Extinction of solutions for some nonlinear parabolic equations

We are dealing with the first vanishing time for solutions of the Cauchy–Neumann problem for the semilinear parabolic equation ∂ _t u − Δu + a(x)u ^q = 0, where a(x) \geqslant d₀exp( - \fracw( | x | )| x |² ) a(x) \geqslant {d_0}\exp \left( { - \frac{{\omega \left( {\left| x \right|} \right)}}{{{{\left| x \right|}^2}}}} \right) , d ₀ > 0, 1 > q > 0, and ω is a positive continuous radial function. We give a Dini-like condition on the function ω which implies that any solution of the above equation vanishes in finite time. The proof is derived from semi-classical limits of some Schr¨odinger operators.     

Nonlinear filtering and large deviations:a pde-control theoretic approach

We consider the asymptotic nonlinear filtering problem dx=f(x)dt + ?^1/2 dw,dy=h(x) dt + ? dv and obtain lim_?→0 ? log q ²(x,t) = -W(x,t) for unnormalized conditional densities q ²(x,t) using PDE methods. HereW(x,t) is the value function for a deterministic optimal control problem arising in Mortensen's deterministic estimation, and is the unique viscosity solution of a Hamilton-Jacobi-Bellman equation. ijab has also studied this filtering problem, and we extend his large deviation result for certain unnormalized conditional measures. The resulting variational problem corresponds to the above control problem     

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.     

Boundedness in nonlinear oscillations

In this paper, the boundedness of all solutions of the nonlinear differential equation (φ_p(x′))′ + αφ_p(x⁺) – βφ_p(x^–) + f(x) = e(t) is studied, where φ_p(u) = |u|^p–2 u, p ≥ 2, α, β are positive constants such that = 2w^–1 with w ∈ ?⁺\?, f is a bounded C⁵ function, e(t) ∈ C⁶ is 2π_p‐periodic, x⁺ = max{x, 0}, x^– = max{–x, 0}. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characterization of polyanalytic functions by meromorphic extensions from chains of circles

Let C _t = {z ∈ ℂ: |z − c(t)| = r(t), t ∈ (0, 1)} be a C ¹-family of circles in the plane such that lim _t→0+ C _t = {a}, lim _t→1− C _t = {b}, a ≠ b, and |c′(t)|² + |r′(t)|² ≠ 0. The discriminant set S of the family is defined as the closure of the set {c(t) + r(t)w(t), t ∈ [0, 1]}, where w = w(t) is the root of the quadratic equation ̅c′(t)w ² + 2r′(t)w + c′(t) = 0 with |w| < 1, if such a root exists.     

Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation u_t + Lu + a(x) |u|^q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb R^N{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò₀¹ s^-1 (meas\nolimits {x ? W: |a(x)| ￡ s })^q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.     

Global existence and time decay of small solutions to the landau-ginzburg type equations

We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|^2/n u = 0,x ∃ Rⁿ,t } 0,u(0,x) = u0(x),x ∃ Rⁿ, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = |∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|^n-1n^n/2 / ((n + 1)|α|² + α^{2 n/2}. Furthermore, we assume that the initial data u₀ ∃ L^p are such that (1 + |x|)^αu0 ∃ L¹, with sufficiently small norm ∃ = (1 + |x|)^α u₀ 1 + u₀ p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L^∞) ∩ C ([0, ∞); L¹ ∩ L^p) satisfying the time decay estimates for allt0 u(t)||_∞ Cɛt^-n/2(1 + η log 〈t〉)^-n/2, if hg = θ^2/n 2_π ℜδ(α, Β) 0; u(t)||_∞ Cɛt^-n/2(1 + Μ log 〈t〉)^-n/4, if η = 0 and Μ = θ^4/n 4π)² (ℑδ(α, Β))² ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||_∞ Cɛt^-n/2(1 + κ log 〈t〉)^-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31).     

On the multiplicity of the eigenvalues of a graph

Given a graph G with characteristic polynomial ϕ(t), we consider the ML-decomposition ϕ(t) = q ₁(t)q ₂(t)² ... q _m (t)^m, where each q _i (t) is an integral polynomial and the roots of ϕ(t) with multiplicity j are exactly the roots of q _j (t). We give an algorithm to construct the polynomials q _i (t) and describe some relations of their coefficients with other combinatorial invariants of G. In particular, we get new bounds for the energy E(G) = |λ_i| of G, where λ₁, λ₂, ..., λ_n are the eigenvalues of G (with multiplicity). Most of the results are proved for the more general situation of a Hermitian matrix whose characteristic polynomial has integral coefficients. This work was done during a visit of the second named author to UNAM.     

Extinction and non-extinction for a polytropic filtration equation with a nonlocal source

In this article, the authors establish the conditions for the extinction of solutions, in finite time, of the fast diffusive polytropic filtration equation u _t ?=?div(|?u ^m | ^p?2?u ^m )?+?a∫_Ω u ^q (y,?t)dy with a, q, m?>?0, p?>?1, m(p???1)?R ^N (N?>?2). More precisely speaking, it is shown that if q?>?m(p???1), any non-negative solution with small initial data vanishes in finite time, and if 0?q?m(p???1), there exists a solution which is positive in Ω for all t?>?0. For the critical case q?=?m(p???1), whether the solutions vanish in finite time or not depends on the comparison between a and μ, where μ?=?∫?_Ωφ ^p?1(x)dx and φ is the unique positive solution of the elliptic problem ?div(|?φ| ^p?2?φ)?=?1, x?∈?Ω; φ(x)?=?0, x?∈??Ω.     

The Generalized Dirichlet to Neumann Map for the KdV Equation on the Half-Line

For the two versions of the KdV equation on the positive half-line an initial-boundary value problem is well posed if one prescribes an initial condition plus either one boundary condition if q _t and q _xxx have the same sign (KdVI) or two boundary conditions if q _t and q _xxx have opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map for the above problems means characterizing the unknown boundary values in terms of the given initial and boundary conditions. For example, if {q(x,0),q(0,t)} and {q(x,0),q(0,t),q _x (0,t)} are given for the KdVI and KdVII equations, respectively, then one must construct the unknown boundary values {q _x (0,t),q _xx (0,t)} and {q _xx (0,t)}, respectively. We show that this can be achieved without solving for q(x,t) by analysing a certain “global relation” which couples the given initial and boundary conditions with the unknown boundary values, as well as with the function Φ ^(t)(t,k), where Φ ^(t) satisfies the t-part of the associated Lax pair evaluated at x=0. The analysis of the global relation requires the construction of the so-called Gelfand–Levitan–Marchenko triangular representation for Φ ^(t). In spite of the efforts of several investigators, this problem has remained open. In this paper, we construct the representation for Φ ^(t) for the first time and then, by employing this representation, we solve explicitly the global relation for the unknown boundary values in terms of the given initial and boundary conditions and the function Φ ^(t). This yields the unknown boundary values in terms of a nonlinear Volterra integral equation. We also discuss the implications of this result for the analysis of the long t-asymptotics, as well as for the numerical integration of the KdV equation.     

On the approximate behavior of the posterior distribution for an extreme multivariate observation

The behavior of the posterior for a large observation is considered. Two basic situations are discussed; location vectors and natural parameters.Let X = (X₁, X₂, …, X_n) be an observation from a multivariate exponential distribution with that natural parameter Θ = (Θ₁, Θ₂, …, Θ_n). Let θ_x^* be the posterior mode. Sufficient conditions are presented for the distribution of Θ − θ_x^* given X = x to converge to a multivariate normal with mean vector 0 as |x| tends to infinity. These same conditions imply that E(Θ | X = x) − θ_x^* converges to the zero vector as |x| tends to infinity.The posterior for an observation X = (X₁, X₂, …, X_n is considered for a location vector Θ = (Θ₁, Θ₂, …, Θ_n) as x gets large along a path, γ, in Rⁿ. Sufficient conditions are given for the distribution of γ(t) − Θ given X = γ(t) to converge in law as t → ∞. Slightly stronger conditions ensure that γ(t) − E(Θ | X = γ(t)) converges to the mean of the limiting distribution.These basic results about the posterior mean are extended to cover other estimators. Loss functions which are convex functions of absolute error are considered. Let δ be a Bayes estimator for a loss function of this type. Generally, if the distribution of Θ − E(Θ | X = γ(t)) given X = γ(t) converges in law to a symmetric distribution as t → ∞, it is shown that δ(γ(t)) − E(Θ | X = γ(t)) → 0 as t → ∞.     

A footnote to mean square value of DirichletL-series

Following appropriate use of approximate functional equation for Hurwitz Zeta function, we obtain upper bounds for } Here fors = σ + it, L(s,x) denotes DirichletL-series for character x(modq). In particular, we obtain S(1/2 +it) ≪q logqt + t^5/8 q^−1/8, which is an improvement in the range q |t| < q^11/7, on hitherto best known result. This incidentally gives S(1/2+ it)≪ q log3q for |t|q^9/5.     

Multiparameter acoustic imaging in the born approximation

Samples of biological tissue are modelled as inhomogeneous fluids with density ?(X) and sound speed c(x) at point x. The samples are contained in the sphere |x| ? δ and it is assumed that ?(x) ? ?₀ = 1 and c(x) ? c₀ = 1 for |x| ? δ, and |γ_n(x)| ? 1 and |?γ?(x)| ? 1 where γ?(x) = ?(x) ? 1 and γ_n(x) = c^?2(x) ? 1. The samples are insonified by plane pulses s(x · θ₀ – t) where x = |θ₀| = 1 and the scattered pulse is shown to have the form |x|^?1 e_s(|x| – t, θ, θ₀) in the far field, where x = |x| θ. The response e_s(τ, θ, θ₀) is measurable. The goal of the work is to construct the sample parameters γn and γ? from e_s(τ, θ, θ₀) for suitable choiches of s, θ and θ₀. In the limiting case of constant density: γ?(x)_? 0 it is shown that Where δ represents the Dirac δ and S² is the unit sphere |θ| = 1. Analogous formulas, based on two sets of measurements, are derived for the case of variable c(x) and ?(x).     

On the Oscillation of Second Order Neutral Differential Equations of Emden-Fowler Type

Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type (a(t)x￠(t))￠+q₁(t)| y(t-s₁)|^a sgn y(t-s₁) +q₂(t)| y(t-s₂)|^b sgn y(t-s₂)=0,    t 3 t₀,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0, where x(t) = y(t) + p(t)y(t − τ), τ, σ₁ and σ₂ are nonnegative constants, α > 0, β > 0, and a, p, q ₁, q₂ ? C([t₀, ￥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R}) . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.     

Moser's second one-dimensional inequality for a class of integral operators

Suppose that $1 < p < \infty $1 < p < \infty , q=p/(p-1)q=p/(p-1), and for non-negative f ? L^p(-￥ ,￥)f\in L^p(-\infty\! ,\infty ) and any real x we let F(x)-F(0)=ò₀^xf(t) dtF(x)-F(0)=\int _0^xf(t)\ dt; suppose in addition that ò_-￥^￥ F(t)exp(-|t|) dt=0\int\limits _{-\infty }^\infty F(t)\exp (-|t|)\ dt=0. Moser's second one-dimensional inequality states that there is a constant C_pC_p, such that ò_-￥^￥ exp[a |F(x)|^q-|x|]  dx ￡ C_p\int\limits _{-\infty }^\infty \exp [a |F(x)|^q-|x|] \ dx\le C_p for each f with ||f||_p ￡ 1||f||_p\le 1 and every a ￡ 1a\le 1. Moreover the value a = 1 is sharp. We replace the operation connecting f with F by a more general integral operation; specifically we consider non-negative kernels K(t,x) with the property that xK(t,x) is homogeneous of degree 0 in t, x. We state an analogue of the inequality above for this situation, discuss some applications and consider the sharpness of the constant which replaces a.     

Blow-up of solutions of nonlinear heat equations in unbounded domains for slowly decaying initial data

Consider the following equations: (E)  u_t-Du=u^p{(E)\ \ u_t-\Delta u=u^p}, (E￠)  u_t-Du=u^p-m | ?u | ^q{(E')\ \ u_t-\Delta u=u^p-\mu\mid\nabla u\mid^q}, (E")  u_t-Du=u^p+a.?(u^q){(E')\ \ u_t-\Delta u=u^p+a.\nabla (u^q)}, in W ì IR^d{\Omega\subset I\!\!R^d}. For any unbounded domain W\Omega, intermediate between a cone and a strip, we obtain a sufficient condition on the decay at infinity of initial data to have blow-up. This condition is related to the geometric nature of W{\Omega}. For instance, if W\Omega is the interior of a revolution surface of the form | x￠_d | < f( | x_d | ){\mid x'_d\mid (x) > Cf( | x | )^-2/(p-1){\Phi (x)>Cf(\mid x\mid )^{-2/(p-1)}} at infinity. Moreover, for a large class of domains W{\Omega}, we prove that those results are optimal (i.e. there exist global solutions with the same order of decay at infinity for their initial data).     

Tangential Convergence of Temperatures and Harmonic Functions in Besov and in Triebel-Lizorkin Spaces

We study the maximal function Mf(x) = sup |f(x + y, t)| when Ω is a region in the (y,t) Ω upper half space R and f(x, t) is the harmonic extension to R₊^N+1 of a distribution in the Besov space B^α_p,q(R^N) or in the Triebel-Lizorkin space F^α_p,q(R^N). In particular, we prove that when Ω= {|_y|^{N/ (N-αp)} < t < 1} the operator M is bounded from F (R^N) into L^p (R^N). The admissible regions for the spaces B (R^N) with p < q are more complicated.     

Unboundedness of the large solutions of some asymmetric oscillators at resonance

In this paper, we consider the unboundedness of solutions of the following differential equation (φ_p(x′))′ + (p ? 1)[αφ_p(x⁺) ? βφ_p(x^?)] = f(x)x′ + g(x) + h(x) + e(t) where φ_p(u) = |u|^{p? 2} u, p > 1, x^± = max {±x, 0}, α and β are positive constants satisfying with m, n ∈ N and (m, n) = 1, f and g are continuous and bounded functions such that lim_x→±∞g(x) ? g(±∞) exists and h has a sublinear primitive, e(t) is 2π_p‐periodic and continuous. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)