Periodic solutions of perturbed isochronous hamiltonian systems at resonance

We look for periodic solutions of planar systems obtained by adding an asymptotically positively homogeneous nonlinear term to an isochronous hamiltonian system. Precise computations of the topological degree are obtained by elementary phase-plane analysis.     

Non-self-similar solutions of multidimensional nonlinear diffusion equations

An original construction of the exact nonnegative solution of multidimensional nonlinear diffusion equations is proposed and studied. On substituting this construction into the original equation, we obtain a system of algebro-differential equations in which the number of equations exceeds the number of unknown functions. It is proved that the resulting system possesses nontrivial solutions. On the basis of this result, we construct exact non-self-similar explicit nonnegative solutions anisotropic with respect to the space variables both of the class of equations of a porous medium (nonstationary filtration) and of the class of equations involving nonlinear thermal conductivity with negative exponent. In particular, this class contains the so-called equations of rapid and limit diffusion. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 250–256, February, 2000.     

Multiple doubly periodic solutions of semi-linear damped beam equations

The purpose of this work is to investigate the weakened Ambrosetti–Prodi type multiplicity results for weak doubly periodic solutions of damped beam equations. By using the topological degree theory, the author obtains a result which is similar to the result for damped wave equations in the literature.     

Existence and uniqueness of periodic solutions for forced Rayleigh-type equations

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T-periodic solutions for a kind of forced Rayleigh equation of the form

x^″+f(x^′(t))+g(t,x(t))=e(t).     

Positive periodic solutions for a system of anisotropic parabolic equations

We consider a system of two degenerate parabolic equations with nonlocal terms and Dirichlet boundary conditions. More precisely, the degeneracy in each equation of the system is of the type r(x)-Laplacian where r(x) is a function depending on x∈Ω, where Ω is a bounded smooth domain of Rn. The system models the diffusion and the interaction between two different biological species sharing the same territory Ω. The paper provides conditions on the parameters of the problem that guarantee the coexistence of a T-periodic non-negative solution (u,v) with both non-trivial u,v.     

Local discontinuous Galerkin approximation of non-Fickian diffusion model in viscoelastic polymers

In this paper, we investigate a fully discrete local discontinuous Galerkin approximation of a non-linear non-Fickian diffusion model in viscoelastic polymers. For the spatial discretization, we adopt local discontinuous Galerkin finite element method and for the time discretization we use backward Euler method. We derive the stability estimate and a priori error estimate for the discrete scheme. Numerical examples are given to verify the theoretical findings.     

Construction of quasi-periodic solutions for nonlinear forced perturbations of dissipative Boussinesq systems

In this paper we consider a class of quasi-periodically forced perturbations of the dissipative Boussinesq systems with an elliptic fixed point (see (1.4)) in two cases: Hamiltonian case and reversible case. We prove the existence and linear stability of quasi-periodic solutions for the system (1.4) with periodic boundary conditions. The method of proof is based on a Nash–Moser iterative scheme in the scale of Sobolev spaces developed by Berti and Bolle in Berti and Bolle (2013, 2012), but we have to be substantially developed to deal with the system (1.4) considered here.     

Existence of sign-changing solutions for some asymptotically linear three-point boundary value problems

In this paper, by using the topological degree theory and the fixed point index theory, the existence of three kinds of solutions (i.e., sign-changing solutions, positive solutions and negative solutions) for asymptotically linear three-point boundary value problems is obtained.     

Behaviors of solutions for a singular diffusion equation

This paper is devoted to some behaviors of solutions of the initial-boundary problem for a singular diffusion equation, namely, localization and large time behavior. After given some special explicit solutions it is proved that solutions of the problem possess the localization property. Next, L² decay estimate as t→∞ is proved by a rather standard energy method. Finally, by comparison with a special solution the expected L^∞ decay estimate is derived.     

Nonstandard perturbation approximation and travelling wave solutions of nonlinear reaction diffusion equations

This paper deals with the construction of a nonstandard numerical method to compute the travelling wave solutions of nonlinear reaction diffusion equations at high wave speeds. Related general properties are studied using the perturbation approximation. At high wave speed the perturbation parameter approaches to zero and the problem exhibits a multiscale character. That is, there are thin layers where the solution varies rapidly, while away from these layers the solution behaves regularly and varies slowly. Most of the conventional methods fail to capture this layer behavior. Thus, the quest for some new numerical techniques that may handle the travelling wave solutions at high wave speeds earns relevance. In this paper, one such parameter robust nonstandard numerical scheme is constructed, in the sense that its numerical solution converges in the maximum norm to the exact solution uniformly well for all finite wave speeds. To overcome the difficulty due to the nonlinearity, the problem is linearized using the quasilinearization process followed by nonstandard finite difference discretization. An extensive amount of analysis is carried out which uses a suitable decomposition of the error into smooth and singular component and a comparison principle combined with appropriate barrier functions. The error estimates are obtained, which ensures uniform convergence of the method. A set of numerical experiment is carried out in support of the predicted theory that validates computationally the theoretical results. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Approximation and existence of periodic solutions for controlled diffusion equations

This paper concerns the existence of control functions such that a system of controlled stochastic differential equations (SDE) with periodic coefficients has a solution which is periodic in distribution. We show that bounded periodic controls acting in the same direction as the Driving Wiener process will achieve this under some nondegeneracy condition on the diffusion part. The main tool is an approximation theorem for solutions of SDEs which enables one to check certain stability conditions on a more suitable differential equation     

A spectral countability condition for almost automorphy of solutions of differential equations

We consider the almost automorphy of bounded mild solutions to equations of the form

with (generally unbounded) -periodic and almost automorphic in a Banach space . Under the assumption that does not contain , the part of the spectrum of the monodromy operator associated with the evolutionary process generated by on the unit circle is countable. We prove that every bounded mild solution of on the real line is almost automorphic.

     

Periodic solutions of non-Newtonian polytropic filtration equations with nonlinear sources

In this paper, the authors first consider the Dirichlet boundary value problem to the non-Newtonian polytropic filtration equation of the form

     

带有热源项的非线性扩散方程的精确解

讨论了带有热源项的非线性扩散方程.通过一种直接简洁的方法得到了几种精确解.该方法可用于更高阶演化方程的求解问题.     

ASYMPTOTIC METHOD OF TRAVELLING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR REACTION DIFFUSION EQUATION

In this article the travelling wave solution for a class of nonlinear reaction diffusion problem;are considered.Using the homotopic method and the theory of travelling wave transform,the approximate solution for the corresponding problem is obtained.