Singular integral equation with Cauchy kernel on a complicated contour

We present a reasonably comprehensive exposition of the theory of a singular integral equation with Cauchy kernel for the case in which the integration contour is a set of disjoint smooth open arcs. We construct numerical schemes for this equation and give an order estimate for the accuracy of the approximate solutions.     

On a Two-Point Boundary-Value Problem with Spurious Solutions

The Carrier–Pearson equation     with boundary conditions   u (−1) = u (1) = 0  is studied from a rigorous point of view. Known solutions obtained from the method of matched asymptotics are shown to approximate true solutions within an exponentially small error estimate. The so-called spurious solutions turn out to be approximations of true solutions, when the locations of their "spikes" are properly assigned. An estimate is also given for the maximum number of spikes that these solutions can have.     

Blow-up and global solutions for nonlinear reaction-diffusion equations with nonlinear boundary condition

This paper deals with the blow-up of positive solutions for a nonlinear reaction-diffusion equation subject to nonlinear boundary conditions. We obtain the conditions under which the solutions may exist globally or blow up in finite time. Moreover, an upper bound of the blow-up time, an upper estimate of the blow-up rate, and an upper estimate of the global solutions are given. At last we give two examples to which the theorems obtained in the paper may be applied.     

Unimprovable Integral Estimate for Solutions of the Dirichlet Problem for Quasilinear Elliptic Equations of the Second Order in a Neighborhood of an Edge

We obtain an exact estimate for the second derivatives of solutions (in the weight Sobolev norm) of the Dirichlet problem for quasilinear second-order elliptic equations of nondivergence type in a neighborhood of an edge of a domain.     

Well-posedness,global solutions and blowup phenomena for a nonlinearly dispersive wave equation

We establish the local well-posedness for a new nonlinearly dispersive wave equation which has solutions that exist for indefinite times as well as solutions that blowup infinite time. We also derive an explosion criterion for the equation, and we give a sharp estimate of the existence time for solutions with smooth initial data.     

Existence of a Global Solution for a Scalar Conservation Law with a Source Term

We are concerned with a scalar conservation law with a source term. This equation is proposed to describe the qualitative behavior of waves for a general system in resonance with the source term by T.P. Liu. In addition to this, the scalar conservation law is used in various areas such as fluid dynamics, traffic problems etc.In the present paper, we prove the global existence and stability of entropy solutions to the Cauchy problem. The difficult point is to obtain the bounded estimate of solutions. To solve it, we introduce some functions as the lower and upper bounds. Therefore, our bounded estimate depends on the space variable. This idea comes from the generalized invariant region theory for the compressible Euler equation. The method is also applicable to other nonlinear problems involving similar difficulties. Finally, we use the vanishing viscosity method to construct approximate solutions and derive the convergence by the compensated compactness.     

Double logarithmic inequality with a sharp constant

We prove a Log Log inequality with a sharp constant. We also show that the constant in the Log estimate is ``almost' sharp. These estimates are applied to prove a Moser-Trudinger type inequality for solutions of a 2D wave equation.

     

Nonstandard characteristics and localized asymptotic solutions of a linearized magnetohydrodynamic system with small viscosity and drag

We describe asymptotic solutions of the Cauchy problem for a linearized system of magnetohydrodynamics with initial conditions localized in a small neighborhood of a curve or a two-dimensional surface. We investigate how a change of the multiplicity of characteristics affects such solutions and prove a uniform estimate of the residual.     

Asymptotic behavior of solutions to a class of nonlinear wave equations of sixth order with damping

We investigate the initial value problem for a class of nonlinear wave equations of sixth order with damping. The decay structure of this equation is of the regularity‐loss type, which causes difficulty in high‐frequency region. By using the Fourier splitting frequency technique and energy method in Fourier space, we establish asymptotic profiles of solutions to the linear equation that is given by the convolution of the fundamental solutions of heat and free wave equation. Moreover, the asymptotic profile of solutions shows the decay estimate of solutions to the corresponding linear equation obtained in this paper that is optimal under some conditions. Finally, global existence and optimal decay estimate of solutions to this equation are also established. Copyright © 2016 John Wiley & Sons, Ltd.     

Estimates for the embedding operator of a sobolev space of periodic functions and for the solutions of differential equations with periodic coefficients

We obtain estimates for the embedding operator of a Sobolev space in the space of continuous periodic functions and use them to estimate the solutions of differential equations with periodic coefficients. We prove a theorem on a necessary and sufficient condition for the invertibility of a differential operator with unbounded operator coefficients.     

Strong unique continuation for higher order elliptic equations with Gevrey coefficients

We address the strong unique continuation problem for higher order elliptic partial differential equations in 2D with Gevrey coefficients. We provide a quantitative estimate of unique continuation (observability estimate) and prove that the solutions satisfy the strong unique continuation property for ranges of the Gevrey exponent strictly including non-analytic Gevrey classes. As an application, we obtain a new upper bound on the Hausdorff length of the nodal sets of solutions with a polynomial dependence on the coefficients.     

Local search with a generalized neighborhood in the optimization problem for pseudo-Boolean functions

In the optimization problem for pseudo-Boolean functions we consider a local search algorithm with a generalized neighborhood. This neighborhood is constructed for a locally optimal solution and includes nearby locally optimal solutions. We present some results of simulations for pseudo-Boolean functions whose optimization is equivalent to the problems of facility location, set covering, and competitive facility location. The goal of these experiments is to obtain a comparative estimate for the locally optimal solutions found by the standard local search algorithm and the local search algorithm using a generalized neighborhood.     

On the a posteriori estimates for inverse operators of linear parabolic equations with applications to the numerical enclosure of solutions for nonlinear problems

We consider the guaranteed a posteriori estimates for the inverse parabolic operators with homogeneous initial-boundary conditions. Our estimation technique uses a full-discrete numerical scheme, which is based on the Galerkin method with an interpolation in time by using the fundamental solution for semidiscretization in space. In our technique, the constructive a priori error estimates for a full discretization of solutions for the heat equation play an essential role. Combining these estimates with an argument for the discretized inverse operator and a contraction property of the Newton-type formulation, we derive an a posteriori estimate of the norm for the infinite-dimensional operator. In numerical examples, we show that the proposed method should be more efficient than the existing method. Moreover, as an application, we give some prototype results for numerical verification of solutions of nonlinear parabolic problems, which confirm the actual usefulness of our technique.     

Finite‐volume schemes for Friedrichs systems in multiple space dimensions: A priori and a posteriori error estimates

We consider a class of finite‐volume schemes on unstructured meshes for symmetric hyperbolic linear systems of balance laws in two and three space dimensions. This class of schemes has been introduced and analyzed by Vila and Villedieu ( 5 ). They have proven an a priori error estimate for approximations of smooth solutions. We extend the results to weak solutions. This is the base to derive an a posteriori error estimate for finite‐volume approximations of weak solutions. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Collapse Rate of Solutions of the Cauchy Problem for the Nonlinear Schrödinger Equation

We prove that solutions of the Cauchy problem for the nonlinear Schrödinger equation with certain initial data collapse in a finite time, whose exact value we estimate from above. We obtain an estimate from below for the solution collapse rate in certain norms.

     

On an estimate of solutions of a linear homogeneous Volterra integro-differential equation of the first order in the critical case on the half-line

We obtain sufficient conditions for a certain estimate to hold on the half-line for all solutions of a linear homogeneous Volterra integro-differential equation of the first order in the critical case. We present some corollaries concerning the absolute integrability of powers of the solutions on the half-line and the convergence of solutions to zero (including the exponential and power-law convergence) as the independent variable tends to infinity. Illustrative examples are given.     

Long-time behavior for a nonlinear plate equation with thermal memory

We consider a nonlinear plate equation with thermal memory effects due to non-Fourier heat flux laws. First we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we use a suitable ?ojasiewicz-Simon type inequality to show the convergence of global solutions to single steady states as time goes to infinity under the assumption that the nonlinear term f is real analytic. Moreover, we provide an estimate on the convergence rate.     

On a class of nonlinear obstacle problems with nonstandard growth

In this paper, we study a class of nonlinear obstacle problems with nonstandard growth. We obtain the L∞ estimate on the solutions and prove the existence and uniqueness of solutions to such problems. Our results are generalizations of the corresponding results in the constant exponent case. Copyright © 2014 John Wiley & Sons, Ltd.     

On the differentiability of the solution to the Hamilton–Jacobi equation with critical fractional diffusion

We prove that the Hamilton–Jacobi equation for an arbitrary Hamiltonian H (locally Lipschitz but not necessarily convex) and fractional diffusion of order one (critical) has classical C1,α solutions. The proof is achieved using a new Hölder estimate for solutions of advection–diffusion equations of order one with bounded vector fields that are not necessarily divergence free.     

Applications of the Kinetic Formulation for Scalar Conservation Laws with a Zero-Flux Type Boundary Condition

The authors are concerned with a zero-flux type initial boundary value problem for scalar conservation laws.Firstly,a kinetic formulation of entropy solutions is established.Secondly,by using the kinet...