Domain identification for harmonic functions

The authors investigate the problem of identifying the domainG of a harmonic functionu such that Cauchy data are given on a known portion of the boundary ofG, while a zero Dirichlet condition is specified on the remaining portion of the boundary, which is to be found. Under certain conditions on the domainG, it is shown that the problem reduces to identifying the coefficients of an elliptic equation which, in turn, is converted into the problem of minimizing a functional. Under certain conditions onG, it is shown that the solution, if it exists, is unique. An application is pointed out for the problem of designing a vessel shape that realizes a given plasma shape.This work was completed with a financial support from the National Basic Research in the Natural Sciences.     

Dynamical behaviors for 1D compressible Navier-Stokes equations with density-dependent viscosity

The dynamical behaviors of vacuum states for one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefficient are considered. It is first shown that a unique strong solution to the free boundary value problem exists globally in time, the free boundary expands outwards at an algebraic rate in time, and the density is strictly positive in any finite time but decays pointwise to zero time-asymptotically. Then, it is proved that there exists a unique global weak solution to the initial boundary value problem when the initial data contains discontinuously a piece of continuous vacuum and is regular away from the vacuum. The solution is piecewise regular and contains a piece of continuous vacuum before the time T_∗>0, which is compressed at an algebraic rate and vanishes at the time T_∗, meanwhile the weak solution becomes either a strong solution or a piecewise strong one and tends to the equilibrium state exponentially.     

Solution of the Two-Dimensional Stefan Problem by the Singularity-Separating Method

In this paper the idea of "singularity-separating" presented in [10] is used to solve a two-dimensional phase-change problem. A difference scheme with second-order accuracy everywhere, including the region near the boundary between two phases, is constructed for the above problem. Through the computation it is shown that the singularity-separating method, whose accuracy is high, is efficient for two-dimensional phase-change problem.     

Wasserstein kernels for one-dimensional diffusion problems

We treat the evolution as a gradient flow with respect to the Wasserstein distance on a special manifold and construct the weak solution for the initial-value problem by using a time-discretized implicit scheme. The concept of Wasserstein kernel associated with one-dimensional diffusion problems with Neumann boundary conditions is introduced. On the basis of this, features of the initial data are shown to propagate to the weak solution at almost all time levels, whereas, in a case of interest, these features even help with obtaining the weak solution. Numerical simulations support our theoretical results.     

Oscillatory integrals generated by the Ostrovsky equation

The Ostrovsky equation governs the propagation of long nonlinear surface waves in the presence of rotation. It is related to the Korteweg-de Vries (KdV) and the Kadomtsev-Petviashvili models. KdV can be obtained from the equation in question when the rotation parameter γ equals zero. A fundamental solution of the Cauchy problem for the linear Ostrovsky equation is presented in the form of an oscillatory Fourier integral. Another integral representation involving Airy and Bessel functions is derived for it. It is shown that its asymptotic expansion as γ → 0 contains the KdV fundamental solution as the zero term. The Airy transform is used to establish some of its properties. Higher-order asymptotics for γ → 0 on a bounded time interval are obtained for both the fundamental solution and the solution of the linear Cauchy problem for the Ostrovsky equation. Received: November 23, 2004; revised: March 13, 2005 Research is supported by US Department of Defense, under grant No. DAAD19-03-1-0204     

Existence and uniqueness of a solution for a two dimensional nonlinear inverse diffusion problem

The problem of identifying the coefficient in a square porous medium is considered. It is shown that under certain conditions of data f,g, and for a properly specified class A of admissible coefficients, there exists at least one a∈A such that (a,u) is a solution of the corresponding inverse problem.     

A singular solution with smooth initial data for a semilinear parabolic equation

We consider the Cauchy problem for a parabolic partial differential equation with a power nonlinearity. Our concern in this paper is the existence of a singular solution with smooth initial data. By using the Haraux-Weissler equation, it is shown that there exist singular forward self-similar solutions. Using this result, we also obtain a sufficient condition for the singular solution with general initial data including smooth initial data.     

An approximation problem in inverse scattering theory

In [3] a new method was introduced for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium. This method is based on the solution of a new class of boundary value problems for the reduced wave equation called interior transmission problems. In this paper it is shown that if there is absorption there exists at most one solution to the interior transmission problem and an approximate solution can be found such that the metaharmonic part is a Herglotz wave function. These results provide the necessary theoretical basis for the inverse scattering method introduced in [3]     

A nonlinear parabolic equation with a nonlocal boundary term

A nonlinear parabolic problem with a nonlocal boundary condition is studied. We prove the existence of a solution for a monotonically increasing and Lipschitz continuous nonlinearity. The approximation method is based on Rothe’s method. The solution on each time step is obtained by iterations, convergence of which is shown using a fixed-point argument. The space discretization relies on FEM. Theoretical results are supported by numerical experiments.     

Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations

The paper considers the Cauchy problem for linear partial differential equations of non-Kowalevskian type in the complex domain. It is shown that if the Cauchy data are entire functions of a suitable order, the problem has a formal solution which is multisummable. The precise bound of the admissible order of entire functions is described in terms of the Newton polygon of the equation.     

Persistence of the bifurcation structure for a semilinear elliptic problem on thin domains

We study a semilinear elliptic problem on thin domains with a bifurcation parameter. It is shown that the set of solutions is upper semicontinuous as the thickness of a domain tends to 0, and that solution branches including bifurcation points persist near those of a one-dimensional limiting equation.     

The global attractor of a competitor–competitor–mutualist reaction–diffusion system with time delays

The aim of this paper is to investigate the asymptotic behavior of time-dependent solutions of a three-species reaction–diffusion system in a bounded domain under a Neumann boundary condition. The system governs the population densities of a competitor, a competitor–mutualist and a mutualist, and time delays may appear in the reaction mechanism. It is shown, under a very simple condition on the reaction rates, that the reaction–diffusion system has a unique constant positive steady-state solution, and for any nontrivial nonnegative initial function the corresponding time-dependent solution converges to the positive steady-state solution. An immediate consequence of this global attraction property is that the trivial solution and all forms of semitrivial solutions are unstable. Moreover, the state–state problem has no nonuniform positive solution despite possible spatial dependence of the reaction and diffusion. All the conclusions for the time-delayed system are directly applicable to the system without time delays and to the corresponding ordinary differential system with or without time delays.     

Singular homogenization with stationary in time and periodic in space coefficients

We study the homogenization problem for a random parabolic operator with coefficients rapidly oscillating in both the space and time variables and with a large highly oscillating nonlinear potential, in a general stationary and mixing random media, which is periodic in space. It is shown that a solution of the corresponding Cauchy problem converges in law to a solution of a limit stochastic PDE.     

Multisummability of formal power series solutions of partial differential equations with constant coefficients

In an earlier paper of the author's, partial differential equations with constant coefficients have been studied. Under a certain (restrictive) assumption upon the equation, those initial conditions were characterized for which the normalized formal solution of a corresponding Cauchy problem is k-summable. Here we treat the general situation and prove an analogous result, using multisummability instead of k-summability. The appropriate multisummability type is shown to depend upon the given PDE only, and can be determined from a corresponding Newton polygon.     

Singular and degenerate anisotropic parabolic equations with a nonlinear source

The Dirichlet problem in arbitrary domain for degenerate and singular anisotropic parabolic equations with a nonlinear source term is considered. We state conditions that guarantee the existence and uniqueness of a global weak solution to the problem. A similar result is proved for the parabolic p-Laplace equation.     

Mildly degenerate Kirchhoff equations with weak dissipation: Global existence and time decay

We consider the hyperbolic-parabolic singular perturbation problem for a degenerate quasilinear Kirchhoff equation with weak dissipation. This means that the coefficient of the dissipative term tends to zero when t→+∞.We prove that the hyperbolic problem has a unique global solution for suitable values of the parameters. We also prove that the solution decays to zero, as t→+∞, with the same rate of the solution of the limit problem of parabolic type.     

Generalized 1D Jörgens theorem

We study the Cauchy problem for a 1D nonlinear wave equation on R. The nonlinearity can depend on the unknown function and its first order spatial derivative. Using the fixed point theorem we prove the existence of a classical solution. Moreover, the existence of periodic and almost periodic solutions are shown.     

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.     

Monotone iterative methods for infinite systems of parabolic functional differential equations

Two monotone iterative methods for an infinite system of parabolic functional differential equations with initial boundary conditions are constructed: the method of direct iterations and the Chaplygin method. By using the first one the existence theorem is proved. Next, it is shown that the Chaplygin sequences converge quadratically to the unique solution of the original problem.     

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.