Initial Value U0 Research Articles

Global asymptotic behavior of solutions of a semilinear parabolic equation

We study the large time behavior of solutions for the semilinear parabolic equation ∆u+V up−ut = 0. Under a general and natural condition on V = V (x) and the initial value u0, we show that global positive solutions of the parabolic equation converge pointwise to positive solutions of the corresponding elliptic equation. As a corollary of this, we recapture the global existence results on semilinear elliptic equations obtained by Kenig and Ni and by F.H. Lin and Z. Zhao. Our method depends on newly found global bounds for fundamental solutions of certain linear parabolic equations.

Nonlinear parabolic problems on manifolds, and a nonexistence result for the noncompact Yamabe problem

We study the Cauchy problem for the semilinear parabolic equations Δ u − R u + u p − u t = 0 \Delta u - R u + u^{p} - u_{t} =0 on M n × ( 0 , ∞ ) \mathbf {M}^{n} \times (0, \infty ) with initial value u 0 ≥ 0 u_{0} \ge 0 , where M n \mathbf {M}^{n} is a Riemannian manifold including the ones with nonnegative Ricci curvature. In the Euclidean case and when R = 0 R=0 , it is well known that 1 + 2 n 1+ \frac {2}{n} is the critical exponent, i.e., if p > 1 + 2 n p > 1 + \frac {2}{n} and u 0 u_{0} is smaller than a small Gaussian, then the Cauchy problem has global positive solutions, and if p > 1 + 2 n p>1+\frac {2}{n} , then all positive solutions blow up in finite time. In this paper, we show that on certain Riemannian manifolds, the above equation with certain conditions on R R also has a critical exponent. More importantly, we reveal an explicit relation between the size of the critical exponent and geometric properties such as the growth rate of geodesic balls. To achieve the results we introduce a new estimate for related heat kernels. As an application, we show that the well-known noncompact Yamabe problem (of prescribing constant positive scalar curvature) on a manifold with nonnegative Ricci curvature cannot be solved if the existing scalar curvature decays “too fast” and the volume of geodesic balls does not increase “fast enough”. We also find some complete manifolds with positive scalar curvature, which are conformal to complete manifolds with positive constant and with zero scalar curvatures. This is a new phenomenon which does not happen in the compact case.

A Finite-Dimensional Attractor for Three-Dimensional Flow of Incompressible Fluids

The asymptotic behavior of the flow for a system of the Navier–Stokes type is investigated. In the considered model, the viscous part of the stress tensor is generally a nonlinear function of the symmetric part of the velocity gradient. Provided that the function describing this dependence satisfies the polynomial (p−1) growth condition, a unique weak solution exists if eitherp⩾(2+n)/2 andu0∈Horp⩾1+2n/(n+2) andu0∈W1, 2(C)n∩H. In the first case, the existence of a global attractor inHis proved. In order to indicate the finite-dimensional behavior of the flow at infinity, the fractal dimension of the new invariant set, composed from all shortδ-trajectories with initial value in the attractor, is estimated in theL2(0, δ; H) topology. Having uniqueness only for more regular data in the second case, many trajectories can start from the initial valueu0∈H. This does not allow one to define a semigroup on the spaceH. Therefore, the set of short trajectoriesXsδ, closed inL2(0, δ; H), is introduced along with a semigroup working on this set. The existence of a global attractor with a finite fractal dimension is then demonstrated.

A semilinear parabolic system in a bounded domain

Consider the system $$\left\{ \begin{gathered} u_t - \Delta u = v^p , in Q = \{ (t, x), t > 0, x \in \Omega \} , \hfill \\ v_t - \Delta v = u^q , in Q , \hfill \\ u(0, x) = u_0 (x) v(0, x) = v_0 (x) in \Omega , \hfill \\ u(t, x) = v(t, x) = 0 , when t \geqslant 0, x \in \partial \Omega , \hfill \\ \end{gathered} \right.$$ (S) where Ω is a bounded open domain in ℝN with smooth boundary, p and q are positive parameters, and functions u0 (x), v0(x) are continuous, nonnegative and bounded. It is easy to show that (S) has a nonnegative classical solution defined in some cylinder QT=(0, T) × Ω with T ⩽ ∞. We prove here that solutions are actually unique if pq ⩾ 1, or if one of the initial functions u0, v0 is different from zero when 0 1, solutions may be global or blow up in finite time, according to the size of the initial value (u0, v0).

Flat blow-up in one-dimensional semilinear heat equations

Consider the Cauchy problem ut=uxx+up, x∈R, t>0, u(x,0)=u0(x), x∈R, where p>1 and u0(x) is continuous, nonnegative and bounded. Assume that u(x,t) blows up at x=0, t=T and set u(x,t)=(T−t)−1/(p−1)φ(y,τ), y=x/T−t−−−−√, τ=−ln(T−t). Here we show that there exist initial values u0(x) for which the corresponding solution is such that two maxima collapse at x=0, t=T. One then has that φ(y,τ)=(p−1)1/(p−1)−C1e−τH4(y)+o(e−τ)asτ→∞,(1) with C1>0, H4(y)=c4H˜4(y/2), where c4=(23(4π)1/4)−1, H˜4(s) is the standard 4th Hermite polynomial, and convergence in (1) takes place in Ck,αloc for any k≥1 and some α∈(0,1). We also show that in this case, limt↑T(T−t)1/(p−1)u(ξ(T−t)1/4,t)=(p−1)(1+C1c4ξn)−1/(p−1),(2) where the convergence is uniform on sets |ξ|≤R with R>0. This asymptotic behaviour is different (and flatter) than that corresponding to solutions spreading from data u0(x) having a single maximum, in which case (3)φ(y,τ)=(p−1)−1/(p−1)−(4π)1/4(p−1)−1/(p−1)2√p⋅H2(y)τ+o(1τ) as τ→∞, and limt↑T(T−t)1/(p−1)u(ξ(T−t)1/2|ln(T−t)|1/2,t)=(p−1)−1/(p−1)(1+(p−1)4pξ2)−1/(p−1).(4)

On the mildly degenerate Kirchhoff string

AbstractLet us consider the Cauchy problem for the abstract evolution equation, which describes the Kirchhoff sring u″ + m(〈Au, u〉) Au = 0, (t > 0), Where ( )′ = d/dt, A is any symmetric isomorphism of a Hilbert space V into its (anti) dual V′ and m is a function of one real variable. Following physical considerations, we allow m(·) to be any non‐decreasing, non‐negative continuous function (a degenerate Kirchhoff string).We assume that the initial value u0 satisfies: m(〈Au0, u0〉)>0 (a mildly degenerate Kirchhoff string), that m is locally Lipschitz continuous in the open region where it does not vanish, and that m has a finite order of vanishing (e.g., m(ρ) = ργ, γ > 0).We establish the local well‐posedness of the Cauchy problem in the class D(Aα/2) × D(A(α−1)/2), For each α ⩾ 3/2. This improves the results of Medeiros and Miranda [15], Ebihara et al.[9] and a result of Yamada [27].We are not able to prove the global existence of the solution; however, we provide a lower estimate for the life span of the solution, which yields the almost global existence (AGE) in the case when m(ρ) = ργ (γ>0).

On the Dynamics of a Semilinear Heat Equation with Strong Absorption

This paper deals with the following initial value problem: (1) ∂u/∂t−Δu+u p=0 on RN, N≥1, 0 0 and α≥0. For any x∈RN, the extinction time of x is defined as follows: TE(x)=sup{t>0,u(x,t)>0}, u(x,t) being the solution of (1). The following results are established. (i) If u0(x)≤A(|x|)+B|x−a|2/(1−p) for some a∈RN where A(|x|)=o(|x| 2/(1−p)) as |x|→+∞ and 0≤B TE(y). (ii) It is assumed that N=1. Let u(x,t) be the solution of (1). For a convenient behaviour of u0(x) as |x|→+∞, then u(x,t) tends to a limit as t→+∞, uniformly in compact sets on R. (iii) Let u0(r) be a nonnegative radial function satisfying (2). Then the solution of (1) with initial value u0(r) is radially symmetric and for a convenient behaviour of u0(r) as r→+∞, u(r,t) tends to a limit as r→+∞, uniformly on compact sets in R. For N≥2, the authors restrict themselves to the radial case.

Continuants and precontinuants

1. It is here expedient to ignore the continued fractions from which they are usually derived and to define the simple continuants determined by the parameters a1, a2, a3, …, an, … as given by the chain of difference equationstogether with the initial values u0 = 0, u1 = 1: and these data lead successively to the functionsand