Blow-up for semidiscrete forms of some nonlinear parabolic equations with convection

This paper concerns the study of the numerical approximation for the following parabolic equations with a nonlinear convection term $$\\ \left\{% \begin{array}{ll} \hbox{$u_t(x,t)=u_{xx}(x,t)-g(u(x,t))u_{x}(x,t)+f(u(x,t)),\quad 0<x<1,\; t>0$,} \\ \hbox{$u_{x}(0,t)=0, \quad u_{x}(1,t)=0,\quad t>0$,} \\ \hbox{$u(x,0)=u_{0}(x) > 0,\quad 0\leq x \leq 1$,} \\ \end{array}% \right. $$ \newline where $f:[0,+\infty)\rightarrow [0,+\infty)$ is $C^3$ convex, nondecreasing function,\\ $g:[0,+\infty)\rightarrow [0,+\infty)$ is $C^1$ convex, nondecreasing function,\newline $\displaystyle\lim_{s\rightarrow +\infty}f(s)=+\infty$, $\displaystyle\lim_{s\rightarrow +\infty}g(s)=+\infty$, $\displaystyle\lim_{s\rightarrow +\infty}\frac{f(s)}{g(s)}=+\infty$\newline and $\displaystyle\int^{+\infty}_{c}\frac{ds}{f(s)}<+\infty$ for $c>0$. We obtain some conditions under which the solution of the semidiscrete form of the above problem blows up in a finite time and estimate its semidiscrete blow-up time. We also prove that the semidiscrete blow-up time converges to the real one, when the mesh size goes to zero. Finally, we give some numerical results to illustrate ours analysis.

On the asymptotic behavior of sums ∑n≤xf(n){x/n}k

Let [Formula: see text] be an arbitrary real-valued positive nondecreasing function, in this paper we prove that [Formula: see text] [Formula: see text] where [Formula: see text] is the sum given in the title, [Formula: see text] and [Formula: see text] is a positive integer. This improves the result of Mercier and Nowak.

Some negative results for the interpolation monotone approximation of functions having a fractional derivative

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function $f є W^r [0,1]$ by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function $f є C^r [0,1] \cap \Delta^0$ where $\Delta^0$ is the set of positive functions on [0,1]. Estimates of the form (1) for positive approximation are known ([1],[2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3],[4],[8]. In [3],[4] is consider $r є , r > 2$. In [8] is consider $r є , r > 2$. It was proved that for monotone approximation estimates of the form (1) are fails for $r є , r > 2$. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5],[6]). In [5] is consider $r є , r > 2$. In [6] is consider $r є , r > 2$. It was proved that for convex approximation estimates of the form (1) are fails for $r є , r > 2$. In this paper the question of approximation of function $f є W^r \cap \Delta^1, r є (3,4)$ by algebraic polynomial $p_n є \Pi_n \cap \Delta^1$ is consider. The main result of the work generalize the result of work [8] for $r є (3,4)$.

Blow-up results for a quasilinear von Karman equation of memory type

In this paper, we consider the blow-up result of solution for a quasilinear von Karman equation of memory type with nonpositive initial energy as well as positive initial energy. For nonincreasing function $g>0$ g > 0 and nondecreasing function f, we prove a finite time blow-up result under suitable condition on the initial data.

Description of the scattering data for Sturm-Liouville operators on the half-line

We describe the set of the scattering data for self-adjoint Sturm-Liouville operators on the half-line with potentials belonging to \(L_1(\mathbb{R}_+,\rho(x)\,\text{d} x)\), where \(\rho:\mathbb{R}_+\to\mathbb{R}_+\) is a monotonically nondecreasing function from some family \(\mathscr{R}\). In particular, \(\mathscr{R}\) includes the functions \(\rho(x)=(1+x)^{\alpha}\) with \(\alpha\geq 1\).

One counterexample for convex approximation of function with fractional derivatives, r>4

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function f \in W^r [0,1] by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function f \in C^r [0,1] \Wedge \Delta^0, where \Delta^0 is the set of positive functions on [0,1]. Estimates of the form (1) for positive approximation are known ([1],[2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3],[4],[8]. In [3],[4] is consider r \in N, r>2. In [8] is consider r \in R, r>2. It was proved that for monotone approximation estimates of the form (1) are fails for r \in R, r>2. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5],[6],[11]). In [5] is consider r \in N, r>2. It was proved that for convex approximation estimates of the form (1) are fails for r \in N, r>2. In [6] is consider r \in R, r\in(2;3). It was proved that for convex approximation estimates of the form (1) are fails for r \in R, r\in(2;3). In [11] is consider r \in R, r\in(3;4). It was proved that for convex approximation estimates of the form (1) are fails for r \in R, r\in(3;4). In [9] is consider r \in R, r>4. It was proved that for f \in W^r [0,1] \Wedge \Delta^2, r>4 estimate (1) is not true. In this paper the question of approximation of function f \in W^r [0,1] \Wedge \Delta^2, r>4 by algebraic polynomial p_n \in \Pi_n \Wedge \Delta^2 is consider. It is proved, that for f \in W^r [0,1] \Wedge \Delta^2, r>4, estimate (1) can be improved, generally speaking.

Ap(ϕ) Weights, BMO(ϕ), and Calderón-Zygmund Operators of ϕ-Type

We introduce new classes of weights and BMO functions associated with a nondecreasing function ϕ of upper type β with β>0 and obtain the weighted norm inequalities for Calderón-Zygmund operators of ϕ-type and their commutators.

Small-time almost-sure behaviour of extremal processes

An rth-order extremal process Δ(r) = (Δ(r)t)t≥0 is a continuous-time analogue of the rth partial maximum sequence of a sequence of independent and identically distributed random variables. Studying maxima in continuous time gives rise to the notion of limiting properties of Δt(r) as t ↓ 0. Here we describe aspects of the small-time behaviour of Δ(r) by characterising its upper and lower classes relative to a nonstochastic nondecreasing function bt > 0 with limt↓bt = 0. We are then able to give an integral criterion for the almost sure relative stability of Δt(r) as t ↓ 0, r = 1, 2, . . ., or, equivalently, as it turns out, for the almost sure relative stability of Δt(1) as t ↓ 0.

Intrinsic Comparative Statics of a Nash Bargaining Solution

A generalization of the class of bargaining problems examined by Engwerda and Douven [(2008) On the sensitivity matrix of the Nash bargaining solution, Int. J. Game Theory 37, 265–279] is studied. The generalized class consists of nonconvex bargaining problems in which the feasible set satisfies the requirement that the set of weak Pareto-optimal solutions can be described by a smooth function. The intrinsic comparative statics of the aforesaid class are derived and shown to be characterized by a symmetric and positive semidefinite matrix, and an upper bound to the rank of the matrix is established. A corollary to this basic result is that a Nash bargaining solution is intrinsically a locally nondecreasing function of its own disagreement point. Other heretofore unknown results are similarly deduced from the basic result.

An FPTAS for Uniform Parallel-Machine Scheduling Problem with Deteriorating Jobs and Rejection

This paper studies uniform parallel-machine scheduling problem with deteriorating jobs and rejection. The processing time of each job is a linear nondecreasing function of its starting time. A job can be rejected by paying a penalty cost. The objective is to minimize the sum of the total load of the accepted jobs on all machines and the total rejection penalties of the rejected jobs. We propose a fully polynomial-time approximation scheme (FPTAS) for this problem.