On solvability of the Cauchy problem for one quasilinear singular functional-differential equation

We consider the Cauchy problem with zero initial conditions for quasilinear singular functional-differential equation of the second order with a delay at singular summand. We obtain sufficient conditions of solvability of the problem.     

The singular Cauchy problem for a non-linear hyperbolic equation

Summary The author demonstrates the existence of a smooth solution to a singular initial value problem for a quasiliuear hyperbolic equation in two independent variables. The problem is transformed into an equivalent system of integral equations for which a solution is obtained by invoking Schauder’s fixed point theorem. Entrata in Redazione il 19 ottobre 1970.     

Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem

Let Ω be a C1,1-bounded domain in Rn for n?2. In this paper, we are concerned with the asymptotic behavior of the unique positive classical solution to the singular boundary-value problem Δu+a(x)u−σ=0 in Ω, u|∂Ω=0, where σ?0, a is a nonnegative function in , 0<α<1 and there exists c>0 such that . Here λ?2, μk∈R, ω is a positive constant and δ(x)=dist(x,∂Ω).     

Asymptotic behavior of solutions of the Cauchy problem for Burgers type equations

We show that the solutions of the initial value problems for a large class of Burgers type equations approach with time to the sum of appropriately shifted wave-trains and of diffusion waves.

Résumé

Nous montrons que les solutions du problème de Cauchy pour une grande classe d'équations de type de Burgers sont approchées en temps grand vers des sommes d'ondes de diffusion et d'ondes progressives adéquatement translatées.     

Asymptotic behavior of a solution of the Cauchy problem for the generalized damped multidimensional Boussinesq equation

This work studies the Cauchy problem for the generalized damped multidimensional Boussinesq equation. By using a multiplier method, it is proven that the global solution of the problem decays to zero exponentially as the time approaches infinity, under a very simple and mild assumption regarding the nonlinear term.     

Qualitative investigation of a singular Cauchy problem for a functional differential equation

We consider the singular Cauchy problem

Cauchy problem for a second-order hyperbolic operator-differential equation with a singular coefficient

In a Hilbert space, we study the well-posedness of the Cauchy problem for a second-order operator-differential equation with a singular coefficient.     

On the Cauchy problem for a reaction-diffusion equation with a singular nonlinearity

We consider the following Cauchy problem with a singular nonlinearity

(P)

     

ASYMPTOTIC SOLUTION OF SINGULARLY PERTURBED PROBLEM FOR A NONLINEAR SINGULAR EQUATION

The singularly perturbed initial value problem for a nonlinear singular equation is considered. By using a simple and special method the asymptotic behavior of solution is studied.     

Cauchy problem for doubly singular parabolic equation with gradient source

This paper examines the Cauchy problem for doubly singular parabolic equation with a source term depending solely on the gradient. We establish the local and global existence of solutions when initial data is merely a function in (). Moreover, the uniform ‐estimates and gradient estimates of solutions are obtained.     

Asymptotic behavior and representation of solutions to a Volterra kind of equation with a singular kernel

Semigroup Forum - Let A be&nbsp;a densely defined closed, linear $$\omega$$ -sectorial operator of angle $$\theta \in [0,\frac{\pi }{2})$$ on a Banach space X, for some $$\omega \in \mathbb...     

Asymptotic solutions of the Cauchy problem for the singularly perturbed Korteweg-de Vries equation with variable coefficients

We propose an algorithm for the construction of an asymptotic solution of the Cauchy problem for the singularly perturbed Korteweg-de Vries equation with variable coefficients and prove a theorem on the estimation of its precision. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 122–132, January, 2007.     

Asymptotic behavior of solutions of stochastic functional-differential equations with Poisson switchings

We obtain conditions of global asymptotic stability of solutions of stochastic functional-differential equations with Poisson switchings. Translated from Ukrainskii Matematicheskii Zhurnal, Vol.50, No. 6, pp. 845–861, June, 1998.     

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.