Periodic solutions of a quasilinear parabolic differential equation

This paper treats the quasilinear, parabolic boundary value problem $\text{u}{}_{\mathrm{xx}}\text{? u}{}_{\mathrm{t}}\text{= ??(x, t, u)}$u(0, t) = ?₁(t); u(l, t) = ?₂(t) on an infinite strip $\text{{(x, t) ¦ 0 < x < l, ?∞ < t < ∞}}$ with the functions $\text{?(x, t, u), ?}{}_1\text{(t), ?}{}_2\text{(t)}$ being periodic in t. The major theorem of the paper gives sufficient conditions on $\text{?(x, t, u)}$ for this problem to have a periodic solution u(x, t) which may be constructed by successive approximations with an integral operator. Some corollaries to this theorem offer more explicit conditions on $\text{?(x, t, u)}$ and indicate a method for determining the initial estimate at which the iteration may begin.     

Weak solutions for a nonlinear dispersive equation

In this paper we study the existence, uniqueness, and regularity of the solutions for the Cauchy problem for the evolution equation $\text{u}{}_{\mathrm{t}}\text{+ (f (u))}{}_{\mathrm{x}}\text{? u}{}_{\mathrm{xxt}}\text{= g(x, t), (}{}^1\text{)}\text{where}\text{u = u(x, t), x}$ is in (0, 1), 0 ? t ? T, T is an arbitrary positive real number,f(s)?C¹$\text{R}$, and g(x, t)?L^∞(0, T; L²(0, 1)). We prove the existence and uniqueness of the weak solutions for (¹) using the Galerkin method and a compactness argument such as that of J. L. Lions. We obtain regular solutions using eigenfunctions of the one-dimensional Laplace operator as a basis in the Galerkin method.     

Series inversion of some convolution transforms

We study degeneration for ? → + 0 of the two-point boundary value problems

$\text{τ?}{}^{\mathrm{\pm }}\text{u := ?((au′)′ + bu′ + cu) ± xu′ ? κu = h, u(±1) = A ± B}$

, and convergence of the operators T_?⁺ and T_?^? on $\text{L}$²(?1, 1) connected with them, T_?^±u := τ_?^±u for all

$\text{u?D(T}{}_{\mathrm{?}}{}^{\mathrm{\pm }}\text{, D(T}{}_{\mathrm{?}}{}^{\mathrm{\pm }}\text{) := {u ? L}{}^2\text{(?1, 1) ∣ u″ ? L}{}^2\text{(?1, 1) \&; u(?1) = u(1) = O}, T}{}_0{}^{\mathrm{+}}\text{u: = xu′}$

for all

$\text{u?D(T}{}_{\mathrm{O}}{}^{\mathrm{+}}\text{), D(T}{}_{\mathrm{O}}{}^{\mathrm{+}}\text{) := {u ? L}{}^2\text{(?1, 1) ∣ xu′ ? L}{}^2\text{(?1, 1) \&; u(?1) = u(1) = O}}$

. Here ? is a small positive parameter, λ a complex “spectral” parameter; a, b and c are real $\text{b}$^∞-functions, a(x) ? γ > 0 for all x? [?1, 1] and h is a sufficiently smooth complex function. We prove that the limits of the eigenvalues of T_?⁺ and of T_?^? are the negative and nonpositive integers respectively by comparison of the general case to the special case in which a  1 and b  c  0 and in which we can compute the limits exactly. We show that (T_?⁺ ? λ)^?1 converges for ? → +0 strongly to (T₀⁺ ? λ)^?1 if $\text{R e λ > ?}\text{1}\text{2}$. In an analogous way, we define the operator T_?⁺, n (n ? $\text{N}$ in the Sobolev space H₀^?n(? 1, 1) as a restriction of τ_?⁺ and prove strong convergence of (T⁺_?,n ? λ)^?1 for ? → +0 in this space of distributions if $\text{R e λ > ?n ?}\text{1}\text{2}$. With aid of the maximum principle we infer from this that, if h?$\text{C}$¹, the solution of τ_?⁺u ? λu = h, u(±1) = A ± B converges for ? → +0 uniformly on [?1, ? ?] ∪ [?, 1] to the solution of xu′ ? λu = h, u(±1) = A ± B for each p > 0 and for each λ ? $\text{C}$ if ? ?$\text{N}$.Finally we prove by duality that the solution of τ_?^?u ? λu = h converges to a definite solution of the reduced equation uniformly on each compact subset of (?1, 0) ∪ (0, 1) if h is sufficiently smooth and if 1 ? ?$\text{N}$.     

A constructive approach to the solution of a class of boundary value problems of mixed type

In this article we discuss the solution of boundary value problems which are described by the linear integrodifferential equation $\text{?xu}\text{?t}\text{(t, x) + u(t, x) ?}\text{1}\text{π}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{∝}{}_{\mathrm{?\infty }}{}^{\mathrm{\infty }}\text{exp}\text{(?y}{}^2\text{) u(t, y) dy = 0}$, where t∈J?$\text{R}$, x∈$\text{R}$. We interpret the equation in functional form as an ordinary differential equation for the mapping u:J→L₂(R,μ), where L₂(R,μ) is a weighted L₂-space. Emphasis is on the constructive aspects of the solution and on finding representations of the relevant isomorphisms.     

On the null-controllability of nonlinear delay systems with restrained controls

Sufficient conditions are developed for the null-controllability of the nonlinear delay process (1) $\text{x}\text{?}\text{(t) = L(t, x}{}_{\mathrm{t}}\text{) + B(t) u(t) + f(t, x}{}_{\mathrm{t}}\text{, u(t))}$ when the values of the control functions u lie in an m-dimensional unit cube C^m of E^m. Conditions are placed on f which guarantee that if the uncontrolled system $\text{x}\text{?}\text{(t) = L(t, x}{}_{\mathrm{t}}\text{)}$ is uniformly asymptotically stable and if the linear control system x(t) = L(t, x_t) + B(t) u(t) is proper, then (1) is null-controllable.     

Asymptotic properties of general symmetric hyperbolic systems

Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem $\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0, u(0, x) = ?(x)}$. Here the A_j's are constant, k × k Hermitian matrices, x = (x₁,…, x_n), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit $\text{E}{}_{\mathrm{M}}\text{(?)}\text{as}\text{¦ t ¦ → ∞}$, where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U₀(t) and such that $\text{U}{}_0\text{(t)? = 0}\text{for}\text{¦ x ¦ ? a ¦ t ¦ ? R, a}\text{and}\text{R}$ depending on ? and that the local energy of nonstatic solutions decays as $\text{¦ t ¦ → ∞}$. More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation $\text{E(x)}\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0,}\text{where}\text{¦ E(x) ? I ¦ = O(¦ x ¦}{}^{\mathrm{?1\; ?\; ?}}\text{)}$ at infinity.     

On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations

Asymptotic properties of solutions of the nonlinear Klein-Gordon equation ?_t²u ? Δu + m²u + f(u) = 0 (NLKG) $\text{u¦}{}_0\text{= θ}$, $\text{?}{}_{\mathrm{t}}\text{u¦}{}_0\text{= Ψ}$, are investigated, which are inherited from the corresponding solutions v of the (linear) Klein-Gordon equation ?_t²v ? Δv + m²v = 0$\text{v¦}{}_0\text{= θ}$, $\text{?}{}_{\mathrm{t}}\text{v¦}{}_0\text{= Ψ}$, (KG) In particular, the finiteness of time-integrals in L_q over R₊ of certain Sobolevnorms in space of the solution is proved to be such a hereditary property. Together with a device by W. A. Strauss and a weak decay result for the (KG) due to R. S. Strichartz, this is used to prove that under suitable restrictions on the nonlinearity, the scattering operator for the (NLKG) is defined on all of L₂¹ × L₂ for n = 3.     

An abstract averaging theorem

For each t ? 0, let A(t) generate a contraction semigroup on a Banach space L. Suppose the solution of u_t = ?A(t)u is given by an evolution operator V_?(t, s). Conditions are given under which $\text{V}{}_{\mathrm{?}}\text{(}\text{(t+s)}\text{?}\text{,}\text{s}\text{?}\text{)}$ converges strongly as ? → 0 to a semigroup T(t) generated by the closure of $\text{A}\text{?}\text{f ≡}\text{lim}{}_{\mathrm{T\to \infty }}\text{(}\text{1}\text{T}\text{) ∝}{}_0{}^{\mathrm{T}}\text{A(t)f dt}$.This result is applied to the following situation: Let B generate a contraction group S(t) and the closure of ?A + B generate a contraction semigroup S_?(t). Conditions are given under which $\text{S(?}\text{t}\text{?}\text{) S}{}_{\mathrm{?}}\text{(}\text{t}\text{?}\text{)}$ converges strongly to a semigroup generated by the closure of $\text{A}\text{?}\text{f ≡}\text{lim}{}_{\mathrm{T\to \infty }}\text{(}\text{1}\text{T}\text{) ∝ S(?t) AS(t)f dt}$. This work was motivated by and generalizes a result of Pinsky and Ellis for the linearized Boltzmann Equation.     

Asymptotic stability of some quasilinear parabolic equations in divergence form. L∞ results

In this paper we study the behavior of solutions of some quasilinear parabolic equations of the form

$\text{(}\text{?u}\text{?t}\text{) ?}\text{∑}\text{i=1}\text{n}\text{(}\text{d}\text{dx}{}_{\mathrm{i}}\text{) a}{}_{\mathrm{i}}\text{(x, t, u, u}{}_{\mathrm{x}}\text{) + a(x, t, u, u}{}_{\mathrm{x}}\text{)u + f(x, t) = O,}$

as t → ∞. In particular, the solutions of these equations will decay to zero as t → ∞ in the L^∞ norm.     

Convergence of the Kato approximants for evolution equations involving functional perturbations

The existence of a unique strong solution of the nonlinear abstract functional differential equation $\text{u′(t) + A(t)u(t) = F(t,u}{}_{\mathrm{t}}\text{), u}{}_0\text{= φεC}{}^1\text{(¦?r,0¦,X),tε¦0, T¦}$, (E) is established. X is a Banach space with uniformly convex dual space and, for $\text{t? ¦0, T¦, A(t)}$ is m-accretive and satisfies a time dependence condition suitable for applications to partial differential equations. The function F satisfies a Lipschitz condition. The novelty of the paper is that the solution u(t) of (E) is shown to be the uniform limit (as n → ∞) of the sequence u_n(t), where the functions u_n(t) are continuously differentiate solutions of approximating equations involving the Yosida approximants. Thus, a straightforward approximation scheme is now available for such equations, in parallel with the approach involving the use of nonlinear evolution operator theory.     

Some asymptotic boundary behavior of solutions of non-linear parabolic initial-boundary value problems

For parabolic initial boundary value problems various results such as $\text{lim}{}_{\mathrm{t\; \downarrow \; 0}}\text{{}\text{(}\text{?u}\text{t6x}\text{)(0, t)}\text{(}\text{?u}{}_{\mathrm{\alpha }}\text{?x}\text{)(0, t)}}\text{= 1}$, where u satisfies $\text{?u}\text{?t}\text{= a(u)(}\text{?}{}^2\text{u}\text{?x}{}^2\text{), 0 < x < 1, 0 < t ? T, u(x, 0) = 0, u(0, t) = |}{}_1\text{(t), 0 < t ? T, u(1, t) = |}{}_2\text{(t), 0 < t ? T, u}{}_{\mathrm{\alpha }}\text{satisfies}\text{(}\text{?u}{}_{\mathrm{\alpha }}\text{?t}\text{) = α(}\text{?}{}^2\text{u}{}_{\mathrm{\alpha }}\text{?x}{}^2\text{), 0 < x < 1, 0 < t ? T, u}{}_{\mathrm{\alpha }}\text{(x, 0) = 0, u}{}_{\mathrm{\alpha }}\text{(0, t) = |}{}_1\text{(t), 0 < t ? T, u}{}_{\mathrm{\alpha }}\text{(1, t) = |}{}_2\text{(t), 0 < t ? T,}\text{and}\text{α = a(0)}$, are demonstrated via the maximum principle and potential theoretic estimates.     

The Cauchy problem for a linear parabolic partial differential equation

Numerical approximation of the solution of the Cauchy problem for the linear parabolic partial differential equation is considered. The problem: (p(x)u_x)_x ? q(x)u = p(x)u_t, 0 < x < 1,0 < t? T; $\text{u(0, t) = ?}{}_1\text{(t), 0 < t ? T}$; $\text{u(1,t) = ?}{}_2\text{(t), 0 < t ? T}$; p(0) u_x(0, t) = g(t), 0 < t₀ ? t ? T, is ill-posed in the sense of Hadamard. Complex variable and Dirichlet series techniques are used to establish Hölder continuous dependence of the solution upon the data under the additional assumption of a known uniform bound for ¦ u(x, t)¦ when 0 ? x ? 1 and 0 ? t ? T. Numerical results are obtained for the problem where the data ?₁, ?₂ and g are known only approximately.     

Boundary layer methods for certain nonlinear singularly perturbed optimal control problems

We shall examine the control problem consisting of the system $\text{dx}\text{dt}\text{= f}{}_1\text{(x, z, u, t, ?)}$$\text{?(}\text{dz}\text{dt}\text{) = f}{}_2\text{(x, z, u, t, ?)}$ on the interval 0 ? t ? 1 with the initial values x(0, ?) and z(0, ?) prescribed, where the cost functional J(?) = π(x(1, ?), z(1, ?), ?) + ∝₀¹V(x(t, ?), z(t, ?), u(t, ?), t, ?) dt is to be minimized. We shall restrict attention to the special problem where the f_i's are linear in z and u, V is quadratic in z and independent of z when ? = 0, π and V are positive semidefinite functions of x and z, and V is a positive definite function of u. Under appropriate conditions, we shall obtain an asymptotic solution of the problem valid as the small parameter ? tends to zero. The techniques of constructing such asymptotic expansions will be stressed.     

Analyticity of solutions of nonlinear evolution equations

Analyticity in t of solutions u(t) of nonlinear evolution equations of the form $\text{u′ + A(t, u)u = ?(t, u), t > 0, u(0) = u}{}_0$, is established under suitable conditions on $\text{A(t, u), ?(t, u),}\text{and}\text{u}{}_0$. An application is given to quasilinear parabolic equations.     

Asymptotic behavior for an almost periodic,strongly dissipative wave equation

This paper deals with asymptotic behavior for (weak) solutions of the equation $\text{u}{}_{\mathrm{tt}}\text{? Δu + β(u}{}_{\mathrm{t}}\text{) ? ?(t, x)}$, on $\text{R}$⁺ × Ω; u(t, x) = 0, on $\text{R}$⁺ × ?Ω. If $\text{?∈L∞(R}{}^{\mathrm{+}}\text{,L}{}^2\text{(Ω))}$ and β is coercive, we prove that the solutions are bounded in the energy space, under weaker assumptions than those used by G. Prouse in a previous work. If in addition $\text{?}{}_{\mathrm{t}}\text{∈S}{}^2\text{(R}{}^{\mathrm{+}}\text{,L}{}^2\text{(Ω))}$ and ? is srongly almost-periodic, we prove for strongly monotone β that all solutions are asymptotically almost-periodic in the energy space. The assumptions made on β are much less restrictive than those made by G. Prouse: mainly, we allow β to be multivalued, and in the one-dimensional case β need not be defined everywhere.     

The Tjon-Wu equation in Banach space settings

The authors investigate the Tjon-Wu (TW) equation: (TW)

$\text{?u}\text{?t}\text{(t, x) + u(t, x) = ∫}{}_{\mathrm{x}}{}^{\mathrm{\infty }}\text{dy}\text{y}\text{∫}{}_0{}^{\mathrm{y}}\text{u(t, y ? z) u(t, z)dz, u(0, x) = u}{}_0\text{(x)}$

, which has been obtained from a classical Boltzmann equation by applying the Abel transform. (TW) is considered as an ordinary differential equation first in the space L²={u:[0,∞)→R|∫_x^∞|u(x)|²e^xdx < + ∞}The authors establish existence and uniqueness of solutions in disks of codimension 2 around 0 and around e^?x. Asymptotic stability of these latter functions is also established. The basic tool is an unusual eigenvalue property of the nonlinear right-hand side of (TW) which leads to a reformulation of (TW) as a differential equation in l². Similar results are established in L¹ working with (TW) directly.     

Adiabatic invariants for linear Hamiltonian systems

n independent adiabatic invariants in involution are found for a slowly varying Hamiltonian system of order 2n × 2n. The Hamiltonian system considered is $\text{?}\text{u}\text{?}\text{= A(t)u}\text{as}\text{? → 0}{}^{\mathrm{+}}$, where A(t) is a 2n × 2n real matrix with distinct, pure imaginary eigen values for each t? [?∞, ∞], and $\text{d}{}^{\mathrm{(j)}}\text{A}\text{dt}{}^{\mathrm{(j)}}\text{? L}{}_{\mathrm{j}}\text{(?∞, ∞)}$, for all j > 0. The adiabatic invariants I_s(u, t), s = 1,…, n are expressed in terms of the eigen vectors of A(t). Approximate solutions for the system to arbitrary order of ? are obtained uniformly for t? [?∞, ∞].     

On the control of certain interacting populations

We study the time optimal control of the system $\text{x}\text{?}{}_1\text{= x}{}_1\text{?}{}_1\text{(x}{}_1\text{, x}{}_2\text{) + u}{}_1\text{(t) g}{}_1\text{(x}{}_1\text{),}\text{x}\text{?}{}_2\text{= x}{}_2\text{?}{}_2\text{(x}{}_1\text{, x}{}_2\text{) + u}{}_2\text{(t)g}{}_2\text{(x}{}_2\text{)}$, where x₁ is the size of the population of one species, x₂ is the population size of the second species, ?₁ and ?₂ are the fractional growth rates of the respective species, g₁ and g₂ are nowhere vanishing functions of class C¹(0, + ∞), and the control u(t) = (u₁(t), u₂(t)) takes on values in a closed rectangle. The functions ?₁ and ?₂ are chosen to represent prey-predator, competitive, and symbiotic interactions.We show, for the various interactions, that a time optimal control, if it exists, must be “bang-bang,” and give sufficient conditions for the controllability, and for the existence, of time optimal controls of the above system.     

Square roots of elliptic operators

Consider an elliptic sesquilinear form defined on $\text{V}$ × $\text{V}$ by $\text{J[u, v] = ∫}{}_{\mathrm{\Omega }}\text{a}{}_{\mathrm{jk}}\text{?u}\text{?x}{}_{\mathrm{k}}\text{}\text{?}\text{t6v}\text{?x}{}_{\mathrm{j}}\text{+ a}{}_{\mathrm{k}}\text{?u}\text{?x}{}_{\mathrm{k}}\text{v}\text{?}\text{+ α}{}_{\mathrm{j}}\text{u}\text{}\text{?}\text{t6v}\text{?x}{}_{\mathrm{j}}\text{+ au}\text{v}\text{?}\text{dx}$, where $\text{V}$ is a closed subspace of $\text{H}{}^1\text{(Ω)}$ which contains $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$, Ω is a bounded Lipschitz domain in $\text{R}$ⁿ, $\text{a}{}_{\mathrm{jk}}\text{, a}{}_{\mathrm{k}}\text{, α}{}_{\mathrm{j}}\text{, a ? L}{}_{\mathrm{\infty }}\text{(Ω),}\text{and Re}\text{a}{}_{\mathrm{jk}}\text{ζ}{}_{\mathrm{k}}\text{ζ}{}_{\mathrm{j}}\text{? κ > 0}$ for all ζ?$\text{C}$ⁿ with ¦ζ¦ = 1. Let L be the operator with largest domain satisfying J[u, v] = (Lu, v) for all υ∈$\text{V}$. Then L + λI is a maximal accretive operator in $\text{L}{}_2\text{(Ω)}$ for λ a sufficiently large real number. It is proved that $\text{(L + λI)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is a bounded operator from $\text{V}$ to $\text{L}{}_2\text{(Ω)}$ provided mild regularity of the coefficients is assumed. In addition it is shown that if the coefficients depend differentiably on a parameter t in an appropriate sense, then the corresponding square root operators also depend differentiably on t. The latter result is new even when the forms J are hermitian.