Convergence of the Dirichlet solutions of the very fast diffusion equation

For any −1<m<0, positive functions f, g and u₀≥0, we prove that under some mild conditions on f, g and u₀ as R→∞ the solution u^R of the Dirichlet problem u_t=(u^m/m)_xx in (−R,R)×(0,∞), u(R,t)=(f(t)|m|R)^1/m, u(−R,t)=(g(t)|m|R)^1/m for all t>0, u(x,0)=u₀(x) in (−R,R), converges uniformly on every compact subset of R×(0,T) to the solution of the equation u_t=(u^m/m)_xx in R×(0,T), u(x,0)=u₀(x) in R, which satisfies some mass loss formula on (0,T) where T is the maximal time such that the solution u is positive. We also prove that the solution constructed is equal to the solution constructed in Hui (2007) [15] using approximation by solutions of the corresponding Neumann problem in bounded cylindrical domains.     

A note on the behavior of blow-up solutions for one-phase Stefan problems

ＡＮＯＴＥＯＮＴＨＥＢＥＨＡＶＩＯＲＯＦＢＬＯＷ┐ＵＰＳＯＬＵＴＩＯＮＳＦＯＲＯＮＥ┐ＰＨＡＳＥＳＴＥＦＡＮＰＲＯＢＬＥＭＳＺＨＵＮＩＮＧＡｂｓｔｒａｃｔ．Ｉｎｔｈｉｓｐａｐｅｒ,ｔｈｅｆｏｌｏｗｉｎｇｏｎｅ－ｐｈａｓｅＳｔｅｆａｎｐｒｏｂｌｅｍｉｓ...     

Multiple positive solutions of four point boundary value problems for finite difference equations†

In this paper, we consider the following second-order four-point boundary-value problems Δ² u(k ? 1)+f(k,u(k), Δu(k)) = 0,k ∈ {1,2,…,T}, u(0) = au(l ₁), u(T+1) = bu(l ₂). We give conditions on f to ensure the existence of at least three positive solutions of the given problem by applying a new fixed-point theorem of functional type in a cone. The emphasis is put on the nonlinear term involved with the first-order difference.     

On the Propagation of Regularity and Decay of Solutions to the k-Generalized Korteweg-de Vries Equation

We study special regularity and decay properties of solutions to the IVP associated to the k-generalized KdV equations. In particular, for datum u ₀ ∈ H ^3/4+ (?) whose restriction belongs to H ^l ((b, ∞)) for some l ∈ ?⁺ and b ∈ ? we prove that the restriction of the corresponding solution u(·, t) belongs to H ^l ((β, ∞)) for any β ∈ ? and any t ∈ (0, T). Thus, this type of regularity propagates with infinite speed to its left as time evolves.     

Identification of the unknown diffusion coefficient in a quasi‐linear parabolic equation by semigroup approach with mixed boundary conditions

In this article, a semigroup approach is presented for the mathematical analysis of the inverse coefficient problems of identifying the unknown diffusion coefficient k(u(x, t)) in the quasi‐linear parabolic equation u_t(x, t)=(k(u(x, t))u_x(x, t))_x, with Dirichlet boundary conditions u_x(0, t)=ψ₀, u(1, t)=ψ₁. The main purpose of this work is to analyze the distinguishability of the input–output mappings Φ[·] : ??→C¹[0, T], Ψ[·] : ??→C¹[0, T] using semigroup theory. In this article, it is shown that if the null space of semigroups T(t) and S(t) consists of only a zero function, then the input–output mappings Φ[·] and Ψ[·] have the distinguishability property. Copyright © 2008 John Wiley & Sons, Ltd.     

The Ext-group of unitary equivalence classes of unital extensions

Let A be a unital separable nuclear C*-algebra which belongs to the bootstrap category N and B be a separable stable C*-algebra. In this paper, we consider the group Ext _u (A, B) consisting of the unitary equivalence classes of unital extensions τ: A→Q(B). The relation between Ext _u (A, B) and Ext(A, B) is established. Using this relation, we show the half-exactness of Ext _u (·, B) and the (UCT) for Ext _u (A, B). Furthermore, under certain conditions, we obtain the half-exactness and Bott periodicity of Ext _u (A, ·).     

Analysis of a semigroup approach in the inverse problem of identifying an unknown coefficient

This article presents a semigroup approach to the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u_x) in the quasi‐linear parabolic equation u_t(x, t)=(k(u_x)u_x(x, t))_x+F(x, t), with Dirichlet boundary conditions u(0, t)=ψ₀, u(1, t)=ψ₁ and source function F(x, t). The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[·]: ?? → C¹[0, T], Ψ[·]: ?? → C¹[0, T] via semigroup theory. Copyright © 2008 John Wiley & Sons, Ltd.     

Convexity and all‐time C∞‐regularity of the interface in flame propagation

We consider the following one‐phase free boundary problem: Find (u, Ω) such that Ω = {u > 0} and with Q_T = ?ⁿ × (0, T). Under the condition that Ω_o is convex and log u_o is concave, we show that the convexity of Ω(t) and the concavity of log u(·, t) are preserved under the flow for 0 ≤ t ≤ T as long as ?Ω(t) and u on Ω(t) are smooth. As a consequence, we show the existence of a smooth‐up‐to‐the‐interface solution, on 0 < t < T_c, with T_c denoting the extinction time of Ω(t). We also provide a new proof of a short‐time existence with C^2,α initial data on the general domain. © 2002 John Wiley & Sons, Inc.     

Analysis for the identification of an unknown diffusion coefficient via semigroup approach

This paper presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u_x) in the inhomogenenous quasi‐linear parabolic equation u_t(x, t)=(k(u_x)u_x(x, t))_x +F(u), with the Dirichlet boundary conditions u(0, t)=ψ₀, u(1, t)=ψ₁ and source function F(u). The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[·]:??→C¹[0, T], Ψ[·]:??→C¹[0, T] via semigroup theory. Copyright © 2009 John Wiley & Sons, Ltd.     

Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

We consider an inverse boundary value problem for the heat equation ? _t u = div (γ? _x u) in (0, T) × Ω, u = f on (0, T) × ?Ω, u| _t=0 = u ₀, in a bounded domain Ω ? ? ⁿ , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time: γ(t, x) = k ² (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω?D(t)). Fix a direction e* ∈ 𝕊 ^n?1 arbitrarily. Assuming that ?D(t) is strictly convex for 0 ≤ t ≤ T, we show that k and sup {e*·x; x ∈ D(t)} (0 ≤ t ≤ T), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ?_ν u(t, x)|_{(0, T)×?Ω}. The knowledge of the initial data u ₀ is not used in the proof. If we know min_0≤t≤T (sup _x∈D(t) x·e*), we have the same conclusion from the local Dirichlet-to-Neumann map. Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. Consider a physical body consisting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking.     

On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities

We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain Ω of R³. We first prove the local existence of solutions (ρ,u) in C([0,T_*]; (ρ^∞ +H³(Ω)) × under the assumption that the data satisfies a natural compatibility condition. Then deriving the smoothing effect of the velocity u in t>0, we conclude that (ρ,u) is a classical solution in (0,T_**)×Ω for some T_** ∈ (0,T_*]. For these results, the initial density needs not be bounded below away from zero and may vanish in an open subset (vacuum) of Ω.     

Parabolic problems for the Anderson model

Summary. This is a continuation of our previous work [6] on the investigation of intermittency for the parabolic equation (∂/∂t)u=Hu on ℝ₊×ℤ ^d associated with the Anderson Hamiltonian H=κΔ+ξ(·) for i.i.d. random potentials ξ(·). For the Cauchy problem with nonnegative homogeneous initial condition we study the second order asymptotics of the statistical moments <u(t,0) ^p > and the almost sure growth of u(t,0) as t→∞. We point out the crucial role of double exponential tails of ξ(0) for the formation of high intermittent peaks of the solution u(t,·) with asymptotically finite size. The challenging motivation is to achieve a better understanding of the geometric structure of such high exceedances which in one or another sense provide the essential contribution to the solution. Received: 10 December 1996 / In revised form: 30 September 1997     

Reaction diffusion equations with non-linear boundary conditions,blowup and steady states

In this paper we study the following problem: u_t−Δu=−f(u) in Ω×(0, T)≡Q_T, ∂u ∂n=g(u) on ∂Ω×(0, T)≡S_T, u(x, 0)=u₀(x) in Ω , where Ω⊂ℝ^N is a smooth bounded domain, f and g are smooth functions which are positive when the argument is positive, and u₀(x)>0 satisfies some smooth and compatibility conditions to guarantee the classical solution u(x, t) exists. We first obtain some existence and non-existence results for the corresponding elliptic problems. Then, we establish certain conditions for a finite time blow-up and global boundedness of the solutions of the time-dependent problem. Further, we analyse systems with same kind of boundary conditions and find some blow-up results. In the last section, we study the corresponding elliptic problems in one-dimensional domain. Our main method is the comparison principle and the construction of special forms of upper–lower solutions using related equations. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Nonlinear degenerate parabolic equations with time dependent singular coefficients

We are concerned with the nonexistence of positive solutions of the nonlinear parabolic partial differential equations in a cylinder Ω × (0, T) with initial condition u(., 0) = u₀(.) ? 0 and vanishing on the boundary ?Ω × (0, T), given by where $\Omega \in \mathbf {R}^NWe are concerned with the nonexistence of positive solutions of the nonlinear parabolic partial differential equations in a cylinder Ω × (0, T) with initial condition u(., 0) = u₀(.) ? 0 and vanishing on the boundary ?Ω × (0, T), given by where $\Omega \in \mathbf {R}^N$ (resp. a Carnot Carathéodory metric ball in $\mathbf {R}^{2N+1})$ with smooth boundary and the time dependent singular potential function V(x, t) ∈ L¹_loc(Ω × (0, T)), $\alpha , \beta \in \mathbf {R}$, 1 < p < N, p ? 1 + α + β > 0. We find the best lower bounds for p + β and provide proofs for the nonexistence of positive solutions. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Smoothing properties for a coupled system of nonlinear evolution dispersive equations

We study the smoothness properties of solutions to the coupled system of equations of Korteweg—de Vries type. We show that the equations dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data (u₀, v₀ possesses certain regularity and sufficient decay as x → ∞, then the solution (u(t). v(t)) will be smoother than (u0, v0) for 0 < t ≤ T where T is the existence time of the solution.     

Localization on the Boundary of Blow-up for Reaction–Diffusion Equations with Nonlinear Boundary Conditions

Abstract

In this work we analyze the existence of solutions that blow-up in finite time for a reaction–diffusion equation u _t  ? Δu = f(x, u) in a smooth domain Ω with nonlinear boundary conditions ?u/?n = g(x, u). We show that, if locally around some point of the boundary, we have f(x, u) = ?βu ^p , β ≥ 0, and g(x, u) = u ^q then, blow-up in finite time occurs if 2q > p + 1 or if 2q = p + 1 and β < q. Moreover, if we denote by T _b the blow-up time, we show that a proper continuation of the blowing up solutions are pinned to the value infinity for some time interval [T, τ] with T _b  ≤ T < τ. On the other hand, for the case f(x, u) = ?βu ^p , for all x and u, with β > 0 and p > 1, we show that blow-up occurs only on the boundary.     

带非局部源的退化半线性抛物方程的解的爆破性质

This paper deals with the blow-up properties of the positive solutions to the nonlocal degenerate semilinear parabolic equation u _t − (x ^a u _x ) _x =∫ ₀ ^a f(u)dx in (0,a) × (0,T) under homogeneous Dirichlet conditions. The local existence and uniqueness of classical solution are established. Under appropriate hypotheses, the global existence and blow-up in finite time of positve solutions are obtained. It is also proved that the blow-up set is almost the whole domain. This differs from the local case. Furthermore, the blow-up rate is precisely determined for the special case: f(u)=u ^p , p>1.     

Existence results for a class of porous medium type equations with a quadratic gradient term

In this paper we study the Dirichlet problem in Q _T = Ω × (0, T) for degenerate equations of porous medium-type with a lower order term:

Special Uniform Approximations of Continuous Vector-Valued Functions. Part I: Special Approximations in CX(T)

In this paper, we give special uniform approximations of functions u from the spaces C_X(T) and C_∞(T,X), with elements of the tensor products C_Γ(T)X, respectively C₀(T,Γ)X, for a topological space T and a Γ-locally convex space X. We call an approximation special, if satisfies additional constraints, namely supp vu⁻¹(X\{0}) and (T) co(u(T)) (resp. co(u(T){0})). In Section 3, we give three distinct applications, which are due exactly to these constraints: a density result with respect to the inductive limit topology, a Tietze–Dugundji's type extension new theorem and a proof of Schauder–Tihonov's fixed point theorem.