On the existence and stability of periodic and almost periodic solutions of quasilinear equations with maxima

We study the problem of existence of periodic and almost periodic solutions of the scalar equation x′ (t) = − δx(t) + pmax _{u∈[t − h, t]} x(u) + f(t) where δ, p ∈ R, with a periodic (almost periodic) perturbation f(t). For these solutions, we establish conditions of global exponential stability and prove uniqueness theorems. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 747–754, June, 1998.     

Global existence and L ∞ estimates of solutions for a quasilinear parabolic system

In this paper, we study the global existence, L^∞ estimates and decay estimates of solutions for the quasilinear parabolic system u_t = div (|∇ u|^m ∇ u) + f(u, v), v_t = div (|∇ v|^m ∇ v) + g(u,v) with zero Dirichlet boundary condition in a bounded domain Ω ⊂ R^N. In particular, we find a critical value for the existence and nonexistence of global solutions to the equation u_t = div (|∇ u|^m ∇ u) + λ |u|^{α - 1} u.     

Existence,uniqueness and regularity of stationary solutions to inhomogeneous Navier-Stokes equations in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

For a bounded domain Ω ⊂ ℝ ⁿ , n ⩾ 3, we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system − Δu + u · ∇u + ∇p = f, div u = k, u _|aΩ = g with u ∈ L ^q , q ⩾ n, and very general data classes for f, k, g such that u may have no differentiability property. For smooth data we get a large class of unique and regular solutions extending well known classical solution classes, and generalizing regularity results. Moreover, our results are closely related to those of a series of papers by Frehse & Růžička, see e.g. Existence of regular solutions to the stationary Navier-Stokes equations, Math. Ann. 302 (1995), 669–717, where the existence of a weak solution which is locally regular is proved.      

On a nonlinear elliptic equation arising in a free boundary problem

 Let p ^* =n/(n−2) and n≥3. In this paper, we first classify all non-constant solutions of

On the growth of mass for a viscous Hamilton-Jacobi equation

We investigate the large time behavior of positive solutions with finite mass for the viscous Hamilton-Jacobi equationu _t = Δu + |Δu| ^p ,t>0,x ∈ ℝ ^N , wherep≥1 andu(0,.)=u ₀≥0,u ₀≢0,u ₀∈L ¹. DenotingI _∞=lim _t→∞‖u(t)‖₁≤∞, we show that the asymptotic behavior of the mass can be classified along three cases as follows:

Existence and non-existence of global solutions for a semilinear heat equation

The existence and non-existence of global solutions and theL ^p blow-up of non-global solutions to the initial value problemu′(t)=Δu(t)+u(t) ^γ onR ⁿ are studied. We consider onlyγ>1. In the casen(γ − 1)/2=1, we present a simple proof that there are no non-trivial global non-negative solutions. Ifn(γ−1)/2≦1, we show under mild technical restrictions that non-negativeL ^p solutions always blow-up inL ^p norm in finite time. In the casen(γ−1)/2>1, we give new sufficient conditions on the initial data which guarantee the existence of global solutions. Research partially supported by NSF grant MCS79-03636.     

Source-type solutions of degenerate diffusion equations with absorption

The object of this paper is to study the existence of a solution of the Cauchy problemu _t=Δu^m−u^p, u(x,0)=δ(x) and when a solution exists, to study its behaviour ast→0.     

Explosive solutions of elliptic equations with absorption and nonlinear gradient term

Letf be a non-decreasing C¹-function such that andF(t)/f ² a(t)→ 0 ast → ∞, whereF(t)=∫ ₀ ^t f(s) ds anda ∈ (0, 2]. We prove the existence of positive large solutions to the equationΔu +q(x)|Δu| ^a =p(x)f(u) in a smooth bounded domain Ω ⊂R^N, provided thatp, q are non-negative continuous functions so that any zero ofp is surrounded by a surface strictly included in Ω on whichp is positive. Under additional hypotheses onp we deduce the existence of solutions if Ω is unbounded.     

Initial boundary value problem for generalized boussinesque type equations with nonlinear source

A comparison principle for solutions of the first initial boundary value problem for the generalized Boussinesque equation with a nonlinear sourceu _t-Δψ(u)-Δu _t+q(u)=0 is established. By using this comparison principle, we prove new existence and nonexistence theorems for solutions of the first initial boundary value problem in the case of power-law functions ψ (ξ) andq (ξ). Translated fromMathematicheskie Zametki, Vol. 65, No. 1, pp. 70–75, January, 1999.     

Asymptotic behavior of solutions to viscous conservation laws with slowly varying external forces

This paper considers the existence and large time behavior of solutions to the convection-diffusion equation u _t −Δu+b(x)·∇(u|u| ^{q −1})=f(x, t) in ℝ ⁿ ×[0,∞), where f(x, t) is slowly decaying and q≥1+1/n (or in some particular cases q≥1). The initial condition u ₀ is supposed to be in an appropriate L ^p space. Uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term.This work is partially supported by an AMO Grant     

Large time behaviour of solutions of the porous media equation with absorption

We study the large time behaviour of nonnegative solutions of the Cauchy problemu _t=Δu ^m −u ^p,u(x, 0)=φ(x). Specifically we study the influence of the rate of decay ofφ(x) for large |x|, and the competition between the diffusion and the absorption term.     

Global solutions for superlinear parabolic equations involving the biharmonic operator for initial data with optimal slow decay

We are interested in stability/instability of the zero steady state of the superlinear parabolic equation u _t + Δ² u = |u| ^p-1 u in , where the exponent is considered in the “super-Fujita” range p > 1 + 4/n. We determine the corresponding limiting growth at infinity for the initial data giving rise to global bounded solutions. In the supercritical case p > (n + 4)/(n−4) this is related to the asymptotic behaviour of positive steady states, which the authors have recently studied. Moreover, it is shown that the solutions found for the parabolic problem decay to 0 at rate t ^−1/(p-1).     

Fast diffusion flow on manifolds of nonpositive curvature

We consider the fast diffusion equation (FDE) u _t = Δu ^m (0 < m < 1) on a nonparabolic Riemannian manifold M. Existence of weak solutions holds. Then we show that the validity of Euclidean–type Sobolev inequalities implies that certain L ^p −L ^q smoothing effects of the type ∥u(t)∥ _q ≤ Ct ^−α ∥u ₀∥^γ _p , the case q = ∞ being included. The converse holds if m is sufficiently close to one. We then consider the case in which the manifold has the addition gap property min σ(−Δ) > 0. In that case solutions vanish in finite time, and we estimate from below and from above the extinction time.      

Initial trace for a porous medium equation: I. The strong absorption case

We study the equation (E): u_t−Δu^m+u^q=0, (m, q>0) in Δ×ℝ₊, in a regular bounded open set Ω, or the whole space. We first prove that when 0<m<q, distributional solutions of (E) have an initial trace which is a Borel measure, then we study existence and uniqueness results with measure initial data. Entrata in Redazione il 12 giugno 1999. Ricevuta versione finale il 5 febbraio 2000.     

Singular solutions of some nonlinear parabolic equations

We consider the existence and uniqueness of singular solutions for equations of the formu ₁=div(|Du|^p−2 Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2. Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the existence of very singular solutions, i.e. singular solutions such that ∫_|x|≤r u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result. In the case ϕ(u)=u ^q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal. Dedicated to Professor Shmuel Agmon     

Asymptotiques d’un systeme dynamique conservatif associe a une equation elliptique non lineaire

LetM be a compact riemannian manifold,h an odd function such thath(r)/r is non-decreasing with limit 0 at 0. Letf(r)=h(r)-γr and assume there exist non-negative constantsA andB and a realp>1 such thatf(r)>Ar ^P-B. We prove that any non-negative solutionu ofu _tt+Δ_gu=f(u) onM x ℝ⁺ satisfying Dirichlet or Neumann boundary conditions on ϖM converges to a (stationary) solution of Δ _g Φ=f(Φ) onM with exponential decay of ‖u-Φ‖C ²(M). For solutions with non-constant sign, we prove an homogenisation result for sufficiently small λ; further, we show that for every λ the map (u(0,·),u _t(0,·))→(u(t,·), u _t(t,·)) defines a dynamical system onW ^1/2(M)⊂C(M)×L ²(M) which possesses a compact maximal attractor.      

Existence of periodic solutions for a fourth-order -Laplacian equation with a deviating argument

In this paper, we study the existence of periodic solutions for a fourth-order p-Laplacian differential equation with a deviating argument as follows:

[φ_p(u^″(t))]^″+f(u^″(t))+g(u(t−τ(t)))=e(t).

Existence and uniqueness of global solutions of onR 2

In this paper, we consider the solutions of the nonlinear Schrödinger equations u/t–iu+|u| ^p u=f andu(x,0)=u ₀(x), whereu is defined onR ⁺×R ². We prove the existence and uniqueness of global weak solutions of the above equations. Lastly, we consider the special case:p=2, and we obtain the strong solutions.     

Analysis of a Free Boundary Problem Modeling Tumor Growth

In this paper, we study a free boundary problem arising from the modeling of tumor growth. The problem comprises two unknown functions： R = R（t）, the radius of the tumor, and u = u（r, t）, the concentration of nutrient in the tumor. The function u satisfies a nonlinear reaction diffusion equation in the region 0 〈 r 〈 R（t）, t 〉 0, and the function R satisfies a nonlinear integrodifferential equation containing u. Under some general conditions, we establish global existence of transient solutions, unique existence of a stationary solution, and convergence of transient solutions toward the stationary solution as t →∞.     

A priori and universal estimates for global solutions of superlinear degenerate parabolic equations

We prove an a priori estimate and a universal bound for any global solution of the nonlinear degenerate reaction-diffusion equation u _t =Δu ^m +u ^p in a bounded domain with zero Dirichlet boundary conditions. Received: October 1, 2001?Published online: July 9, 2002