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.     

Interaction of Three Conormal Waves for Semilinear Wave Equations

In this paper, we consider the interaction of triple of conormal waves with different singularities for semilinear wave equations. We will show that if three characteristic hyperplanes carrying different conormal singularities intersect transversally at the origin, then the solution will be conormal with respect to the three hyperplanes, and a new singularity will be produced on the surface of the light cone at later times. We can also prove here that the strength of the new singularity will be dependent only on the weakest one and strongest one in the three hyperplanes.     

On the singularities of solutions to 4-D semilinear dispersive wave equations

In this note, we are concerned with the global singularity structures of weak solutions to 4 - D semilinear dispersive wave equations whose initial data are chosen to be singular at a single point, Combining Strichartz＇s inequality with the commutator argument techniques, we show that the weak solutions stay globally conormal if the Cauchy data are conormal     

Evolution d'une singularite ponctuelle dans un fluide compressible

In this work, we study the regularity of the solution of compressible Euler system when the Cauchy data is conormal to this origin. We prove that, outside the origin, the solution is conormal to the disjoint union of a smooth curve and a smooth hypersurface.     

Global singularity structure of weak solutions to 3-D semilinear dispersive wave equations with discontinuous initial data

We study the global singularity structure of solutions to 3-D semilinear wave equations with discontinuous initial data. More precisely, using Strichartz’ inequality we show that the solutions stay conormal after nonlinear interaction if the Cauchy data are conormal along a circle.     

A spectral solution method for a boundary-value problem for a mixed-type equation with two singularity lines

We consider a boundary-value problem for a mixed-type equation with two perpendicular singularity lines given in a domain whose elliptic part is a rectangle, while the hyperbolic one is a vertical half-strip. This problem differs from the Dirichlet one by the fact that at the left boundary of the rectangle and the half-strip we specify the vanishing order of the desired function rather than its value. We find a solution to the problem by a spectral method with the use of the Fourier–Bessel series and prove the uniqueness of the solution. We substantiate the uniform convergence of the corresponding series under certain requirements to the problem statement.     

Singularity in a mathematical model of emulsion liquid membrane

An earlier paper of ours presented a mathematical model to study metal recovery from wastewater with emulsion liquid membrane and the analytical solution resulting from the model. In this paper, we point out that in a certain parameter range the eigenvalue equation has a singularity that gives rise to an additional term in the analytical solution, whose impact is strongest in the initial phase of metal recovery. This paper examines the origin and consequence of the additional term associated with the singularity.     

A nonlocal boundary-value problem with conormal derivative for a mixed-type equation with two inner degeneration lines and various orders of degeneracy

We prove the unique solvability of the boundary value problem with conormal derivative for a mixed-type equation with two inner degeneration lines and with various orders of degeneracy.     

Singularities of Solutions to Cauchy Problems for Semilinear Wave Equations in Two Space Dimensions

This paper concerns the Cauchy problem for semilinear wave equations with two space variables, of which the initial data have conormal singularities on finite curves intersecting at one point on the initial plane. It is proved that the solution is of conormal distribution type, and its singularities are contained in the union of the characteristic surfaces through these curves and the characteristic cone issuing from the intersection point.     

Asymptotics of Small Exterior Navier-Stokes Flows with Non-Decaying Boundary Data

We prove the existence of unique solutions for the 3D incompressible Navier-Stokes equations in an exterior domain with small boundary data which do not necessarily decay in time. As a corollary, the existence of unique small time-periodic solutions is shown. We next show that the spatial asymptotics of the periodic solution is given by the same Landau solution at all times. Lastly we show that if the boundary datum is time-periodic and the initial datum is asymptotically self-similar, then the solution converges to the sum of a time-periodic vector field and a forward self-similar vector field as time goes to infinity.     

Blowup,Global Fast and Slow Solutions for a Semilinear Combustible System

In this paper, we investigate a semilinear combustible system $u_t-du_{xx}=v^p, v_t-dv_{xx}=u^q$ with double fronts free boundary, where p ≥ 1, q ≥ 1. For such a problem, we use the contraction mapping theorem to prove the local existence and uniqueness of the solution. Also we study the blowup and global existence property of the solution. Our results show that when pq › 1 blowup occurs if the initial datum is large enough and the solution is global and slow, whose decay rate is at most polynomial if the initial value is suitably large, while when p › 1, q › 1 there is a global and fast solution, which decays uniformly at an exponential rate if the initial datum is small.     

On mean curvature flow of singular Riemannian foliations: Noncompact cases

In this paper we investigate the mean curvature flow (MCF) of a regular leaf of a closed generalized isoparametric foliation as initial datum, generalizing previous results of Radeschi and the first author. We show that, under bounded curvature conditions, any finite time singularity is a singular leaf, and the singularity is of type I. The new techniques also allow us to discuss the existence of basins of attraction, how cylinder structures can affect convergence of basic MCF of immersed submanifolds and assure convergence of MCF of non-closed leaves of generalized isoparametric foliation on compact manifold.     

A Free Boundary Problem of a Semilinear Combustible System with Higher Dimension and Heterogeneous Environment

In this paper, we investigate a free boundary problem of a semilinear combustible system with higher dimension and heterogeneous environment. Such a problem is usually used as a model to describe heat propagation in a two-component combustible mixture in which the free boundary is described by Stefan-like condition. For simplicity, we assume that the environment and solutions are radially symmetric. We use the contraction mapping theorem to prove the local existence and uniqueness of the solution. Also we study the blowup property and the long time behavior of the solution. Our results show that when pq › 1 blowup occurs if the initial datum is large enough and the solution is global and slow, whose decay rate is at most polynomial if the initial value is suitably large, while when p › 1, q › 1 there is a global and fast solution, which decays uniformly at an exponential rate if the initial datum is small.     

Analysis of Velázquez’s solution to the mean curvature flow with a type II singularity

Velázquez in 1994 used the degree theory to show that there is a perturbation of Simons’ cone, starting from which the mean curvature flow develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution around the origin, the rescaled flow converges in the C⁰ sense to a minimal hypersurface which is tangent to Simons’ cone at infinity. In this paper, we prove that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form.     

非等熵Chaplygin气体极限黎曼解 关于扰动的依赖性

主要研究了非等熵Chaplygin气体黎曼问题初值扰动后解的结构,分析了经典的黎曼问题和扰动问题解的结构及极限结构,发现后者的极限解在δ质量权趋于零时不同于前者解的结构.该结果表明对非等熵Chaplygin气体而言,经典的黎曼问题与带δ初值的黎曼问题有着本质的区别.     

带尖角的障碍声波散射区域的反演

对带尖角的障碍声波散射区域进行了反演,其前提条件是整体场满足奇次Dirichlet边界条件．在用Nystrom方法解正问题的过程中,由于采用等距网格积分给尖角处带来很差的收敛性,这是因为双层位势的积分算子的核在尖角处有Mellin型奇性,不再是紧算子;为此采用梯度网格,数值例子表明该处理方法的有效可靠性．     

Dirichlet Problem for a Mixed Type Equation of the Second Kind in Exceptional Cases

We prove that the Dirichlet problem for a mixed-type equation of the second kind with an integer negative coefficient has a solution in a rectangular domain. The corresponding estimates are constructed and used to determine sufficient conditions for the problem to have a solution. The cases of nonuniqueness of the solution are distinguished, and solvability conditions in these cases are obtained.     

Global solvability for quasilinear parabolic equation with inhomogeneous density and a source

We study the Cauchy problem for quasilinear parabolic equation with inhomogeneous density and a source. We show that this problem has a global solution under the assumptions that initial datum is small enough in the integral sense and the source term has overcritical behaviour. The sharp estimates of a solution is obtained as well.     

Solvability of the Dirichlet problem for a mixed-type equation of the second kind

We obtain sufficient conditions for the solvability of the Dirichlet problem for a mixed-type equation of the second kind in a rectangular domain. The solution is represented by a convergent series constructed from the problem data. Some cases of nonuniqueness of the solution are described.     

On solvability of the Dirichlet problem to the semilinear Schrödinger equation with singular potential

We prove the existence of a positive solution of the Dirichlet problem for the Schrödinger equation whose potential possesses a critical singularity. It is assumed that the boundary of the domain is average concave in a neighborhood of the origin. Bibliography: 13 titles.