Conditions for the Existence of Solutions of a Periodic Boundary-Value Problem for an Inhomogeneous Linear Hyperbolic Equation of the Second Order. I

We consider the periodic boundary-value problem u _tt − u _xx = g(x, t), u(0, t) = u(π, t) = 0, u(x, t + ω) = u(x, t). By representing a solution of this problem in the form u(x, t) = u ⁰(x, t) + ũ(x, t), where u ⁰(x, t) is a solution of the corresponding homogeneous problem and ũ(x, t) is the exact solution of the inhomogeneous equation such that ũ(x, t + ω) u x = ũ(x, t), we obtain conditions for the solvability of the inhomogeneous periodic boundary-value problem for certain values of the period ω. We show that the relation obtained for a solution includes known results established earlier. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 912–921, July, 2005.     

Smooth solutions to some differential-difference equations of neutral type

The paper is devoted to the scalar linear differential-difference equation of neutral type

Problem with periodicity conditions for a degenerating equation of mixed type

For the equation K(t)u _xx + u _tt − b ² K(t)u = 0 in the rectangular domain D = “(x, t)‖ 0 < x < 1, −α < t < β”, where K(t) = (sgnt)|t| ^m , m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability of the boundary value problem u(0, t) = u(1, t), u _x (0, t) = u _x (1, t), −α ≤ t ≤ β, u(x, β) = φ(x), u(x,−α) = ψ(x), 0 ≤ x ≤ 1.     

Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation u_t = (uⁿ)_xx + lf(u)/(ò_-1¹ f(u)dx)²{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s ^n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}.     

The Generalized Dirichlet to Neumann Map for the KdV Equation on the Half-Line

For the two versions of the KdV equation on the positive half-line an initial-boundary value problem is well posed if one prescribes an initial condition plus either one boundary condition if q _t and q _xxx have the same sign (KdVI) or two boundary conditions if q _t and q _xxx have opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map for the above problems means characterizing the unknown boundary values in terms of the given initial and boundary conditions. For example, if {q(x,0),q(0,t)} and {q(x,0),q(0,t),q _x (0,t)} are given for the KdVI and KdVII equations, respectively, then one must construct the unknown boundary values {q _x (0,t),q _xx (0,t)} and {q _xx (0,t)}, respectively. We show that this can be achieved without solving for q(x,t) by analysing a certain “global relation” which couples the given initial and boundary conditions with the unknown boundary values, as well as with the function Φ ^(t)(t,k), where Φ ^(t) satisfies the t-part of the associated Lax pair evaluated at x=0. The analysis of the global relation requires the construction of the so-called Gelfand–Levitan–Marchenko triangular representation for Φ ^(t). In spite of the efforts of several investigators, this problem has remained open. In this paper, we construct the representation for Φ ^(t) for the first time and then, by employing this representation, we solve explicitly the global relation for the unknown boundary values in terms of the given initial and boundary conditions and the function Φ ^(t). This yields the unknown boundary values in terms of a nonlinear Volterra integral equation. We also discuss the implications of this result for the analysis of the long t-asymptotics, as well as for the numerical integration of the KdV equation.     

Maximal solutions of semilinear elliptic equations with locally integrable forcing term

We study the existence of a maximal solution of −Δu+g(u)=f(x) in a domain Ω ∈ ℝ ^N with compact boundary, assuming thatf ∈ (L _loc ¹ (Ω))₊ and thatg is nondecreasing,g(0)≥0 andg satisfies the Keller-Osserman condition. We show that if the boundary satisfies the classicalC _1,2 Wiener criterion, then the maximal solution is a large solution, i.e., it blows up everywhere on the boundary. In addition, we discuss the question of uniqueness of large solutions. This research was partially supported by an EC Grant through the RTN Program “Front-Singularities”, HPRN-CT-2002-00274.     

The multi-resolution method applied to the sideways heat equation

We consider the sideways heat equation uxx(x,t)=ut(x,t), 0?x<1, t?0. The solution u(x,t) on the boundary x=1 is a known function g(t). This is an ill-posed problem, since the solution—if it exists—does not depend continuously on the boundary, i.e., small changes on the boundary may result in big changes in the solution. In this paper, we shall use the multi-resolution method based on the Shannon MRA to obtain a well-posed approximating problem and obtain an estimate for the difference between the exact solution and the solution of the approximating problem defined in Vj.     

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.     

Linear periodic boundary-value problem for a second-order hyperbolic equation. I

In three spaces, we obtain exact classical solutions of the boundary-value periodic problem u _tt−a ² u _xx=g(x,t), u(0,t)=u(π,t)=0, u(x,t+T)=u(x,t)=0, x,t∈ĝ Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1537–1544, November, 1998.     

Normal forms for stochastic differential equations

Summary.   We address the following problem from the intersection of dynamical systems and stochastic analysis: Two SDE dx _t = ∑ _{j =0} ^m f _j (x _t )∘dW _t ^j and dx _t =∑ _{j =0} ^m g _j (x _t )∘dW _t ^j in ℝ ^d with smooth coefficients satisfying f _j (0)=g _j (0)=0 are said to be smoothly equivalent if there is a smooth random diffeomorphism (coordinate transformation) h(ω) with h(ω,0)=0 and Dh(ω,0)=id which conjugates the corresponding local flows,

Uniqueness of Solutions of Some Nonlocal Boundary-Value Problems for Operator-Differential Equations on a Finite Segment

For the equation L ₀ x(t) + L ₁ x ⁽¹⁾(t) + ... + L _n x ⁽ⁿ⁾(t) = 0, where L _k, k = 0, 1, ... , n, are operators acting in a Banach space, we formulate conditions under which a solution x(t) that satisfies some nonlocal homogeneous boundary conditions is equal to zero.     

Symmetry breaking for a class of semilinear elliptic problems and the bifurcation diagram for a 1-dimensional problem

We study the behaviour of the positive solutions to the Dirichlet problem IR ⁿ in the unit ball in IR ^R wherep<(N+2)/(N−2) ifN≥3 and λ varies over IR. For a special class of functionsg viz.,g(x)=u ₀ ^p (x) whereu ₀ is the unique positive solution at λ=0, we prove that for certain λ’s nonradial solutions bifurcate from radially symmetric positive solutions. WhenN=1, we obtain the complete bifurcation diagram for the positive solution curve.     

Neumann boundary condition for a non-autonomous Hamilton-Jacobi equation in a quarter plane

We consider Hamilton-Jacobi equation u _t +H(u, u _x ) = 0 in the quarter plane and study initial boundary value problems with Neumann boundary condition on the line x = 0. We assume that p → H(u, p) is convex, positively homogeneous of degree one. In general, this problem need not admit a continuous viscosity solution when s → H(s, p) is non increasing. In this paper, explicit formula for a viscosity solution of the initial boundary value problem is given for the cases s → H(s, p) is non decreasing as well as s → H(s, p) is non increasing.     

Diffusion approximation for the convection-diffusion equation with random drift

We consider the asymptotic behavior of the solutions ofscaled convection-diffusion equations ∂ _t u _ɛ (t, x) = κΔ _x (t, x) + 1/ɛV(t/ɛ²,xɛ) ·∇ _x u _ɛ (t, x) with the initial condition u _ɛ(0,x) = u ₀(x) as the parameter ɛ↓ 0. Under the assumptions that κ > 0 and V(t, x), (t, x) ∈R ^d is a d-dimensional,stationary, zero mean, incompressible, Gaussian random field, Markovian and mixing in t we show that the laws of u _ɛ(t,·), t≥ 0 in an appropriate functional space converge weakly, as ɛ↓ 0, to a δ-type measureconcentrated on a solution of a certain constant coefficient heat equation. Received: 23 March 2000 / Revised version: 5 March 2001 / Published online: 9 October 2001     

Properties of the Flows Generated by Stochastic Equations with Reflection

We consider the properties of a random set ϕ _t (ℝ ₊ ^d ), where ϕ _t (x) is a solution of a stochastic differential equation in ℝ ₊ ^d with normal reflection from the boundary that starts from a point x. We characterize inner and boundary points of the set ϕ _t (ℝ ₊ ^d ) and prove that the Hausdorff dimension of the boundary ∂ϕ _t (ℝ ₊ ^d ) does not exceed d − 1. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1069 – 1078, August, 2005.     

Existence of the mild solution for some fractional differential equations with nonlocal conditions

We are concerned in this paper with the existence of mild solutions to the Cauchy Problem for the fractional differential equation with nonlocal conditions: D ^q x(t)=Ax(t)+t ⁿ f(t,x(t),Bx(t)), t∈[0,T], n∈ℤ⁺, x(0)+g(x)=x ₀, where 0<q<1, A is the infinitesimal generator of a C ₀-semigroup of bounded linear operators on a Banach space X.     

Formula for a solution ofu t +H(u,Du) =g

We study the continuous as well as the discontinuous solutions of Hamilton-Jacobi equationu _t +H(u,Du) =g in ℝ ⁿ x ℝ₊ withu(x, 0) =u ₀(x). The HamiltonianH(s,p) is assumed to be convex and positively homogeneous of degree one inp for eachs in ℝ. IfH is non increasing ins, in general, this problem need not admit a continuous viscosity solution. Even in this case we obtain a formula for discontinuous viscosity solutions.     

Asymptotic behaviour of solutions for the wave equation with an effective dissipation around the boundary

We consider the initial-boundary value problem for the wave equation with a dissipation a(t,x)u_t in an exterior domain, whose boundary meets no geometrical condition. We assume that the dissipation a(t,x)u_t is effective around the boundary and a(t,x) decays as |x|→∞. We shall prove that the total energy does not in general decay, and the solution is asymptotically free as the time goes to infinity. Further, we shall show that the local energy decays like O(t⁻¹) (t→∞).     

Long-time existence for signed solutions of the heat equation with a noise term

Summary.   Let ? be the circle [0,J] with the ends identified. We prove long-time existence for the following equation.