Simultaneous determination of the source terms in a linear hyperbolic problem from the final overdetermination: weak solution approach

The problem of determining the pair w:={F(x, t);f(t)} of sourceterms in the hyperbolic equation u_tt = (k(x)u_x)_x + F(x, t) andin the Neumann boundary condition k(0)u_x(0, t) = f(t) from themeasured data µ(x):=u(x, T) and/or (x):=u_t(x, t) at thefinal time t = T is formulated. It is proved that both componentsof the Fréchet gradient of the cost functionals J₁(w)= ||u(x, t;w) – µ(x)||₀² and J₂(w) = ||u_t(x, T;w)– (x)||₀² can be found via the solutions of correspondingadjoint hyperbolic problems. Lipschitz continuity of the gradientis derived. Unicity of the solution and ill-conditionednessof the inverse problem are analysed. The obtained results permitone to construct a monotone iteration process, as well as toprove the existence of a quasi-solution.     

Asymptotic simplification for a reaction-diffusion problem with a nonlinear boundary condition

We study non-negative solutions of the porous medium equationwith a source and a nonlinear flux boundary condition, u_t =(u^m)_xx + u^p in (0, ), x (0, T); – (u^m)_x (0, t) = u^q (0,t) for t (0, T); u (x, 0) = u₀ (x) in (0, ), where m > 1,p, q > 0 are parameters. For every fixed m we prove thatthere are two critical curves in the (p, q-plane: (i) the criticalexistence curve, separating the region where every solutionis global from the region where there exist blowing-up solutions,and (ii) the Fujita curve, separating a region of parametersin which all solutions blow up from a region where both globalin time solutions and blowing-up solutions exist. In the caseof blow up we find the blow-up rates, the blow-up sets and theblow-up profiles, showing that there is a phenomenon of asymptoticsimplification. If 2q < p + m the asymptotics are governedby the source term. On the other hand, if 2q > p + m theevolution close to blow up is ruled by the boundary flux. If2q = p + m both terms are of the same order.     

Stationary Critical Points of the Heat Flow in Spaces of Constant Curvature

The paper considers stationary critical points of the heat flowin sphere S^N and in hyperbolic space H^N, and proves severalresults corresponding to those in Euclidean space R^N which havebeen proved by Magnanini and Sakaguchi. To be precise, it isshown that a solution u of the heat equation has a stationarycritical point, if and only if u satisfies some balance lawwith respect to the point for any time. In Cauchy problems forthe heat equation, it is shown that the solution u has a stationarycritical point if and only if the initial data satisfies thebalance law with respect to the point. Furthermore, one point,say x₀, is fixed and initial-boundary value problems are consideredfor the heat equation on bounded domains containing x₀. It isshown that for any initial data satisfying the balance law withrespect to x₀ (or being centrosymmetric with respect to x₀)the corresponding solution always has x₀ as a stationary criticalpoint, if and only if the domain is a geodesic ball centredat x₀ (or is centrosymmetric with respect to x₀, respectively).     

Adaptive numerical schemes for a parabolic problem with blow-up

In this paper we present adaptive procedures for the numericalstudy of positive solutions of the following problem: u_t = u_xx (x, t) (0, 1) x [0, T), u_x(0, t) = 0 t [0, T), u_x(1, t) = u^p(1, t) t [0, T), u(x, 0) = u₀(x) x (0, 1), with p > 1. We describe two methods. The first one refinesthe mesh in the region where the solution becomes bigger ina precise way that allows us to recover the blow-up rate andthe blow-up set of the continuous problem. The second one combinesthe ideas used in the first one with moving mesh methods andmoves the last points when necessary. This scheme also recoversthe blow-up rate and set. Finally, we present numerical experimentsto illustrate the behaviour of both methods.     

Uniform l1 behaviour for time discretization of a Volterra equation with completely monotonic kernel: I. stability

This paper is the first of two papers on the time discretizationof the equation u_t + ^t ₀ ß (t — s) Au (s) ds= 0, t > 0, u (0) = u₀, where A is a self-adjoint denselydefined linear operator on a Hilbert space H with a completeeigensystem {_m, _m}_{m = 1}, and ß (t) is completely monotonicand locally integrable, but not constant. The equation is discretizedin time using first-order differences in combination with order-oneconvolution quadrature. The stability properties of the timediscretization are derived in the l¹_t (0, ; H) norm.     

Long-Time Behavior of Solutions of the Fast Diffusion Equations with Critical Absorption Terms

This paper is devoted to the long-time behavior of solutionsto the Cauchy problem of the porous medium equation u_t = (u^m)– u^p in Rⁿ x (0,) with (1 – 2/n)₊ < m < 1and the critical exponent p = m + 2/n. For the strictly positiveinitial data u(x,0) = O(1 + |x|)^–k with n + mn(2 –n + nm)/(2[2 – m + mn(1 – m)]) k < 2/(1 –m), we prove that the solution of the above Cauchy problem convergesto a fundamental solution of u_t = (u^m) with an additional logarithmicanomalous decay exponent in time as t .     

On approximation theorems for controllability of non-linear parabolic problems

^** Email: anil{at}math.iitb.ac.in^*** Email: mcj{at}math.iitb.ac.in^**** Email: akp{at}math.iitb.ac.in In this paper, we consider the following control system governedby the non-linear parabolic differential equation of the form: [graphic: see PDF] where A is a linear operator with dense domain and f(t, y)is a non-linear function. We have proved that under Lipschitzcontinuity assumption on the non-linear function f(t, y), theset of admissible controls is non-empty. The optimal pair (u^*,y^*) is then obtained as the limit of the optimal pair sequence{(u_n^*, y_n^*)}, where u_n^* is a minimizer of the unconstrainedproblem involving a penalty function arising from the controllabilityconstraint and y_n^* is the solution of the parabolic non-linearsystem defined above. Subsequently, we give approximation theoremswhich guarantee the convergence of the numerical schemes tooptimal pair sequence. We also present numerical experimentwhich shows the applicability of our result.     

On a theorem of K. Schmidt

Let Y be a locally compact group, Aut(Y) be the group of topologicalautomorphisms of Y and (Y) be the set of continuous positivedefinite functions on Y which have unit value at the identity.A function (Y²) is said to be of product type if there aresuch functions _j (Y) that (u, v) = ₁(u)₂(v). Define the mappingT: Y² Y² by the formula T(u, v) = (A₁ uA₂ v, A₃ u A₄ v), whereA_j Aut(Y), and assume that T is a one-to-one transform. K.Schmidt proved: (i) if both (u, v) and (T(u, v)) are of producttype, then the functions _j are infinitely divisible; (ii) ifY is Abelian, both (u, v) and (T(u, v)) are of product type,and (u, v) 0, then the functions _j are Gaussian. We show thatstatement (i), generally, is not valid, but K. Schmidt's proofholds true if (u, v) 0. We also give another proof of statement(ii). Our proof uses neither the Levy–Khinchin formulafor a continuous infinitely divisible positive definite functionnor (i) on which K. Schmidt's proof is based.     

Differentiability Properties of an Abstract Autonomous Composition Operator

The autonomous composition operator is the nonlinear map whichtakes a pair of functions into its composite function. The compositionoperator often appears in problems of nonlinear analysis andto analyse such problems it is often important to know whetherthe composition operator is continuous or differentiable. Afairly large number of papers in the literature have been devotedto the study of composition operators. For fullscale references,we refer the reader to the extensive monographs of Appell andZabrejko [1] and Runst and Sickel [8]. To exemplify a typicalsituation, we consider the semilinear Dirichlet boundary valueproblem where denotes a sufficiently regular bounded open subset ofR^N, and h₀ a map of R to R, and where u is the unknown of theproblem. We assume that we know that a certain function u₀ belongingto a certain function space X solves (1.1). Then if we wishto know whether by perturbing h₀ in a certain function space,say Y, the solutions u depend on h continuously, with differentiability,with analyticity or bifurcate, we could set G[h, u] u+h u,recast problem (1.1) into the abstract form G[h, u] (1.2) and study the solution set of equation (1.2) around the pair(h₀, u₀) by means of the implicit function theorem or by localbifurcation theorems in a Banach space setting.     

Quantum Markov Processes with a Christensen-Evans Generator in a Von Neumann Algebra

Let A be a unital von Neumann algebra of operators on a complexseparable Hilbert space H₀, and let {T_t, t 0} be a uniformlycontinuous quantum dynamical semigroup of completely positiveunital maps on A. The infinitesimal generator L of {T_t} is abounded linear operator on the Banach space A. For any Hilbertspace K, denote by B(K) the von Neumann algebra of all boundedoperators on K. Christensen and Evans [3] have shown that Lhas the form [formula] where is a representation of A in B(K) for some Hilbert spaceK, R: H₀ K is a bounded operator satisfying the ‘minimality’condition that the set {(RX–(X)R)u, uH₀, XA} is totalin K, and K₀ is a fixed element of A. The unitality of {T_t}implies that L(1) = 0, and consequently K₀=iH–R^*R, whereH is a hermitian element of A. Thus (1.1) can be expressed as [formula] We say that the quadruple (K, , R, H) constitutes the set ofChristensen–Evans (CE) parameters which determine theCE generator L of the semigroup {T_t}. It is quite possible thatanother set (K', ', R', H') of CE parameters may determine thesame generator L. In such a case, we say that these two setsof CE parameters are equivalent. In Section 2 we study thisequivalence relation in some detail. 1991 Mathematics SubjectClassification 81S25, 60J25.     

An Oscillation Theorem for Self-Adjoint Differential Systems and the Rayleigh Principle for Quadratic Functionals

In this note an oscillation theorem on self-adjoint differentialsystems of the form = Ax + Bu, = (C – C₀)x – A^Tu is obtained,complementing, in particular, results of M. Morse. The applicationof this oscillation result yields the Rayleigh principle forquadratic functionals, respectively, the existence theorem forcorresponding self-adjoint eigenvalue problems, under the centralassumptions that the pair (A, B) is controllable (or identicallynormal) and the triple (A, B, C₀) is strongly observable.     

Correction: the Embedding of Radical Rings in Simple Radical Rings

As G. M. Bergman has pointed out, in the proof of the lemmaon p. 187, we cannot conclude that $$\stackrel{\&macr;}{S}$$is universal in the sense stated. However, the proof can becompleted as follows: Any element of $$\stackrel{\&macr;}{S}$$can be obtained as the first component of the solution u ofa system (A–I)u+a = 0, (1) where A S_n, a ⁿS and A–I has an inverse over L. SinceS is generated by R and k{s}, A can (by the last part of Lemma3.2 of [1]) be taken to be linear in these arguments, say A= A₀ + sA1, where A₀ R_n, A₀ R_n, A₁ K_n. Multiplying by (I–sA₁)^–1,we reduce this equation to the form (S_vB_v–I)u+a=0, (2) with the same solution u as before, where B_v R_n, s_v k{s}¹and a ⁿS. Now consider the retraction S k{s} (3) obtained by mapping R 0. If we denote its effect by x x^*,then (2) goes over into an equation –I.v + a^* 0, (4) which clearly has a unique solution v in k{s}; therefore theretraction (3) can be extended to a homomorphism $$\stackrel{\&macr;}{S}$$ k{s}, again denoted by x x^*, provided we can show that u₁^*does not depend on the equation (1) used to define it. Thisamounts to showing that if an equation (1), or equivalently(2), has the solution u₁ = 0, then after retraction we get v₁= 0 in (4), i.e. a₁^* = 0. We shall use induction on n; if u₁= 0 in (2), then by leaving out the first row and column ofthe matrix on the left of (2), we have an equation for u₂,...,u_n and by the induction hypothesis, their values after retractionare uniquely determined. Now from (2) we have where B = (b_ij^v). Applying ^* and observing that b_ij^vR, we seethat a₁ ^* = 0, as we wished to show. The proof still appliesfor n = 1, so we have a well-defined mapping $$\stackrel{\&macr;}{S}$$ k{s}, which is a homomorphism. Now the proof of the lemma canbe completed as before.     

Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes

This paper is devoted to the study of an error estimate of thefinite volume, approximation to the solution u L(R_N x R) ofthe equation u_t + div(Vf(u)) = 0, where v is a vector functiondepending on time and space. A 'h' error estimate for an initialvalue in BV(R^N) is shown for a large variety of finite volumemonotonous flux schemes, with an explicit or implicit time discretization.For this purpose, the error estimate is given for the generalsetting of approximate entropy solutions, where the error isexpressed in terms of measures in R^N and R^N x R. The study ofthe implicit schemes involves the study of the existence anduniqueness of the approximate solution. The cases where an 'h'error estimate can be achieved are also discussed.     

Linear Independence of Hecke Operators in the Homology of X0(N)

In Merel's recent proof [7] of the uniform boundedness conjecturefor the torsion of elliptic curves over number fields, a keystep is to show that for sufficiently large primes N, the Heckeoperators T₁, T₂, ..., T_D are linearly independent in theiractions on the cycle e from 0 to i in H₁(X₀(N) (C), Q). In particular,he shows independence when max(D⁸, 400D⁴) < N/(log N)⁴. Inthis paper we use analytic techniques to show that one can chooseD considerably larger than this, provided that N is large.     

On time regularity and related conditions for power-bounded operators

Let T be a bounded linear operator in a complex Banach space.Our main result gives various characterizations of the condition:T is power-bounded and an estimate ||(I – T)Tⁿ || cn^–1/2 holds for all positive integers n. In particular, this conditionholds if and only if T = β S + (1 – β)I, forsome β (0, 1) and some power-bounded operator S; or ifand only if T is power-bounded and the discrete semigroup (Tⁿ)is dominated by the continuous semigroup (e^{– t(I –T)})_{t 0} in a natural sense. As a consequence of our main results,for 1/2 < 1 we characterize the condition that T is power-boundedand ||(I – T)Tⁿ || c n^– for all n, in terms ofestimates on the semigroup e^{–t(I – T)}.     

On the Error Term for the Fourth Moment of the Riemann Zeta-Function

Let E₂(T) denote the error term in the asymptotic formula for^T₀|(+it)|⁴dt. It is proved that there exist constants A>0,B>1 such that for TT₀>0 every interval [T, BT] containspoints T₁, T₂ for which and that ^T₀|E₂(t)|^adt>>T^1+(a/2) for any fixed a1. Theseresults complement earlier results of Motohashi and Ivi that^T₀E₂(t)dt<<T^3/2 and that ^T₀E²₂(t)dt<<T²log^CT forsome C>0. Omega-results analogous to the above ones are obtainedalso for the error term in the asymptotic formula for the Laplacetransform of |(+it)|⁴.     

Algebraic decay and variable speeds in wavefront solutions of a scalar reaction-diffusion equation

Wavefront solutions of scalar reaction-diffusion equations havebeen intensively studied for many years. There are two basiccases, typified by quadratic and cubic kinetics. An intermediatecase is considered in this paper, namely, u_l = u_xx + u²(1 –u). It is shown that there is a unique travelling-wave solution,with a speed 1/2, for which the decay to zero ahead of the waveis exponential with x. Moreover, numerical evidence is presentedwhich suggests that initial conditions with such exponentialdecay evolve to this travelling-wave solution, independentlyof the half-life of the initial decay. It is then shown thatfor all speeds greater than 1/2 there is also a travelling-wavesolution, but that these faster waves decay to zero algebraically,in proportion to 1/x. The numerical evidence suggests that initialconditions with such a decay rate evolve to one of these fasterwaves; the particular speed depends in a simple way on the detailsof the initial condition. Finally, initial conditions decayingalgebraically for a power law other than 1/x are considered.It is shown numerically that such initial conditions evolveeither to an algebraically decaying travelling wave, or in somecases to a wavefront whose shape and speed vary as a functionof time. This variation is monotonic and can be quite pronounced,and the speed is a function of u as well as of time. Using asimple linearization argument, an approximate formula is derivedfor the wave speed which compares extremely well with the numericalresults. Finally, the extension of the results to the more generalcase of u_l = u_xx + u^m(1 – u), with m > 1, is discussed.     

Generalized Ramsey Theory for Graphs V. the Ramsey Number of a Digraph

Continuing this series of papers on generalized Ramsey theoryfor graphs, we define the Ramsey number r(D_l, D₂) of two digraphsD₁ and D₂ as the minimum p such that every 2-colouring of thearcs (directed lines) of DK_P (the complete symmetric digraphof order p) contains a monochromatic D₁ or D₂. It is shown(Theorem1) that this number exists if and only if D₁ or D₂ is acyclic.Then r(D), the diagonal Ramsey number of a given acyclic digraphD, is defined as r(D, D). Notation: D' is the converse of D,GD is the underlying graph of D, DG is the symmetric digraphof G, and T_p is the transitive tournament of order p. Let r(m,n) be the traditional Ramsey number of the two complete graphsK_m and K_n. Finally, let S_n be the star with n arcs from onepoint u to n points v_i. Assuming the Ramsey numbers under discussionexist, we prove the following results: THEOREM 2. r(D₁, D₂) = r(D₁' D₂'). THEOREM 3. r(D₁, D₂) r(GD₁, GD₂). THEOREM 4. r(D₁, D₂) r(T_P1, T_P2) if both D₁ and D₂ (with p₁and p₂ points respectively) are acyclic. THEOREM 5. r(T_m, T_n) = r(m, n). THEOREM 6. r(m, ri) r(T_m, DK_n) r(2^m–1, n). THEOREM 7. r(S_m, S_n) = r(S_m, S_n') = m+n. Finally, we establish all Ramsey numbers r(D₁, D₂) for digraphswithout isolates and with less than four points, and all diagonalRamsey numbers r(D) of acyclic digraphs without isolates withless than five points.     

Rate of convergence of numerical approximations to homoclinic bifurcation points

For x=f (x, ), x Rⁿ, R, having a hyperbolic or semihyperbolicequilibrium p(), we study the numerical approximation of parametervalues ^* at which there is an orbit homoclinic to p(). We approximate^* by solving a finite-interval boundary value problem on J=[T_–,T₊], T_–<0<T₊, with boundary conditions that sayx(T_–) and x(T₊) are in approximations to appropriate invariantmanifolds of p(). A phase condition is also necessary to makethe solution unique. Using a lemma of Xiao-Biao Lin, we improve,for certain phase conditions, existing estimates on the rateof convergence of the computed homoclinic bifurcation parametervalue , to the true value ^*. The estimates we obtain agree withthe rates of convergence observed in numerical experiments.Unfortunately, the phase condition most commonly used in numericalwork is not covered by our results.     

On Positive Multipeak Solutions of a Nonlinear Elliptic Problem

In this paper we continue our investigation in [5, 7, 8] onmultipeak solutions to the problem –²u+u=Q(x)|u|^q–2u, xR^N, uH¹(R^N) (1.1) where = ^N_i=1²/x²_i is the Laplace operator in R^N, 2 < q < for N = 1, 2, 2 < q < 2N/(N–2) for N3, and Q(x)is a bounded positive continuous function on R^N satisfying thefollowing conditions. (Q₁) Q has a strict local minimum at some point x₀R^N, that is,for some > 0 Q(x)>Q(x₀) for all 0 < |x–x₀| < . (Q₂) There are constants C, > 0 such that |Q(x)–Q(y)|C|x–y| for all |x–x₀| , |y–y₀| . Our aim here is to show that corresponding to each strict localminimum point x₀ of Q(x) in R^N, and for each positive integerk, (1.1) has a positive solution with k-peaks concentratingnear x₀, provided is sufficiently small, that is, a solutionwith k-maximum points converging to x₀, while vanishing as 0 everywhere else in R^N.