An asymptotic solution of the Cauchy problem for the Davey-Stewartson-I equation

We construct and asymptotic solution of the Cauchy problem for the Davey-Stewartson-I equation as t→∞. The solution, which is of order t⁻¹, is rapidly oscillating. The envelope of the oscillations can be defined by an integral equation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 114, No. 1, pp. 104–114, January, 1998.     

Numerical solution of a time-like Cauchy problem for the wave equation

Let D ? ?ⁿ be a bounded domain with piecewise-smooth boundary, and q(x,t) a smooth function on D × [0, T]. Consider the time-like Cauchy problem Given g, h for which the equation has a solution, we show how to approximate u(x,t) by solving a well posed fourth-order elliptic partial differential equation (PDE). We use the method of quasi-reversibility to construct the approximating PDE. We derive error estimates and present numerical results.     

Cauchy problem for a degenerate parabolic equation with non-divergence form

In this article, it is shown that there exists a unique viscosity solution of the Cauchy problem for a degenerate parabolic equation with non-divergence form.     

An asymptotic Cauchy problem for the Laplace equation

The Cauchy problem for the Laplace operator $$\sum\limits_{k = 1}^\infty {\frac{{\left| {\hat f(n_k )} \right|}}{k}} \leqslant const\left\| f \right\|1$$ is modified by replacing the Laplace equation by an asymptotic estimate of the form $$\begin{gathered} \Delta u(x,y) = 0, \hfill \\ u(x,0) = f(x),\frac{{\partial u}}{{\partial y}}(x,0) = g(x) \hfill \\ \end{gathered} $$ with a given majoranth, satisfyingh(+0)=0. Thisasymptotic Cauchy problem only requires that the Laplacian decay to zero at the initial submanifold. It turns out that this problem has a solution for smooth enough Cauchy dataf, g, and this smoothness is strictly controlled byh. This gives a new approach to the study of smooth function spaces and harmonic functions with growth restrictions. As an application, a Levinson-type normality theorem for harmonic functions is proved.     

On a semilinear wave equation: The Cauchy problem and the asymptotic behavior of solutions

The semilinear wave equation

$\text{□u + m}{}^2\text{u + ¦u¦}{}^{\mathrm{p\; ?\; 2}}\text{u(V1 ¦u¦}{}^{\mathrm{p}}\text{) = 0}$

in $\text{Ω= R}{}^3\text{, ?∞ < t < ∞}$, is studied where □ denotes the d'Alembertian operator and 1 means spatial convolution. Under mild assumptions on the real-valued function V and 2 ? p ? 3 the well-posedness of the Cauchy problem is proved. Furthermore, some properties of the solutions of the equation are analyzed such as the asymptotic behavior of local energy as $\text{¦t¦ → + ∞}$ in the case of zero mass. Our results extend that of Perla Menzala and Strauss, where case p = 2 was studied.     

Cauchy problem for a degenerate heat equation in Banach spaces

We study the solvability of a degenerate heat equation with closed linear operators B multiplying the time derivative and A multiplying the Laplace operator in the class of generalized functions in Banach spaces. Under various assumptions on the operator pencil λB-A (it can be Fredholm of index zero, Fredholm, spectrally bounded, sectorial, or radial), we construct the fundamental operator function for the differential operator Bδ′(t) × δ ( $ \bar x $ ) ? Aδ(t) × Δδ ( $ \bar x $ ) and use it for the closed-form construction of the desired generalized solution of the Cauchy problem for the equation in question. We single out uniqueness classes for these solutions and analyze the relationship between continuous and generalized solutions.     

On existence and asymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions

We study the global existence of solutions of the nonlinear degenerate wave equation (ρ?0)

     

Propagation of singularities in the Cauchy problem for a class of degenerate hyperbolic operators

We consider second order degenerate hyperbolic Cauchy problems, the degeneracy coming either from low regularity (less than Lipschitz continuity) of the coefficients with respect to time, or from weak hyperbolicity. In the weakly hyperbolic case, we assume an intermediate condition between effective hyperbolicity and the Levi condition. We construct the fundamental solution and study the propagation of singularities using an unified approach to these different kinds of degeneracy.     

Spectral aspects of regularization of the Cauchy problem for a degenerate equation

We study the Cauchy problem for an equation whose generating operator is degenerate on some subset of the coordinate space. To approximate a solution of the degenerate problem by solutions of well-posed problems, we define a class of regularizations of the degenerate operator in terms of conditions on the spectral properties of approximating operators. We show that the behavior (convergence, compactness, and the set of partial limits in some topology) of the sequence of solutions of regularized problems is determined by the deficiency indices of the degenerate operator. We define an approximative solution of the degenerate problem as the limit of the sequence of solutions of regularized problems and analyze the dependence of the approximative solution on the choice of an admissible regularization.     

Bilateral estimates for the support of a solution of the Cauchy problem for an anisotropic quasilinear degenerate equation

We establish exact-order bilateral estimates for the size of the support of a solution of the Cauchy problem for a doubly nonlinear parabolic equation with anisotropic degeneration in the case where the initial data are finite and have finite mass. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1477–1486, November, 2006.     

The Cauchy problem for a degenerate equation of plate vibration type

We study the Cauchy problem with zero initial conditions for an equation of plate vibration type that is degenerate on the plane of the given initial data. We obtain conditions for the existence and uniqueness of a generalized solution in the case when the rate of degeneration is at least linear.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 79–81.     

The asymptotic expansion of the solution to the Cauchy problem for a parabolic equation in a perforated space

We consider the Cauchy problem for a second-order parabolic equation with rapidly oscillating coefficients of the forma(ɛ ⁻ x,ɛ ^−2k t) in a perforated space ℝ _ɛ ⁿ . We construct a complete asymptotic expansion for the solution of the said problem and obtain an estimate for the remainder term in this expansion. Bibliography: 8 titles. To my dear Teacher, Olga Arsenievna Oleinik This research was supported in part by Grant MIE000 from the International Science Foundation. Translated from Trudy Seminara imeni I. G. Petrovskogo. No. 19, pp. 000-000. 0000.     

The asymptotic behavior of a solution to the boundary-value problem in a degenerate domain with rapidly oscillating boundary

We construct and justify the asymptotics (as ε → +0) of a solution of the mixed boundary-value problem for the Poisson equation in the domain obtained by joining two sets Ω₊ and Ω_- by a large number of thin (of width O (ε)) curvilinear strips (a hub and a rim with a large number of spokes). As a resulting limit problem describing the principal terms of exterior expansions (in Ω_± and in the set ω occupied by the strips) we take the problem of conjugating the partial differential equations and an ordinary differential equation depending on a parameter. Bibliography: 16 titles; Illustrations: 1 figure. Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995, pp. 63–90.     

ON THE CAUCHY PROBLEM OF NONLINEAR DEGENERATE PARABOLIC EQUATION

ＯＮＴＨＥＣＡＵＣＨＹＰＲＯＢＬＥＭＯＦＮＯＮＬＩＮＥＡＲＤＥＧＥＮＥＲＡＴＥＰＡＲＡＢＯＬＩＣＥＱＵＡＴＩＯＮ￥ＹＡＮＧＪＩＮＳＨＵＮＡｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒ，ｗｅｐｒｏｖｅｔｈｅｅｘｉｓｔｅｎｃｅｏｆｓｏｌｕｔｉｏｎｏｆｔｈｅＣａ...     

Cauchy problem for a semilinear wave equation. II

The existence and the uniqueness of a weak solution of the Cauchy problem, for a second-order semilinear pseudodifferential hyperbolic equation on a smooth Riemannian manifold (of dimension n3) without boundary, is proved. In particular, for the equation ü–u+u^p–1=0 one has unique solvability for the Cauchy problem when 1(n+2)/(n–2).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 182, pp. 38–85, 1990.