On the behavior of the solution of the Burgers equation as the viscosity goes to zero

The solution of the initial boundary-value problem u_?′ ? ?D²u_? + u_?Du_? = f on (a, b) x(0, T), u_?(a, t) = u_?(b, t) = 0 and u_?(x, 0) = 0 on (a, b), is shown to converge to the solution of the limiting equation as the viscosity tends to zero. Estimates on the rate of convergence are given.     

Control problems governed by Sobolev partial differential equations

Let G be a bounded subset of Rⁿ with a smooth boundary and Q = G × (0, T]. We consider a control problem governed by the Sobolev initial-value problem My_t(u) + Ly(u) = u in L²(Q), y(·, 0; u) = 0 in L₂(G), where M = M(x) and L = L(x) are symmetric uniformly strongly elliptic operators of orders 2m and 2l, respectively. The problem is to find the control u₀ of L²(Q)-norm at most b that steers to within a prescribed tolerance ? of a given function Z in L²(G) and that minimizes a certain energy functional. Our main results establish regularity properties of u₀. We also give results concerning the existence and uniqueness of the optimal control, the controllability of Sobolev initial-value problems, and properties of the Lagrange multipliers associated with the problem constraints.     

Nonlinear parabolic equations with p-growth and unbounded data

The purpose of this paper is to prove the existence of a solution for a nonlinear parabolic equation in the form u_t - div(a(t, x, u, Du)) = H(t, x, u, Du) - div(g(t, x)) in Q_T =]0,T[×Ω, Ω ⊂ R^N, with an initial condition u(0) = u₀, where u₀ is not bounded, |H(t,x, u, ξ)⩽ β|ξ|^p + f(t,x) + βe^λ₁^|u|f, |g|^p/(p-1) ∈ L^r(Q_T) for some r = r{N) ⩾ 1, and - div(a(t,x,u, Du)) is the usual Leray-Lions operator.     

An Asymptotic Analysis of Localized Solutions for Some Reaction-Diffusion Models in Multidimensional Domains

In the limit ? → 0, a spike-layer solution is constructed for the reaction-diffusion equation where b > 0 and D is a bounded convex domain. Here Q(u) is such that there exists a unique radially symmetric function u_c(?^?1 r) satisfying ?²Δu_c + Q(u_c) = 0 in all of ?^N, with u_c(ρ) decaying exponentially at infinity. The spike-layer solution has the form u ~ u_c [?|x ? x₀|], where the spike-layer location x₀ ? D is to be determined subject to the condition that dist(x₀, ?D) as ? → D. The determination of x₀ is shown to be exponentially ill conditioned and asymptotic estimates for the exponentially small eigenvalues and the corresponding eigenfunctions associated with the linearized problem are obtained. These spectral results are used together with a limiting solvability condition to derive an equation for x₀. For a strictly convex domain, it is shown that there is an x₀ that is located at an O(?) distance away from the point in D that is furthest from ?D. Finally, hot-spot solutions to Bratu's equation are constructed asymptotically in a singularly perturbed limit.     

Entrainment of frequency in evolution equations

The existence of periodic solutions near resonance is discussed using elementary methods for the evolution equation ·u = Au + ?f(t, u) when the linear problem is totally degenerate (e^2πA = I) and the period of f is entrained with ? (T = 2π(1 + ?μ)). The approach is to solve the periodicity equation u(T,p,?) = p for an element p(?) in D, the domain of A, as a perturbation from an approximate solution p₀. p₀ is a solution of the nonlinear boundary value problem 2πμAp + ∝₀^2πe^?Asf(s, e^Asp) ds = 0 obtained from the periodicity equation by dividing by ?, applying the entrainment assumption, and letting ? → 0. Once p₀ is known, the conventional inverse function theorem is applied in a slightly unconventional manner. Two particular cases where results are obtained are u_t = u_x + ?{g(u) ? h(t, x)} with g strongly monotone and

$\text{d}\text{dt}\text{v}\text{w}\text{=}\text{0}\text{d}\text{dx}\text{d}\text{dx}\text{0}\text{v}\text{w}\text{+ ?}\text{v}{}^3\text{h(t,x)}$

, where in both cases D is a certain class of 2π-periodic functions of x.     

Existence of positive solutions to some nonlinear equations on locally finite graphs

Let G =(V,E) be a locally finite graph,whose measure μ(x) has positive lower bound,and A be the usual graph Laplacian.Applying the mountain-pass theorem due to Ambrosetti and Rabinowitz(1973),we establish existence results for some nonlinear equations,namely △u+hu=f(x,u),x∈V.In particular,we prove that if h and f satisfy certain assumptions,then the above-mentioned equation has strictly positive solutions.Also,we consider existence of positive solutions of the perturbed equation △u+hu=f(x,u)+∈g.Similar problems have been extensively studied on the Euclidean space as well as on Riemannian manifolds.     

Uniqueness of solutions to an Abel type nonlinear integral equation on the half line

We consider a convolution-type integral equation u = k ? g(u) on the half line (???; a), a ?? ?, with kernel k(x) = x ^???1, 0 < ??, and function g(u), continuous and nondecreasing, such that g(0) = 0 and 0 < g(u) for 0 < u. We concentrate on the uniqueness problem for this equation, and we prove that if ?? ?? (1, 4), then for any two nontrivial solutions u ₁, u ₂ there exists a constant c ?? ? such that u ₂(x) = u ₁(x +c), ??? < x. The results are obtained by applying Hilbert projective metrics.     

Analytic-numerical solutions with a priori error bounds for a class of strongly coupled mixed partial differential systems

This paper deals with the construction of analytic-numerical solutions with a priori error bounds for systems of the type u_t = Au_xx, u(0,t) + u_x(0,t) = 0, Bu(1,t) + Cu_x(1,t) = 0, 0 < x < 1, t > 0, u(x,0) = f(x). Here A, B, C are matrices for which no diagonalizable hypothesis is assumed. First an exact series solution is obtained after solving appropriate vector Sturm-Liouville-type problems. Given an admissible error ε and a bounded subdomain D, after appropriate truncation an approximate solution constructed in terms of data and approximate eigenvalues is given so that the error is less than the prefixed accuracy ε, uniformly in D.     

The asymptotic distribution of weighted empirical distribution functions

Let G_n denote the empirical distribution based on n independent uniform (0, 1) random variables. The asymptotic distribution of the supremum of weighted discrepancies between G_n(u) and u of the forms 6w_v(u)D_n(u)6 and 6w_v(G_n(u))D_n(u)6, where D_n(u) = G_n(u)?u, w_v(u) = (u(1?u))^?1+v and 0 ? v < $\text{1}\text{2}$ is obtained. Goodness-of-fit tests based on these statistics are shown to be asymptotically sensitive only in the extreme tails of a distribution, which is exactly where such statistics that use a weight function w_v with $\text{1}\text{2}$ ? v ? 1 are insensitive. For this reason weighted discrepancies which use the weight function w_v with 0 ? v < $\text{1}\text{2}$ are potentially applicable in the construction of confidence contours for the extreme tails of a distribution.     

Stability and bifurcation of equilibrium solutions of reaction-diffusion equations on an infinite space interval

Under the condition that f(x, y, z, α) and its partial derivatives decay sufficiently fast as $\text{¦x¦ → ∞}$ we will study the (linear) stability and bifurcation of equilibrium solutions of the scalar problem u_t = u_xx + f(x, u, u_x, α), u_x(?∞, t) = u_x(∞, t) = 0 (1) where α is a real bifurcation parameter. After introducing appropriate function spaces X and Y the problem (1) can be rewritten $\text{d}\text{dt}\text{u = G(u, α)}$, (7) where G:X×R → Y is given by G(u, α)(x) = u″(x) + f(x, u(x), u′(x), α). It will be shown, for each (u, α)?X × R, that the Fréchet derivative G_u(u,a): X → Y is not a Fredholm operator. This difficulty is due to the fact that the domain of the space variable x, is infinite and cannot be eliminated by making another choice of X and Y. Since G_u(u, α) is not Fredholm, the hypotheses of most of the general stability and bifurcation results are not satisfied. If (u₀, α₀?S = {(u, α): G(u, α) = 0}, (i.e., (u₀,α₀) is an equilibrium solution of (7)), a necessary condition on the spectrum of G_u(u₀, α₀) for a change in the stability of points in S to occur at G_u(u₀, α₀) will be given. When this condition is met, the principle of exchange of stability which means, in a neighborhood of (u₀, α₀), that adjacent equilibrium solutions for the same α have opposite stability properties in a weakened sense will be established. Also, when G_u or its first order partial derivatives, evaluated at (u₀, α₀), are not too degenerate, the shape of S in a neighborhood of (u₀, α₀) will be described and a strenghtened form of the principle of exchange of stability will be obtained.     

The Cauchy problem for electromagnetic wave propagation in globally perturbed nonselfadjoint media

Let G be a subset of a locally convex separated topological vector space E with int(G) ≠ Ø, cl(G) convex and quasi-complete. Let f: cl(G) → E be a continuous condensing multifunction with compact and convex values and with a bounded range. It is shown that for each w? int(G), there exists a u = u(w) ??(cl(G)) such that p(f(u) ? u) = inf{p(x ? y): x?f(u), y? cl(G)}, where p is the Minkowski's functional of the set (cl(G) ? w). Several fixed point results are obtained as a consequence of this result.     

Applications of the Fourier—Wiener transform to differential equations on infinite dimensional spaces,II

In this paper, we investigate the existence and regularity of solutions for Cauchy problems associated with the following two equations: (1) u_t(x, t) = (?1)^{k + 1}Δ^ku(x, t), and (2) u_tt(x, t) = (?1)^{k + 1}Δ^ku on infinite dimensional spaces. The nonhomogeneous cases associated with (1) and (2) are also considered.     

Scattering length and perturbations of −Δ by positive potentials

This paper extends a result of Fujita [On the blowing up of solutions to the Cauchy problem for u_t = Δu + u^{1 + a}, J. Faculty Science, U. of Tokyo 13 (1966), 109–124] to show that solutions u = u(t, x) for t > 0 and x?R² to the equation u_t = Δu + u² with u(0, x) = a(x) must grow at a rate faster than exp(∥x∥²) at some finite time t, as long as a(x) is nonnegative and not almost everywhere zero.     

Homogenization of Hamilton‐Jacobi‐Bellman equations with respect to time‐space shifts in a stationary ergodic medium

We consider a family {u_? (t, x, ω)}, ? < 0, of solutions to the equation ?u_?/?t + ?Δu_?/2 + H (t/?, x/?, ?u_?, ω) = 0 with the terminal data u_?(T, x, ω) = U(x). Assuming that the dependence of the Hamiltonian H(t, x, p, ω) on time and space is realized through shifts in a stationary ergodic random medium, and that H is convex in p and satisfies certain growth and regularity conditions, we show the almost sure locally uniform convergence, in time and space, of u_?(t, x, ω) as ? → 0 to the solution u(t, x) of a deterministic averaged equation ?u/?t + H?(?u) = 0, u(T, x) = U(x). The “effective” Hamiltonian H? is given by a variational formula. © 2007 Wiley Periodicals, Inc.     

Stable viscosity matrices for systems of conservation laws

A natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension, u₁ + f(u)_x = 0, u?R^m, is studied. These matrices are admissible in the sense that small-amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system u_t + f(u)_x = v(Du_x)_x as ifv → 0 if D is in this class. The class is determined by a linearized stability requirement: The Cauchy problem for the equation u₁ + f′(u₀) u_x = vDu_xx should be well posed in L² uniformly in v as v → 0. Previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion.     

On the generalized Burgers equation

The present paper is concerned with the initial boundary value problem for the generalized Burgers equation u _t + g(t, u)u _x + f(t, u) = εu _xx which arises in many applications. We formulate a condition guaranteeing the a priori estimate of max |u _x | independent of ε and t and give an example demonstrating the optimality of this condition. Based on this estimate we prove the global existence of a unique classical solution of the problem and investigate the behavior of this solution for ε → 0 and t → + ∞. The Cauchy problem for this equation is considered as well.     

On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium

We prove the uniqueness (as well as the existence and regularity) of solutions of the Cauchy problem and of the first and mixed boundary value problems for the equation u_t = φ(u)_xx + b(u)_x. (E) φ and b are assumed to belong to a large class of functions, including, in particular, cases φ(u) = u^m, b(u) = u^λ, m ⩾ 1 and λ > 0.     

Determination of the Unknown Cauchy Data in a Linear Parabolic Problem from the Measured Data at the Final Time

The problem of determining the initial value u(x, 0) = μ ₀(x) in the parabolic equation u _t = (k(x)u _x (x, t)) _x + F(x, t) from the final overdetermination μ _T (x) = u(x, T) is formulated. It is proved that the Fréchet derivative of the cost functional ${{J(\mu_0) = \|\mu_T(x) - u(x, T)\|_0^2}}$ can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. The existence of a quasisolution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method.     

On Dirichlet Type Problems of Polynomial Dirac Equations with Boundary Conditions

Let ${{\bf D}_{\bf x} := \sum_{i=1}^n \frac{\partial}{\partial x_i} e_i}$ be the Euclidean Dirac operator in ${\mathbb{R}^n}$ and let P(X) = a _m X ^m + . . . + a ₁ X +  a ₀ be a polynomial with real coefficients. Differential equations of the form P(D _x )u(x) = 0 are called homogeneous polynomial Dirac equations with real coefficients. In this paper we treat Dirichlet type problems of the a slightly less general form P(D _x )u(x) = f(x) (where the roots are exclusively real) with prescribed boundary conditions that avoid blow-ups inside the domain. We set up analytic representation formulas for the solutions in terms of hypercomplex integral operators and give exact formulas for the integral kernels in the particular cases dealing with spherical and concentric annular domains. The Maxwell and the Klein–Gordon equation are included as special subcases in this context.     

Existence and non‐existence of global solutions for a class of non‐linear wave equations

This paper studies the existence and the non‐existence of global solutions to the initial boundary value problems for the non‐linear wave equation u_tt + u_xxxx = σ(u_x)_x + f(x, t) and the Boussinesq‐type equation u_tt + u_xxxx = σ(u)_xx + f(x, t). The paper proves that every above‐mentioned problem has a unique global solution under rather mild confining conditions, and arrives at some sufficient conditions of blow‐up of the solutions in finite time. Finally, a few examples are given. Copyright © 2000 John Wiley & Sons, Ltd.