Weak solutions to a nonlinear variational wave equation and some related problems

In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and some related problems. We first introduce the main tools, the L ^p Young measure theory and related compactness results, in the first section. Then we use the L ^p Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed. In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic equation, which is also the so-called vortex density equation arising from sup-conductivity.     

Best Lp approximate solutions of nonlinear integrodifferential equations

In this paper best L_p approximate solutions are shown to exist for a wide class of integrodifferential equations. Using approximation theory techniques, a local existence theorem for solutions is established, and the convergence of the best approximate solutions to a solution is shown.     

On solvability of one class of nonlinear integral equations on whole line with a weak singularity at zero

In present work we investigate one class of nonlinear integral equations with singularity at zero and boundary value conditions at ±∞. Above mentioned class of equations has direct applications in string theory and in the case of concrete structure of the kernel it describes the dynamics of the open-closed p-adic string for the scalar tachyon field. We prove the existence of nontrivial solution in a certain weight class of functions.With an additional restriction on the kernel the uniqueness of the obtained solution is proved.     

Multiplicity results for p(x)-biharmonic equations with Navier boundary conditions

This paper deals with the existence of solutions for a class of p(x)-biharmonic equations with Navier boundary conditions. The approach is based on variational methods and critical point theory. Indeed, we investigate the existence of two solutions for the problem under some algebraic conditions with the classical Ambrosetti–Rabinowitz condition on the nonlinear term. Moreover, by combining two algebraic conditions on the nonlinear term which guarantee the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of the third solution for the problem.     

Almost automorphic solutions for some partial functional differential equations

In this work, we study the existence of almost automorphic solutions for some partial functional differential equations. We prove that the existence of a bounded solution on R⁺ implies the existence of an almost automorphic solution. Our results extend the classical known theorem by Bohr and Neugebauer on the existence of almost periodic solutions for inhomegeneous linear almost periodic differential equations. We give some applications to hyperbolic equations and Lotka-Volterra type equations used to describe the evolution of a single diffusive animal species.     

Maximum principle and existence of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic terms

We study L^p-viscosity solutions of fully nonlinear, second-order, uniformly elliptic partial differential equations (PDE) with measurable terms and quadratic nonlinearity. We present a sufficient condition under which the maximum principle holds for L^p-viscosity solution. We also prove stability and existence results for the equations under consideration.     

Positive Solutions of One-Dimensional <Emphasis Type="Italic">p</Emphasis>-Laplacian Boundary Value Problems for Fourth-Order Differential Equations with Deviating Arguments

This paper considers the existence of positive solutions of four-point boundary value problems for fourth-order ordinary differential equations with deviating arguments and p-Laplacian. We discuss such problems in the cases when the deviating arguments are delayed or advanced, what may concern optimization issues related to some technical problems. To obtain the existence results, a fixed point theorem for cones due to Avery and Peterson is applied. According to the Author’s knowledge, the results are new. It is a first paper where a fixed point theorem for cones is applied to fourth-order differential equations with deviating arguments and p-Laplacian. An example is included to verify the theoretical results.     

On the domain dependence of solutions to the Navier–Stokes equations of a two‐dimensional compressible flow

We consider the Navier–Stokes equations for compressible, barotropic flow in two space dimensions, with pressure satisfying p(?)=a?log^d(?) for large ?, here d>1 and a>0. After introducing useful tools from the theory of Orlicz spaces, we prove a compactness result for the solution set of the equations with respect to the variation of the underlying bounded spatial domain. Especially, we get a general existence theorem for the system in question with no restrictions on smoothness of the bounded spatial domain. Copyright © 2009 John Wiley & Sons, Ltd.     

Applications of Schochet''s methods to parabolic equations

Methods used by S. Schochet in [32] enable one to find a lower bound for the life span of solutions of hyperbolic PDEs with a small parameter. We prove a similar theorem for such equations where a diffusion term has been added, with the minimal assumption on the Sobolev regularity of the initial data ( in the d-dimensional torus). When the data is smooth and under a “small divisor” assumption on the perturbation, the first term of an asymptotic expansion of the solution is computed. Those results are then applied to prove global existence theorems, for arbitrary initial data, in the case of the primitive system of the quasigeostrophic equations, followed by the rotating fluid equations. We finally prove a more precise existence theorem for the latter, using anisotropic Sobolev and Besov spaces.     

Thermo-chemical modelling of the formation of a composite material

We consider a composite material composed of carbon or glass fibres included in a resin which becomes solid when it is heated up (the reaction of reticulation).

A mathematical model of the cure process is given by a kinetic equation describing the evolution of the reaction of reticulation coupled with the heat equation. The geometry of the composite material is periodic, with a small period ? >0, thus we get a coupled system of nonlinear partial differential equations.

First we prove the existence and uniqueness of a solution by using a fixed point theorem and we obtain a priori estimates. Then we derive the homogenized problem which describes the macroscopic behaviour of the material. We prove the convergence of the solution of the problem to the solution of the homogenized problem when ? tends to zero as well as the estimates for the difference of the exact and the approximate solutions.     

Infinitely many solutions for polyharmonic equations of p(x)-Laplace type

We show the existence of infinitely many weak solutions to a class of quasilinear elliptic p(x)-polyharmonic Kirchhoff equations via the mountain pass principle without the (AR) condition. Furthermore, we obtain infinitely many solutions to this equation based on the genus theory, introduced by Krasnoselskii and the abstract critical point theorem (a variant of Ljusternik-Schnirelman theory) under Cerami condition.     

A study on periodic solutions of higher order nonlinear functional difference equations with p-Laplacian†

The study on the existence of periodic solutions of higher order nonlinear functional difference equations with p-Laplacian is made. Sufficient conditions for the existence of at least one periodic solution of these equations are established, respectively. We give no restriction on the deviating function, which is the significance of the paper. Our result is based on Mawhin's continuation theorem. The methods we used to estimate a priori bound on periodic solutions are also new.     

Periodic problems and problems with discontinuities for nonlinear parabolic equations

In this paper we study nonlinear parabolic equations using the method of upper and lower solutions. Using truncation and penalization techniques and results from the theory of operators of monotone type, we prove the existence of a periodic solution between an upper and a lower solution. Then with some monotonicity conditions we prove the existence of extremal solutions in the order interval defined by an upper and a lower solution. Finally we consider problems with discontinuities and we show that their solution set is a compact R -set in (CT, L ²(Z)).     

On the Existence of Periodic Solutions for a Nonlinear System of Ordinary Differential Equations

Abstract This paper is concerned with the existence of periodic solutions for a nonlinear system of ordinary differential equations. We obtain a Nagumo-type a priori bound for the periodic solutions and then by using this a priori bound we prove the existence of at least one T-periodic solution under some general conditions Research supported by the NNSF of China and the RFDP of China.     

Self-similar p-adic fractal strings and their complex dimensions

We develop a geometric theory of self-similar p-adic fractal strings and their complex dimensions. We obtain a closed-form formula for the geometric zeta functions and show that these zeta functions are rational functions in an appropriate variable. We also prove that every self-similar p-adic fractal string is lattice. Finally, we define the notion of a nonarchimedean self-similar set and discuss its relationship with that of a self-similar p-adic fractal string. We illustrate the general theory by two simple examples, the nonarchimedean Cantor and Fibonacci strings. The text was submitted by the authors in English.     

Existence and uniqueness results for a nonlinear stationary system

We prove a few existence results of a solution for a static system with a coupling of thermoviscoelastic type. As this system involves L⊃ coupling terms we use the techniques of renormalized solutions for elliptic equations with L⊃ data. We also prove partial uniqueness results.AMS Subject Classification: 32D05, 74D99.     

Multiple Solutions for a Differential Inclusion Problem with Nonhomogeneous Boundary Conditions

In this paper, we use a mountain pass theorem with Cerami type conditions for locally Lipschitz functions to investigate the existence of at least one nontrivial solution for a differential inclusion problem involving the p-Laplacian and with nonlinear and nonsmooth boundary conditions. Moreover, by a symmetric version of the mountain pass theorem, we prove the existence of infinitely many solutions.     

Solvability of a nonlinear model Boltzmann equation in the problem of a plane shock wave

We consider a nonlinear system of integral equations describing the structure of a plane shock wave. Based on physical reasoning, we propose an iterative method for constructing an approximate solution of this system. The problem reduces to studying decoupled scalar nonlinear and linear integral equations for the gas temperature, density, and velocity. We formulate a theorem on the existence of a positive bounded solution of a nonlinear equation of the Uryson type. We also prove theorems on the existence and uniqueness of bounded positive solutions for linear integral equations in the space L ₁[?r, r] for all finite r < +∞. For a more general nonlinear integral equation, we prove a theorem on the existence of a positive solution and also find a lower bound and an integral upper bound for the constructed solution.     

Global solution on Cauchy problem in nonlinear non-simple thermoelastic materials

We prove a theorem about global existence (in time) of the solution to the initial-value problem for a nonliear system of coupled partial differential equations of fourth order describing the thermoelasticity of non-simple materias. We consider such the case of thim system in which some nonlinear coeffcients can depend not only on the temperature and the gradient of displacement and also on the second derivative of displacement. The corresponding global existence theorem has been proved using the L ^p – L ^q time decay estmates for the solution of the associated linearized problem. Next, we proved the energy estimate in the Sobolev space with constant independent of time. Such an energy estimate allows us to apply the standard continuation argument and to continue the local solution to one desired for all t ∈ (0, ∞)     

Positive solutions and multiple solutions for periodic problems driven by scalar p ‐Laplacian

In this paper we study a nonlinear second order periodic problem driven by a scalar p ‐Laplacian and with a nonsmooth, locally Lipschitz potential function. Using a variational approach based on the nonsmooth critical point theory for locally Lipschitz functions, we first prove the existence of nontrivial positive solutions and then establish the existence of a second distinct solution (multiplicity theorem) by strengthening further the hypotheses. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)