On Gevrey solutions of threefold singular nonlinear partial differential equations

We study Gevrey asymptotics of the solutions to a family of threefold singular nonlinear partial differential equations in the complex domain. We deal with both Fuchsian and irregular singularities, and allow the presence of a singular perturbation parameter. By means of the Borel–Laplace summation method, we construct sectorial actual holomorphic solutions which turn out to share a same formal power series as their Gevrey asymptotic expansion in the perturbation parameter. This result rests on the Malgrange–Sibuya theorem, and it requires to prove that the difference between two neighboring solutions is exponentially small, what in this case involves an asymptotic estimate for a particular Dirichlet-like series.     

Hölder continuity of weak solutions of a class of linear equations with boundary degeneracy

This paper deals with a class of linear equations with boundary degeneracy. According to the degenerate ratio, the equations are divided into weakly degenerate ones and strongly degenerate ones, which should be supplemented by different Dirichlet boundary value conditions. After establishing some necessary existence, nonexistence and comparison principles, we investigate the optimal Hölder continuity of weak solutions in these two cases utilizing the Harnack inequality and the Morrey theorem, respectively.     

Potential analysis for a class of diffusion equations: A Gaussian bounds approach

We axiomatically develop a potential analysis for a general class of hypoelliptic diffusion equations under the following basic assumptions: doubling condition and segment property for an underlying distance and Gaussian bounds of the fundamental solution. Our analysis is principally aimed to obtain regularity criteria and uniform boundary estimates for the Perron-Wiener solution to the Dirichlet problem. As an example of application, we also derive an exterior cone criterion of boundary regularity and scale-invariant Harnack inequality and Hölder estimate for an important class of operators in non-divergence form with Hölder continuous coefficients, modeled on Hörmander vector fields.     

On the solutions of the Cauchy problem for a class of partial differential equations with double characteristic at a point

We give an explicit representation of the solutions of the Cauchy problem, in terms of series of hypergeometric functions, for the following class of partial differential equations with double characteristic at the origin:

(xkt_∂+ax_∂)(xkt_∂+bx_∂)u+cxk−1t_∂u=0,     

A numerical scheme for a class of nonlinear Fredholm integral equations of the second kind

In this paper an iterative approach for obtaining approximate solutions for a class of nonlinear Fredholm integral equations of the second kind is proposed. The approach contains two steps: at the first one, we define a discretized form of the integral equation and prove that by considering some conditions on the kernel of the integral equation, solution of the discretized form converges to the exact solution of the problem. Following that, in the next step, solution of the discretized form is approximated by an iterative approach. We finally on some examples show the efficiency of the proposed approach.     

Diffusive approximations for a class of nonlinear parabolic equations

We study a class of discrete velocity type approximations to nonlinear parabolic equations with source. After proving existence results and estimates on the solution to the relaxation system, we pass into the limit towards a weak solution, which is the unique entropy solution if the coefficients of the parabolic equation are constant.     

Decay rates at low and high frequencies for a plate equation with feedback concentrated in interior curves

In this paper we study the stabilization of plate vibrations by means of piezoelectric actuators. In this situation the geometric control condition of Bardos, Lebeau and Rauch [6] is not satisfied. We prove that we have exponential stability for the low frequencies but not for the high frequencies. We give an explicit decay rate for regular initial data at high frequencies while clarifying the behavior of the constant which intervenes in this estimation there function of the frequency of cut n. The method used is based on some trace regularity which reduces stability to some observability inequalities for the corresponding undamped problem. Moreover, we show numerically at low frequencies, that the optimal location of the actuator is the center of the domain Ω. Research supported by the RIP program of Oberwolfach Institut and by the Tunisian Ministry for Scientific Research and Technology (MRST) under Grant 02/UR/15-01. Research supported by the RIP program of Oberwolfach Institut. (Received: September 17, 2003; revised: February 26, 2004)     

Linking method applied to a class of a system of critical growth wave equations

We show the existence of at least two solutions for a class of a system of critical growth wave equations, with periodic condition on t and the Dirichlet boundary condition

     

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.     

Initial value problems for a system of nonlinear equations on a half-line

The paper deals with initial value problems for the system of nonlinear equations on a half-line where ₁, ₁ are real constant connections.The first problem for the nonlinear Schrödinger equation (NLS) is obtained from the system (A) when . In this case, the boundary problem associated with this NLS does not have a discrete spectrum, and the solution of the NLS is uniquely found from the given initial condition. Further, we show two remarkable classes of matrix potentials, in which the inverse scattering problem associated with system (A) can be solved exactly. In the case where the given scattering data for (A) consist of only two simple poles, the exact soliton solution of system (A) is presented. This happens if and only if the time evolution of the scattering data for (A) obeys some time evolution equations. In this case, the initial value problem for the NLS which is obtained from (A) when is solved exactly.     

The blow-up property for a system of heat equations with nonlinear boundary conditions

ＴＨＥＢＬＯＷ┐ＵＰＰＲＯＰＥＲＴＹＦＯＲＡＳＹＳＴＥＭＯＦＨＥＡＴＥＱＵＡＴＩＯＮＳＷＩＴＨＮＯＮＬＩＮＥＡＲＢＯＵＮＤＡＲＹＣＯＮＤＩＴＩＯＮＳＬＩＮＺＨＩＧＵＩ,ＸＩＥＣＨＵＮＨＯＮＧＡＮＤＷＡＮＧＭＩＮＧＸＩＮＡｂｓｔｒａｃｔ．Ｔｈｉｓｐａｐ...     

Hölder estimate for non-uniform parabolic equations in highly heterogeneous media

Uniform bound for the solutions of non-uniform parabolic equations in highly heterogeneous media is concerned. The media considered are periodic and they consist of a connected high permeability sub-region and a disconnected matrix block subset with low permeability. Parabolic equations with diffusion depending on the permeability of the media have fast diffusion in the high permeability sub-region and slow diffusion in the low permeability subset, and they form non-uniform parabolic equations. Each medium is associated with a positive number ?$?$, denoting the size ratio of matrix blocks to the whole domain of the medium. Let the permeability ratio of the matrix block subset to the connected high permeability sub-region be of the order ?^2τ${?}^{2\tau }$ for τ∈(0,1]$\tau \in (0,1]$. It is proved that the Hölder norm of the solutions of the above non-uniform parabolic equations in the connected high permeability sub-region are bounded uniformly in ?$?$. One example also shows that the Hölder norm of the solutions in the disconnected subset may not be bounded uniformly in ?$?$.     

Blow-up phenomena for a system of semilinear heat equations with nonlinear boundary flux

This paper deals with the blow-up for a system of semilinear r-Laplace heat equations with nonlinear boundary flux. It is shown that, under certain conditions on the nonlinearities and data, blow-up will occur at some finite time, and when blow-up does occur upper and lower bounds for the blow-up time are obtained.     

Ultra-analytic effect of Cauchy problem for a class of kinetic equations

The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous nonlinear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker-Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat equation. In the second part, we study a model of spatially inhomogeneous linear Landau equation with Maxwellian molecules, and show the analytic effect of the Cauchy problem.     

Probabilistic interpretation for a system of quasilinear parabolic partial differential equation combined with algebra equations

In this paper, we study a kind of system of second order quasilinear parabolic partial differential equation combined with algebra equations. Introducing a family of coupled forward–backward stochastic differential equations, and by virtue of some delicate analysis techniques, we give a probabilistic interpretation for it in the viscosity sense.     

Existence and uniqueness for a nonlinear discrete hyperbolic system

The existence, uniqueness and asymptotic behaviour of the solutions to a nonlinear discrete hyperbolic system subject to some extreme conditions and initial data are investigated in a real Hilbert space.     

The critical curve for a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux

This paper deals with the critical curve of the non-Newtonian polytropic filtration equation coupled via nonlinear boundary flux. The critical global existence curve is obtained by constructing various self-similar supersolutions and subsolutions. Furthermore, we get some new results on the critical Fujita curve.     

A variational framework for nonlinear viscoelastic models in finite deformation regime

This work presents a general framework for constitutive viscoelastic models in the finite deformation regime. The approach is qualified as variational since the constitutive updates consist of a minimization problem within each load increment. The set of internal variables is strain-based and uses a multiplicative decomposition of strain in elastic and viscous components. Spectral decomposition is explored in order to accommodate, into analytically tractable expressions, a wide set of specific models. Moreover, it is shown that, through appropriate choices of the constitutive potentials, the proposed formulation is able to reproduce results obtained elsewhere in the literature. Finally, numerical examples are included to illustrate the characteristics of the present formulation.