The Blow-up Rate for a System of Heat Equations with Non-trivial Coupling at the Boundary

We study the blow-up rate of positive radial solutions of a system of two heat equations, (u₁)_t=Δu₁(u₂)_t=Δu₂, in the ball B(0, 1), with boundary conditions Under some natural hypothesis on the matrix P=(p_ij) that guarrantee the blow-up of the solution at time T, and some assumptions of the initial data u_0i, we find that if ∥x₀∥=1 then u_i(x₀, t) goestoinfinitylike(T−t), where the α_i<0 are the solutions of (P−Id)(α₁,α₂)^t=(−1,−1)^t. As a corollary of the blow-up rate we obtain the loclaization of the blow-up set at the boundary of the domain. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

Global solutions for operator Riccati equations with unbounded coefficients: A non-linear semigroup approach

Let X be a Banach space of real-valued functions on [0, 1] and let ?(X) be the space of bounded linear operators on X. We are interested in solutions R:(0, ∞) → ?(X) for the operator Riccati equation where T is an unbounded multiplication operator in X and the B_i(t)'s are bounded linear integral operators on X. This equation arises in transport theory as the result of an invariant embedding of the Boltzmann equation. Solutions which are of physical interest are those that take on values in the space of bounded linear operators on L¹(0, 1). Conditions on X, R(0), T, and the coefficients are found such that the theory of non-linear semigroups may be used to prove global existence of strong solutions in ?(X) that also satisfy R(t) ? ?(L¹(0,1)) for all t ≥ 0.     

Scattering techniques for a one dimensional inverse problem in geophysics

A one dimensional problem for SH waves in an elastic medium is treated which can be written as v_tt = A^?1 (Av_y)_y, A = (?μ)^1/2, ? = density, and μ = shear modulus. Assume A ? C¹ and A′/A ? L¹; from an input v_y(t, 0) = ?(t) let the response v(t, 0) = g(t) be measured (v(t, y) = 0 for t < 0). Inverse scattering techniques are generalized to recover A(y) for y > 0 in terms of the solution K of a Gelfand-Levitan type equation, .     

On Local Invertible Operators in L2 (ℝ1, H)

We study operators of the form Lu = — G(t) u(t) in L²([t₀ — δ, t₀ + δ], H) with = L² ([t₀— δ, t₀ + δ], H ) in the neighbourhood [t₀— δ, t₀ + δ] of a point t₀ ∈ ℝ¹. Such problems arise in questions on local solvability of partial differential equations (see [6] and [7]). For these operators,one of the major questions is if they are invertible in a neighbourhood of a point t ∈ ℝ¹. To solve this problem we establish needed commutator estimates. Using the commutator estimates and factorization theorems for nonanalytic operator-functions we give additional conditions for the nonanalytic operator -function G(t) and show that the operator L (or ) with some boundary conditions is local invertible.     

Existence for nonoscillatory solutions of higher order nonlinear neutral differential equations

Consider the forced higher-order nonlinear neutral functional differential equation

Existence of singular solutions of a degenerate equation in R 2

Let a₁,a₂, . . . ,a_m ∈ ℝ², 2≤f ∈ C([0,∞)), g_i ∈ C([0,∞)) be such that 0≤g_i(t)≤2 on [0,∞) ∀i=1, . . . ,m. For any p>1, we prove the existence and uniqueness of solutions of the equation u_t=Δ(logu), u>0, in satisfying and logu(x,t)/log|x|→−f(t) as |x|→∞, logu(x,t)/log|x−a_i|→−g_i(t) as |x−a_i|→0, uniformly on every compact subset of (0,T) for any i=1, . . . ,m under a mild assumption on u₀ where We also obtain similar existence and uniqueness of solutions of the above equation in bounded smooth convex domains of ℝ² with prescribed singularities at a finite number of points in the domain.     

Multiplicity results near the principal eigenvalue for boundary‐value problems with periodic nonlinearity

Let us consider the boundary‐value problem where g: ? → ? is a continuous and T ‐periodic function with zero mean value, not identically zero, (λ, a) ∈ ?² and ∈ C [0, π ] with ∫^π ₀ (x) sin x dx = 0. If λ ₁ denotes the first eigenvalue of the associated eigenvalue problem, we prove that if (λ, a) → (λ ₁, 0), then the number of solutions increases to infinity. The proof combines Liapunov–Schmidt reduction together with a careful analysis of the oscillatory behavior of the bifurcation equation. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the convergence rate of vanishing viscosity approximations

Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound ‖u(t, ·) ? u^?(t, ·)‖ = O(1)(1 + t) · |ln ?| on the distance between an exact BV solution u and a viscous approximation u^?, letting the viscosity coefficient ? → 0. In the proof, starting from u we construct an approximation of the viscous solution u^? by taking a mollification u * and inserting viscous shock profiles at the locations of finitely many large shocks for each fixed ?. Error estimates are then obtained by introducing new Lyapunov functionals that control interactions of shock waves in the same family and also interactions of waves in different families. © 2004 Wiley Periodicals, Inc.     

Higher moments for kinetic equations: The Vlasov–Poisson and Fokker–Planck cases

Let us consider a solution f(x,v,t)?L¹(?^2N × [0,T]) of the kinetic equation where |v|^α+1 f_o,|v|^α ?L¹ (?2N × [0, T]) for some α< 0. We prove that f has a higher moment than what is expected. Namely, for any bounded set K_x, we have We use this result to improve the regularity of the local density ρ(x,t) = ∫?dν for the Vlasov–Poisson equation, which corresponds to g = E?, where E is the force field created by the repartition ? itself. We also apply this to the Bhatnagar-Gross-;Krook model with an external force, and we prove that the solution of the Fokker-Pianck equation with a source term in L² belongs to L²([0, T]; H^1/2(?)).     

Multiplicity Results for Second Order Nonlinear Problems with Maximum and Minimum

A functional differential equation of the type where F: C¹(J) → L₁(J) is a unbounded operator, is considered. Sufficient conditions for the existence of at least two different solutions satisfying boundary conditions min{x(t): t ? J} = α, max{x(t): t ? J} = β are given.     

Persistence, contractivity and global stability in logistic equations with piecewise constant delays

We establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium N^*=1/(a+∑_i=0^mb_i) of the following differential equation with piecewise constant arguments:

where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, ∑_i=0^mb_i>0, b_i0, i=0,1,2,…,m, and a+∑_i=0^mb_i>0. These new conditions depend on a,b₀ and ∑_i=1^mb_i, and hence these are other type conditions than those given by So and Yu (Hokkaido Math. J. 24 (1995) 269–286) and others. In particular, in the case m=0 and r(t)≡r>0, we offer necessary and sufficient conditions for the persistence and contractivity of solutions. We also investigate the following differential equation with nonlinear delay terms:

where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, 1−ax−g(x,x,…,x)=0 has a unique solution x^*>0 and g(x₀,x₁,…,x_m)C¹[(0,+∞)×(0,+∞)××(0,+∞)].     

Semilinear wave equation with time dependent potential

We consider the following semilinear wave equation: (1) for (t,x) ∈ ?_t × ?. We prove that if the potential V(t,x) is a measurable function that satisfies the following decay assumption: ∣V(t,x)∣?C(1+t)(1+∣x∣) for a.e. (t,x) ∈ ?_t × ? where C, σ₀>0 are real constants, then for any real number λ that satisfies there exists a real number ρ(f,g,λ)>0 such that the equation has a global solution provided that 0<ρ?ρ(f,g,λ). Copyright © 2004 John Wiley & Sons, Ltd.     

On the oscillation of functional differential equations

In this paper we present certain criteria for the oscillation of functional differential equations of the form where δ = ±1, p, g: [t₀, ∞) → IR, H: [t₀,∞) × IR → IR are continuous, p(t) ≥ 0 for t ≥ t₀ and lim_t → ∞ g(t) — ∞. We like to point out that condition of the form will not be employed.     

Second‐order backward stochastic differential equations and fully nonlinear parabolic PDEs

For a d‐dimensional diffusion of the form dX_t = μ(X_t)dt + σ(X_t)dW_t and continuous functions f and g, we study the existence and uniqueness of adapted processes Y, Z, Γ, and A solving the second‐order backward stochastic differential equation (2BSDE) If the associated PDE has a sufficiently regular solution, then it follows directly from Itô's formula that the processes solve the 2BSDE, where ?? is the Dynkin operator of X without the drift term. The main result of the paper shows that if f is Lipschitz in Y as well as decreasing in Γ and the PDE satisfies a comparison principle as in the theory of viscosity solutions, then the existence of a solution (Y, Z,Γ, A) to the 2BSDE implies that the associated PDE has a unique continuous viscosity solution v and the process Y is of the form Y_t = v(t, X_t), t ∈ [0, T]. In particular, the 2BSDE has at most one solution. This provides a stochastic representation for solutions of fully nonlinear parabolic PDEs. As a consequence, the numerical treatment of such PDEs can now be approached by Monte Carlo methods. © 2006 Wiley Periodicals, Inc.     

On Marcel Riesz's Ultrahyperbolic Kernel

Let t = (t₁, …, t_n) be a point of ?ⁿ. We shall write . We put by definition R_α(u) = u^(α?n)/2/K_n(α); here α is a complex parameter, n the dimension of the space, and K_n(α) is a constant. First we evaluate □R_α(u) = R_α(u), where □ the ultrahyperbolic operator. Then we obtain the following results: R_?2k(u) = □^kδ; R₀(u) = δ; and □^kR_2k(u) = δ, k = 0, 1, …. The first result is the n-dimensional ultrahyperbolic correlative of the well-known one-dimensional formula . Equivalent formulas have been proved by Nozaki by a completely different method. The particular case µ = 1 was solved previously.     

Constant‐sign solutions for singular systems of Fredholm integral equations

We consider the system of Fredholm integral equations where T>0 is fixed and the nonlinearities H_i(t, u₁, u₂, …, u_n) can be singular at t=0 and u_j=0 where j∈{1, 2, …, n}. Criteria are offered for the existence of constant‐sign solutions, i.e. θ_iu_i(t)≥0 for t∈[0, 1] and 1≤i≤n, where θ_i∈{1,?1} is fixed. We also include an example to illustrate the usefulness of the results obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

The method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form , , and (1−t)^α(1+t)^βT(t)T^*(t), with T satisfying T^′=(2Bt+A)T, T(0)=I, T^′=(A+B/t)T, T(1)=I, and T^′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (P_n)_n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F₂, F₁ and F₀ (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F₀ are simultaneously triangularizable (but without making any commutativity assumption).     

Gevrey Asymptotic Representation of the Solutions of Equations with One Turning Point

An ordinary differential equation of the type with parameterξ ? IRⁿ and smooth coefficients a_j,a ? C^∞([-T,T]) is studied. It is assumed that all the characteristic roots of the equation vanish at t = 0 while for t ≠ 0 they are real and distinct. The constructions of real-valued phase functions ?pH_kl (k,l = 1., m) and of amplitude functions A_jkl such that for a given s ? [-T, T] every solution u(t, ξ) of the equation can be represented as where Ψ_j(s, ξ)= D^j_tu(s,ξ), j = 0,m-1 are given.     

Perfect matchings in random uniform hypergraphs

Let ??_k(n, p) be the random k‐uniform hypergraph on V = [n] with edge probability p. Motivated by a theorem of Erd?s and Rényi 7 regarding when a random graph G(n, p) = ??₂(n, p) has a perfect matching, the following conjecture may be raised. (See J. Schmidt and E. Shamir 16 for a weaker version.) Conjecture. Let k|n for fixed k ≥ 3, and the expected degree d(n, p) = p(). Then (Erd?s and Rényi 7 proved this for G(n, p).) Assuming d(n, p)/n^1/2 → ∞, Schmidt and Shamir 16 were able to prove that ??_k(n, p) contains a perfect matching with probability 1 ? o(1). Frieze and Janson 8 showed that a weaker condition d(n, p)/n^1/3 → ∞ was enough. In this paper, we further weaken the condition to A condition for a similar problem about a perfect triangle packing of G(n, p) is also obtained. A perfect triangle packing of a graph is a collection of vertex disjoint triangles whose union is the entire vertex set. Improving a condition p ≥ cn^?2/3+1/15 of Krivelevich 12 , it is shown that if 3|n and p ? n^?2/3+1/18, then © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 111–132, 2003