Diffusion instability of a uniform cycle bifurcating from a separatrix loop

We consider the boundary value problem

Basic boundary-value problems for one equation with fractional derivatives

We prove certain properties of solutions of the equation

Estimates of the rate of decay of solutions to the impedance mixed problem for the wave equation in a region with noncompact boundary

The paper is devoted to the study of the behavior of the following mixed problem for large values of time:

Approximation and growth of generalized axisymmetric potentials

The regular solutions of generalized axisymmetric potential equation , a>−1/2 are called generalized axisymmetric potentials. In this paper, the characterizations of lower order and lower type of entire GASP in terms of their approximation error {E_n} have been obtained.     

Compactification of supports of energy weak solutions to quasilinear parabolic equations of nonstationary filtration type with strong convection

The paper deals with localization properties of solutions to the Cauchy problem with the initial data u₀(x) ∈ L₂(ℝ_n) for a wide class of equations in the divergence form. This class contains, e.g., the following equation:

Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt ＋ α1 uδ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x,t）∈ E R^2, and δu/δt ＋ α2 δu/δx δ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x, t） ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0（x） ∈ H^s（R）, and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.     

Mean value theorem and a maximum principle for Kolmogorov's equation

For an equation of the form $$\begin{gathered} \frac{{\partial u}}{{\partial t}} - \sum\nolimits_{ij = 1}^n {{\text{ }}\alpha ^{ij} } \frac{{\partial ^2 u}}{{\partial x^i \partial x^j }} + \sum\nolimits_{ij = 1}^n {\beta _j^i x^i } \frac{{\partial u}}{{\partial x^i }} = 0, \hfill \\ {\text{ }}x \in R^n ,{\text{ }}t \in R^1 , \hfill \\ \end{gathered}$$ where α=(α^ij) is a constant nonnegative matrix andΒ=(Β _i ⁱ ) is a constant matrix, subject to certain conditions, we construct a fundamental solution, similar in its structure to the fundamental solution of the heat conduction equation; we prove a mean value theorem and show that u(x₀, t₀) can be represented in the form of the mean value of u(x, t) with a nonnegative density over a level surface of the fundamental solution of the adjoint equation passing through the point (x₀, t₀); finally, we prove a parabolic maximum principle.     

Local and Global Existence of Solutions to Initial Value Problems of Modified Nonlinear Kawahara Equations

This paper is devoted to studying the initial value problem of the modified nonlinear Kawahara equation the first partial dervative of u to t ,the second the third ＋α the second partial dervative of u to x ,the second the third ＋β the third partial dervative of u to x ,the second the thire ＋γ the fifth partial dervative of u to x = 0,（x,t）∈R^2.We first establish several Strichartz type estimates for the fundamental solution of the corresponding linear problem. Then we apply such estimates to prove local and global existence of solutions for the initial value problem of the modified nonlinear Karahara equation. The results show that a local solution exists if the initial function uo（x） ∈ H^s（R） with s ≥ 1/4, and a global solution exists if s ≥ 2.     

Défaut d’estimation a priori pour un problème de dirichlet

Résumé  Dans un célèbre papier ([3]), B. GIDAS et J.SPRUCK ont utilisé-sous des hypothèses adéquates- la technique du “blow up” pour montrer que les solutionsu ∈C ⁰ ∩C ¹ (Ω) du problème admettent une estimation a priori dansC ⁰ . Dans ce travail, on montre que, si les solutionsu sont juste supposéesC ⁰ , une telle estimation a priori n’existe plus. In a famous paper ([3]), B. GIDAS and J. SPRUCK used a “blow-up” argument to show that, under appropriate assumptions, all the solutionsu ∈C ⁰ ∩C ¹ (Ω) of the problem admit an a priori estimate inC ⁰ . In this work, we show that, if one supposes the solutions are only inC ⁰ , such an a priori estimate does not hold.     

Null-sets ofH-solutions

We considerC ²-solutionsf=u+iv+jw of the system

Brownian motion with negative drift and convex level sets in space-time

Suppose a, b, and are reals witha<b and consider the following diffusion equation

On initial value problems for quaternionic valued functions

In [2], [6], [7], methods are discussed for solving initial value problems

A differential equation without a solution

A direct construction is given of a functionf(x₁, x₂) ∈ C°, such that the equation $$\frac{{\partial u}}{{\partial x_1 }} + ix_1^{2k - 1} \frac{{\partial u}}{{\partial x_2 }} = f$$ has no solution in any neighborhood of the origin; the functionf and all its derivatives vanish for x₁=0.     

On Inversion Of Bessel Potentials Associated With The Laplace–Bessel Differential Operator

Explicit inversion formulas of Balakrishnan–Rubin type and a characterization of Bessel potentials associated with the Laplace–Bessel differential operator are obtained. As an auxiliary tool the B-metaharmonic semigroup is introduced and some of its properties are investigated.     

Some properties of solutions of a hyperbolic modification of the Bürgers equation

The results of dispersion analysis of the equation and relevant computer-assisted experiments are presented. The existence of solutions with sharpenings (collapses) and solutions of oscillatory type is discovered. Bibliography: 6 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 13–18.     

Another method for computing the densities of integrals of motion for the Korteweg-de Vries equation

In the first section of this article a new method for computing the densities of integrals of motion for the KdV equation is given. In the second section the variation with respect to q of the functional ∫ ₀ ^π w (x,t,x,;q)dx (t is fixed) is computed, where W(x, t, s; q) is the Riemann function of the problem $$\begin{gathered} \frac{{\partial ^z u}}{{\partial x^2 }} - q(x)u = \frac{{\partial ^2 u}}{{\partial t^2 }} ( - \infty< x< \infty ), \hfill \\ u|_{t = 0} = f(x), \left. {\frac{{\partial u}}{{\partial t}}} \right|_{t = 0} = 0. \hfill \\ \end{gathered} $$      

An initial value problem for a singular parabolic equation of even order

At first Cauchy-problem for the equation: \(L[u(X,t)] \equiv \sum\limits_{i = 1}^n {\frac{{\partial ^2 u}}{{\partial x_1^2 }} + \frac{{2v}}{{\left| X \right|^2 }}} \sum\limits_{i = 1}^n {x_i \frac{{\partial u}}{{\partial x_i }} - \frac{{\partial u}}{{\partial t}} = 0} \) wheren≥1,v—an arbitrary constant,t>0,X=(x ₁, …, x_n)∈E ⁿ/{0}, |X|= =(x ₁ ² +…+x _n ² )^1/2, with 0 being a centre of coordinate system, is studied. Basing on the above, the solution of Cauchy-Nicolescu problem is given which consist in finding a solution of the equationL ^p [u (X, t)]=0, withp∈N subject the initial conditions \(\mathop {\lim }\limits_{t \to \infty } L^k [u(X,t)] = \varphi _k (X)\) ,k=0, 1,…,p?1 and ?_k(X) are given functions.     

Su un problema di tipo nuovo relativo alle equazioni paraboliche d'ordine superiore al secondo

Sunto Si studia il problema della determinazione di una soluzione dell'equazione a_k(x)∂^ku/∂x^k=f(x, y) entro la semistriscia a≤x≤b, y≥0, che assuma assegnati valori per y=0 e per x=a, x₁, x₂, b (a<x₁<x₂<b). Analogamente si studia il problema della determinazione di una soluzione dell' equazione a_k(x)∂^ku/∂x^k+b(x)∂u/∂y=f(x,y), entro la medesima semistriscia, cha assuma assegnati valori per y=0 e per x=a, x₁, x₂, b e la cui ∂/∂y assuma assegnati valori per y=0. A Giovanni Sansone nel suo 70^mo compleanno.     

On the Correctness of the Dirichlet Problem in a Characteristic Rectangle for Fourth Order Linear Hyperbolic Equations

It is proved that the Dirichlet problem is correct in the characteristic rectangle D _ab = [0, a] × [0, b] for the linear hyperbolic equation

The study of boundary-value problems for a singular <Emphasis Type="Italic">B</Emphasis>-elliptic equation by the method of potentials

In this paper we apply the method of potentials for studying the Dirichlet and Neumann boundary-value problems for a B-elliptic equation in the form