Approximate solution of an inhomogeneous abstract differential equation

Recently, we have developed the necessary and sufficient conditions under which a rational function F(hA) approximates the semigroup of operators exp(tA) generated by an infinitesimal operator A. The present paper extends these results to an inhomogeneous equation u′(t) = Au(t) + f(t).     

Nonhomogeneous heat equation: Identification and regularization for the inhomogeneous term

We study the nonhomogeneous heat equation under the form ut−uxx=φ(t)f(x), where the unknown is the pair of functions (u,f). Under various assumptions about the function φ and the final value in t=1, i.e., g(x), we propose different regularizations on this ill-posed problem based on the Fourier transform associated with a Lebesgue measure. For φ?0 the solution is unique.     

The integrable nonlinear degenerate diffusion equationu t=[f(u)u x −1 x and its relatives

The nonlinear diffusion equationu _t=[f(u)g(u _x )] arises in recent models of turbulent transport and of stress dissipation in rock blasting. A Lie point symmetry analysis produces many similarity reductions of exponential and power-law forms, and reveals that for all choices off the equation is always integrable wheng(u _x )=1/u _x . We identify the functionsf(u) which guarantee equivalence to the linear heat equation. For all other choices off, the linear canonical form leads to a self-adjoint differential equation by separation of variablesx andt. We construct a number of explicit solutions with simple boundary conditions, which illustrate behavior in the vicinity of the degenerate region withu _x =. If zero flux and constant concentration are maintained on free boundaries, then steep concentration gradients may evolve from smooth initial conditions. For other boundary conditions, unlike the examples of strong degeneracy, smoothing will occur at initial step discontinuities.     

On the time continuity of entropy solutions

We show that any entropy solution u of a convection diffusion equation ?_t u + div F(u)-Df(u) = b{\partial_t u + {\rm div} F(u)-\Delta\phi(u) =b} in Ω × (0, T) belongs to C([0,T),L¹_loc(W)){C([0,T),L^1_{\rm loc}({\Omega}))} . The proof does not use the uniqueness of the solution.     

The minimum number of subgraphs in a graph and its complement

For a graphb F without isolated vertices, let M(F; n) denote the minimum number of monochromatic copies of F in any 2-coloring of the edges of K_n. Burr and Rosta conjectured that when F has order t, size u, and a automorphisms. Independently, Sidorenko and Thomason have shown that the conjecture is false. We give families of graphs F of order t, of size u, and with a automorphisms where . We show also that the asymptotic value of M(F; n) is not solely a function of the order, size and number of automorphisms of F. © 1929 John Wiley & Sons, Inc.     

KAM tori for higher dimensional beam equation with a fixed constant potential

In this paper, we consider the higher dimensional nonlinear beam equation:utt + △2u + σu + f(u)=0 with periodic boundary conditions, where the nonlinearity f(u) is a real-analytic function of the form f(u)=u3+ h.o.t near u=0 and σ is a positive constant. It is proved that for any fixed σ>0, the above equation admits a family of small-amplitude, linearly stable quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system.     

A symmetry classification for a class of (2+1)-nonlinear wave equation

In this paper, a symmetry classification of a (2+1)-nonlinear wave equation u_tt−f(u)(u_xx+u_yy)=0 where f(u) is a smooth function on u, using Lie group method, is given. The basic infinitesimal method for calculating symmetry groups is presented, and used to determine the general symmetry group of this (2+1)-nonlinear wave equation.     

Frequency modules and nonexistence of quasi-periodic solutions of nonlinear evolution equations

This paper gives lower estimates for the frequency modules of almost periodic solutions to equations of the form , where A generates a strongly continuous semigroup in a Banach space , F(t,x) is 2π-periodic in t and continuous in (t,x), and f is almost periodic. We show that the frequency module ℳ(u) of any almost periodic mild solution u of (*) and the frequency module ℳ(f) of f satisfy the estimate e ^{2π iℳ(f)}⊂e ^{2π iℳ(u)}. If F is independent of t, then the estimate can be improved: ℳ(f)⊂ℳ(u). Applications to the nonexistence of quasi-periodic solutions are also given.     

Probabilistic viscosity algorithm for the scalar conservation law

We derive an algorithm for solving the initial value problem for u_t = ½σ²u_xx + f(u)u_x. The approach is based on the representation of the solution to the above equation in the form of the functional of Brownian motion. For small σ we get the approximation for u_t = f(u)u_x. A comparison with the random choice method is illustrated by the numerical example.     

Beyond quenching for singular reaction-diffusion problems

Let f(u) be twice continuously differentiable on [0, c]) for some constant c such that f(0) > 0,f′ ? 0,f″ ? 0, and lim_u→cf(u) = ∞. Also, let χ(S) be the characteristic function of the set S. This article studies all solutions u with non-negative u_t, in the region where u < c and with continuous u_x for the problem: u_xx – u_t = ? f(u)χ({u < c}), 0 < x < a, 0 < t < ∞, subject to zero initial and first boundary conditions. For any length a larger than the critical length, it is shown that if ∫f(u) du < ∞, then as t tends to infinity, all solutions tend to the unique steady-state profile U(x), which can be computed by a derived formula; furthermore, increasing the length a increases the interval where U(x) ? c by the same amount. For illustration, examples are given.     

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)}.     

Existence and multiplicity of positive solutions for an elastic beam equation

This paper investigates the boundary value problem for elastic beam equation of the form

Circular spectrum and bounded solutions of periodic evolution equations

We consider the existence and uniqueness of bounded solutions of periodic evolution equations of the form u^′=A(t)u+?H(t,u)+f(t), where A(t) is, in general, an unbounded operator depending 1-periodically on t, H is 1-periodic in t, ? is small, and f is a bounded and continuous function that is not necessarily uniformly continuous. We propose a new approach to the spectral theory of functions via the concept of “circular spectrum” and then apply it to study the linear equations u^′=A(t)u+f(t) with general conditions on f. For small ? we show that the perturbed equation inherits some properties of the linear unperturbed one. The main results extend recent results in the direction, saying that if the unitary spectrum of the monodromy operator does not intersect the circular spectrum of f, then the evolution equation has a unique mild solution with its circular spectrum contained in the circular spectrum of f.     

A simultaneous approach to inverse source problem by Green's function

In this paper, we consider an inverse source problem of identification of F(t) function in the linear parabolic equation u_t = u_xx + F(t) and u₀(x) function as the initial condition from the measured final data and local boundary data. Based on the optimal control framework by Green's function, we construct Fréchet derivative of Tikhonov functional. The stability of the minimizer is established from the necessary condition. The CG algorithm based on the Fréchet derivative is applied to the inverse problem, and results are presented for a test example. Copyright © 2014 John Wiley & Sons, Ltd.     

Explosive solutions of elliptic equations with absorption and nonlinear gradient term

Letf be a non-decreasing C¹-function such that andF(t)/f ² a(t)→ 0 ast → ∞, whereF(t)=∫ ₀ ^t f(s) ds anda ∈ (0, 2]. We prove the existence of positive large solutions to the equationΔu +q(x)|Δu| ^a =p(x)f(u) in a smooth bounded domain Ω ⊂R^N, provided thatp, q are non-negative continuous functions so that any zero ofp is surrounded by a surface strictly included in Ω on whichp is positive. Under additional hypotheses onp we deduce the existence of solutions if Ω is unbounded.     

Stability of a traveling-wave solution to the Cauchy problem for the Korteweg-de Vries-Burgers equation

The asymptotic behavior of the solution to the Cauchy problem for the Korteweg-de Vries-Burgers equation u _t + (f(u)) _x + au _xxx − bu _xx = 0 as t → ∞ is analyzed. Sufficient conditions for the existence and local stability of a traveling-wave solution known in the case of f(u) = u ² are extended to the case of an arbitrary sufficiently smooth convex function f(u).     

Action functional and quasi‐potential for the burgers equation in a bounded interval

Consider the viscous Burgers equation u_t + f(u)_x = εu_xx on the interval [0,1] with the inhomogeneous Dirichlet boundary conditions u(t,0) = ρ₀, u(t,1) = ρ₁. The flux f is the function f(u) = u(1 − u), ε > 0 is the viscosity, and the boundary data satisfy 0 < ρ₀ < ρ₁ < 1. We examine the quasi‐potential corresponding to an action functional arising from nonequilibrium statistical mechanical models associated with the above equation. We provide a static variational formula for the quasi‐potential and characterize the optimal paths for the dynamical problem. In contrast with previous cases, for small enough viscosity, the variational problem defining the quasi‐potential admits more than one minimizer. This phenomenon is interpreted as a nonequilibrium phase transition and corresponds to points where the superdifferential of the quasi‐potential is not a singleton. © 2011 Wiley Periodicals, Inc.     

Polynomials Annihilating the Witt Ring

Let F be a non-formally real field of characteristic not 2 and let W(F) be the Witt ring of F. In certain cases generators for the annihilator ideal are determined. Aim the primary decomposition of A(F) is given. For formally d fields F, as an analogue the primary decomposition of A_t(F) = {f(X) ∈ Z[X]| f(ω) = 0 for all ω ∈ W_t(F)}, where W_t(F) is the torsion part of the Witt group, is obtained.     

Solution of mixed problems with boundary elastic-force control for the telegraph equation

In terms of a finite-energy generalized solution of the telegraph equation, for any time interval T, we consider the problem on the boundary elastic-force control u _x (0, t) = μ(t) at the endpoint x = 0 for the process described by the Klein-Gordon-Fock equation under the condition that the other endpoint x = l is either fixed, or free, or is controlled by an elastic force. For any time interval T, we obtain the solution u(x, t) in closed form.     

On a Universality of the Heat Equation

We prove that there are solutions u(t,x) of the heat equation u_t = u_xx such that every continuous function f : [a, b] → ? can be uniformly approximated by a subsequence of u (n, ·), n? ?.