The asymptotic behavior of solutions of a system of reaction-diffusion equations which models the Belousov-Zhabotinskii chemical reaction

We investigate the boundary value problem $\text{?u}\text{?t}\text{=}\text{?}{}^2\text{u}\text{?x}{}^2\text{+ u(1 ? u ? rv)}$, $\text{?v}\text{?t}\text{=}\text{?}{}^2\text{v}\text{?x}{}^2\text{? buv}$, u(?∞, t) = v(∞, t) = 0, u(∞, t) = 1, and v(?∞, t) = γ ?t > 0 where r > 0, b > 0, γ > 0 and x?R. This system has been proposed by Murray as a model for the propagation of wave fronts of chemical activity in the Belousov-Zhabotinskii chemical reaction. Here u and v are proportional to the concentrations of bromous acid and bromide ion, respectively. We determine the global stability of the constant solution (u, v) ≡ (1,0). Furthermore we introduce a moving coordinate and for each fixed x?R we investigate the asymptotic behavior of u(x + ct, t) and v(x + ct, t) as t → ∞ for both large and small values of the wave speed c ? 0.     

Instability of certain traveling wave solutions to the Fitz Hugh-Nagumo Nerve Axon equations

A study is made of the existence and stability of certain traveling wave solutions to the Fitz Hugh-Nagumo system $\text{?u}\text{?t}\text{=}\text{?}{}^2\text{u}\text{?x}{}^2\text{+ u(u ? a)(1 ? u) ? v,}\text{?v}\text{?t}\text{= bu}$. In particular it is shown that for the wave speed c sufficiently small such waves are unstable in the temporal linearized sense. The study complements recent work of J. M. Greenberg and D. A. Larson.     

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 ? ?$\text{N}$.     

Series expansions of means

A mean M(u, v) is defined to be a homogeneous symmetric function of two positive real variables satisfying min(u, v) ? M(u, v) ? max(u, v) for all u and v. Setting M(u, v) = uM(1, vu^?1) = uM(1, 1 ? t), 0 ? t < 1, we determine power series expansions in t of various generalized means, including $\text{μ}{}_{\mathrm{p}}\text{(1, 1 ? t) = [}\text{1}\text{2}\text{+}\text{(1 ? t)}{}^{\mathrm{p}}\text{2}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}\text{, m}{}_{\mathrm{p}}\text{(u, v) = [}\text{(v}{}^{\mathrm{p\; +\; 1}}\text{? u}{}^{\mathrm{p\; +\; 1}}\text{)}\text{(v ? u)(p + 1)}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}$ (Stolarsky's mean), $\text{M}{}_{\mathrm{p}}\text{(u, v) =}\text{(u}{}^{\mathrm{p}}\text{+ v}{}^{\mathrm{p}}\text{)}\text{(u}{}^{\mathrm{p?\; 1}}\text{+ v}{}^{\mathrm{p\; ?\; 1}}\text{)}$ (Lehmer's mean), $\text{E(r, s; u, v) = [}\text{r(u}{}^{\mathrm{s}}\text{? v}{}^{\mathrm{s}}\text{)}\text{s(u}{}^{\mathrm{r}}\text{? v}{}^{\mathrm{r}}\text{)}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>(s\; ?\; r)</mtext>}}$ (Leach and Sholander's mean), and $\text{G(r, s; u, v) = [}\text{(u}{}^{\mathrm{s}}\text{+ v}{}^{\mathrm{s}}\text{)}\text{(u}{}^{\mathrm{r}}\text{+ v}{}^{\mathrm{r}}\text{)}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>(s\; ?\; r)</mtext>}}$ (Gini's mean). The explicit power series coefficients and recurrence relations for these coefficients are found. Finally, applications are shown by proving a theorem that generalizes one due to Lehmer.     

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.     

Lp estimates for the wave equation

Let u(x, t) be the solution of u_tt ? Δ_xu = 0 with initial conditions $\text{u(x, 0) = g(x)}\text{and}\text{u}{}_{\mathrm{t}}\text{(x, 0) = ?;(x)}$. Consider the linear operator $\text{T: ?; → u(x, t)}$. (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in L^p if and only if . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ < (n ? 1)}{}^{\mathrm{?1}}$. (b) If n = 2k ? 1, the result is valid for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ ? (n ? 1)}$. This result are sharp in the sense that for p such that $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ > (n ? 1)}{}^{\mathrm{?1}}$ we prove the existence of $\text{?; ? L}{}^{\mathrm{P}}$ in such a way that $\text{T?; ? L}{}^{\mathrm{P}}$. Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers $\text{m(ξ) = ψ(ξ) e}{}^{\mathrm{i¦\xi ¦}}\text{¦ ξ ¦}{}^{\mathrm{?b}}$ and finally we get that the convolution against the kernel $\text{K(x) = ?(x)(1 ? ¦ x ¦)}{}^{\mathrm{?1}}$ is bounded in H¹.     

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.     

Superexponential decay of solutions of a semilinear wave equation

Consider a smooth solution of $\text{u}{}_{\mathrm{tt}}\text{? Δu + q(x) ¦ u ¦}{}^{\mathrm{p?1}}\text{u = 0 x ? R}{}^3\text{, q ? 0}$ and is C¹, and 1 < p < 5. Assume that the initial data decay sufficiently rapidly at infinity, $\text{q(x) ? a}\text{exp}\text{(?b ¦ x ¦}{}^{\mathrm{c}}\text{), a, b > 0, c > 1}$, and for simplicity, q_r ? 0. Then the local energy decays faster than exponentially.     

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.     

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 the nonexistence of global solutions to a nonlinear Euler-Poisson-Darboux equation

In this paper, we show that the initial boundary value problem for the (singular) nonlinear EPD (Euler-Poisson-Darboux) equation

does not possess global solutions for arbitrary choices of $\text{u(x, 0). (x ? Ω ? R}{}^{\mathrm{n}}\text{, Ω}\text{bounded}\text{, Δ}{}_{\mathrm{n}}\text{= n}\text{dimensional Laplacian}$) when 0 < k ? 1 for a wide class of nonlinearities $\text{T}$, which includes all the even powers of u and the functions u^{2n + 1}, n = 1, 2,…. The solutions are assumed to vanish on the “walls” of the spacetime cylinder and satisfy $\text{?u}\text{?t(x, 0)}\text{= 0, x ? Ω}$. The result is independent of the space dimension.     

Large time behaviour of the solutions of a semilinear parabolic equation in RN

The asymptotic behaviour as t tends to +∞ of the solution of $\text{(?u}\text{?t)}\text{? Δu + u¦u¦}{}^{\mathrm{p\; ?\; 1}}\text{= 0}$ in $\text{R}$^N × $\text{R}$⁺, p > 1, was studied. It was proved that the behaviour depends strongly on the sign of $\text{(N + 2)}\text{N ? p}$ and also on the rate of decay of the admissible initial data u(0, x) as $\text{¦x¦}$ tends to +∞.     

One-sided continuous dependence of maximal solutions

We consider the maximal solution of $\text{u′(s) = ?(s, u(s))}$, where ? satisfies a one-sided variant of Carathéodory's conditions. A best-possible condition is proved for the dependence of u on ?. Also we show that a function v satisfies $\text{v(t) ? v(r) ? ?}{}_{\mathrm{r}}{}^{\mathrm{t}}\text{?(s, v(s))ds}$ if and only if v is dominated by the maximal solution u.     

Oscillation theory at a finite singularity

It is well known that every weak solution (with boundary values 0) of a semilinear equation $\text{Au + ?(x, u) = g}$ is a regular solution if ? fulfils the growth condition $\text{(1) ¦?(x, u)¦? c ¦u¦}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; ?\; ?</mtext>}}$. Here 2m is the order of A. In this paper we weaken this condition to $\text{c ¦u ¦}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; +\; 1</mtext>}}\text{? ?(x, u)u ? ?c ¦u ?}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; +\; 1\; ?\; ?</mtext>}}$. This requires a technique completely different from that which may be applied in case (1).     

Decay of solutions of nonlinear wave equations in three space dimensions

Given a solution of the Cauchy problem for nonlinear wave equations of the type $\text{?}{}^2\text{u}\text{?t}{}^2\text{? Δu + f(u) = 0}$ in three space dimensions the asymptotic behaviour in time is considered. It is shown that for nonlinearities which behave like powers $\text{¦u¦}{}^{\mathrm{\sigma \; ?\; 1}}\text{u}$ uniform decay holds with a certain rate depending on σ if $\text{5 > σ >}\text{1}\text{2}\text{+}\text{1}\text{2}\text{√13}$, and moreover scattering states exist if σ is not too small. This improves former results of W. A. Strauss (J. Funct. Anal. 2 (1968), 409–457).     

Remarks on an S-shaped bifurcation curve

We study the non-negative solutions of the boundary value problem $\text{?Δu = λ [}\text{exp}\text{αu}\text{(α + u)}\text{]; x ? Ω. u = 0; x ? ?Ω,}\text{where}\text{α > 0, λ ? 0, Ω ? R}{}^{\mathrm{n}}$ is bounded with smooth boundary ?Ω.This problem arises in the theory of combustion. We study the estimates on the supremum norm of the solutions and estimates on the critical values of λ.     

The number of positive integers ≤x and free of prime factors >y

The number defined by the title is denoted by Ψ(x, y). Let $\text{u =}\text{log}\text{x}\text{log}\text{y}$ and let ?(u) be the function determined by ?(u) = 1, 0 ≤ u ≤ 1, u?′(u) = ? ?(u ? 1), u > 1. We prove the following:Theorem. For x sufficiently large and log y ≥ (log log x)², Ψ(x,y) ? x?(u) while for 1 + log log x ≤ log y ≤ (log log x)², and ε > 0, $\text{Ψ(x, y) ?}{}_{\mathrm{\epsilon }}\text{x?(u)}\text{exp}\text{(?u e}\text{xp}\text{(?(}\text{log}\text{y)}{}^{\mathrm{<mtext>(</mtext><mtext>3</mtext><mtext>5</mtext><mtext>\; ?\; \epsilon )</mtext>}}\text{))}$.The proof uses a weighted lower approximation to Ψ(x, y), a reinterpretation of this sum in probability terminology, and ultimately large-deviation methods plus the Berry-Esseen theorem.     

Concrete model theory for a class of operators with unitary part

Let m and v_t, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral $\text{H}$ = ⊕L²(v^t) dm(t) and the operator $\text{(L?)(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{?(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∫}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∫}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ on $\text{H}$, where e(s, t) = exp ∫_s^t ∫_Tdv_λ(θ) dm(λ). Let μ_t be the measure defined by $\text{∫}{}_{\mathrm{T}}\text{?(x) dμ}{}_{\mathrm{t}}\text{(x) = ∫}{}_0{}^{\mathrm{t}}\text{∫}{}_{\mathrm{T}}\text{?(x) dv}{}_{\mathrm{s}}\text{dm(s)}$ for all continuous ?, and let ?_t(z) = exp[?∫ (e^iθ + z)(e^iθ ? z)^?1dμ_t(gq)]. Call {v_t} regular iff for all $\text{t, ¦?}{}_{\mathrm{t}}\text{(e}{}^{\mathrm{i\theta }}\text{)¦ = ¦?}{}_{\mathrm{2\pi }}\text{(e}{}^{\mathrm{i\theta }}\text{)¦}$ for 1 a.e.     

On Lp estimates for the wave equation

New and more elementary proofs are given of two results due to W. Littman: (1) Let $\text{n ? 2, p ?}\text{2n}\text{(n ? 1)}$. The estimate $\text{∫∫ (¦▽u¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{¦}{}^{\mathrm{p}}\text{) dx dt ? C ∫∫ ¦□u¦}{}^{\mathrm{p}}\text{dx dt}$ cannot hold for all u?C₀^∞(Q), Q a cube in $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">n</mi>}}\text{× R}$, some constant C. (2) Let n ? 2, p ≠ 2. The estimate $\text{∫ (¦▽(t)¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{(t)¦}{}^{\mathrm{p}}\text{) dx ? C(t) ∫ (¦▽u(0)¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{(0)¦}{}^{\mathrm{p}}\text{) dx}$ cannot hold for all C^∞ solutions of the wave equation □u = 0 in $\text{R}{}^{\mathrm{n}}\text{x R}$; all t ?$\text{R}$; some function C: $\text{R}$ → $\text{R}$.     

Periodic solutions of nonlinear delay equations

Existence and uniqueness of 2π-periodic solutions of $\text{d}{}^{\mathrm{j}}\text{x(t)}\text{dt}{}^{\mathrm{j}}\text{+}\text{grad}\text{G(x(t ? τ)) = e(t, x(t), x(t ? τ)) (j = 1, 2)}$, where x(t) is in $\text{R}$ⁿ and e(t, u, v) is a given vector function, 2π-periodic in t, are shown under conditions on the spectrum of the Hessian of G. The equation is studied using a fixed point theorem in the space L²(0, 2π). One feature of this approach is that no relationship between the delay and the period is necessary.