representations. This Einstein metric is not uniquely determined by the initial data g_Q and L_Q; in fact, one may prescribe initial derivatives of higher degree, and examples show that this degree can be arbitrarily high. The proof involves a blend of ODE techniques and representation theory of the principal and singular isotropy groups.     

On the behavior of the solution of the dissipative Camassa–Holm equation with the arbitrary dispersion coefficient

In this paper, we consider the dissipative Camassa–Holm equation with arbitrary dispersion coefficient and compactly supported initial data. We demonstrate the simple conditions on the initial data that lead to finite time blow-up of the solution in finite time or guarantee that the solution exists globally. Also, propagation speed for the equation under consideration is investigated.     

Nonlinear Superposition of Delta Waves in Multi-dimensional Space

We study the behavior of the solution u^ε to the semilinear wave equation with initial data aξ + u_i(i = 1, 2) in multidimensional space, where u_i is a classical function and aξ is smooth and converges to a distribution a_i as ε → 0. In some circumstances one can prove the convergence of u^ε, and our results express a striking superposition principle. The singular part of the solution propagates linearly. The classical part shows the nonlinear effects. And, the limit of the nonlinear solution u^ε, delta wave, as the data become more singular is the sum of the two parts.     

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.