The DeGiorgi-Nash-Moser estimate plays a crucial role in the study of quasilinear elliptic and parabolic equations. In the present paper we shall show that this fundamental estimate holds for solutions of a linear parabolic Volterra inte-grodiierential equation: @u @t = @ @x i a ij (x; t) @u @x j + Z t 0 @ @x i b ij (x; t;) @u @x j dd; where fa ij g and fb ij g are only assumed to be measurable...

We analyze the interplay of nonlinearity and dispersion in a dispersion-decreasing photonic bandgap Bragg fiber as a new platform for generating parabolic pulses. A suitably designed linearly tapered, low-index-contrast, solid-core Bragg fiber – amenable to fabrication by conventional modified chemical vapor deposition technology – is shown to yield stable parabolic pulses. The fiber design was...

The paper suggests a modification of the contracting mapping method for non-linear and non-local parabolic equations. This modification is based on weighted in time energy estimates for the L2-norm of the solution of a parabolic equation via a weighted version of the H−1-norm of the free term such that the inverse matrix of the higher order coefficients of the parabolic equation is included int...

The paper is concerned with Lp error analysis of semi-discrete Galerkin FEMs for nonlinear parabolic equations. The classical energy approach relies heavily on the strong regularity assumption of the diffusion coefficient, which may not be satisfied in many physical applications. Here we focus our attention on a general nonlinear parabolic equation (or system) in a convex polygon or polyhedron ...

Abstract. A (Patlak-) Keller-Segel model in two space dimensions with an additional crossdiffusion term in the equation for the chemical signal is analyzed. The main feature of this model is that there exists a new entropy functional, yielding gradient estimates for the cell density and chemical substance. This allows one to prove, for arbitrarily small cross diffusion, the global existence of ...

In order to extend the blow-up criterion of solutions to the Euler equations, Kozono and Taniuchi [12] have proved a logarithmic Sobolev inequality by means of isotropic (elliptic) BMO norm. In this paper, we show a parabolic version of the Kozono-Taniuchi inequality by means of anisotropic (parabolic) BMO norm. More precisely we give an upper bound for the L∞ norm of a function in terms of its...

On the rectangular space-time domain G = D (0; T ] with the boundary S = GnG, where D = (0; d), we consider the Dirichlet problem for the singularly perturbed parabolic equation: " 2 @ @x a(x; t) @ @x u(x; t) ! ? c(x; t)u(x; t) ? p(x; t) @ @t u(x; t) = f(x; t); (1a) The perturbation parameter " may take any small positive value, say " 2 (0; 1]. If " = 0, the parabolic equation (1a) degenerates ...

We study the structure of generalized Verma modules over a semi-simple complex finite-dimensional Lie algebra, which are induced from simple modules over a parabolic subalgebra. We consider the case when the annihilator of the starting simple module is a minimal primitive ideal if we restrict this module to the Levi factor of the parabolic subalgebra. We show that these modules correspond to pr...

this paper presents a class of theoretical and iterative method for linear partial differential equations. an algorithm and analytical solution with a initial condition is obtained using the reduced differential transform method. in this technique, the solution is calculated in the form of a series with easily computable components. there test modeling problems from mathematical mechanic, physi...

in this paper, we discuss the geometric properties of solution and lower bound estimate of ∆um−1 of the cauchy problem for a degenerate parabolic equation with periodic source term ut =∆um+ upsint. our objective is to show that: (1)with continuous variation of time t, the surface ϕ = [u(x,t)]mδq is a complete riemannian manifold floating in space rn+1and is tangent to the space rn at ∂h0(t); (2...