Existence of fast positive wavefronts for a non-local delayed reaction-diffusion equation

We establish the existence of a continuous family of fast positive wavefronts u(t,x)=?(x+ct), ?(−∞)=0, ?(+∞)=κ, for the non-local delayed reaction-diffusion equation . Here 0 and κ>0 are fixed points of g∈C²(R₊,R₊) and the non-negative K is such that is finite for every real λ. We also prove that the fast wavefronts are non-monotone if .     

Local existence of solutions to some degenerate parabolic equation associated with the p-Laplacian

The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|^q−2u(x,t)=f(x,t), (x,t)∈Ω×(0,T), where 2?p<q<+∞, Ω is a bounded domain in RN, is given and Δp denotes the so-called p-Laplacian defined by Δpu:=∇⋅(|∇u|^p−2∇u), with initial data u₀∈Lr(Ω) is proved under r>N(q−p)/p without imposing any smallness on u₀ and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, Lr-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0,T₀] in which the problem admits a solution. More precisely, T₀ depends only on Lr|u₀| and f.     

Carleson embeddings for Sobolev spaces via heat equation

Let u(t,x) be the solution of the heat equation (∂_t-Δ_x)u(t,x)=0 on subject to u(0,x)=f(x) on Rⁿ. The main goal of this paper is to characterize such a nonnegative measure μ on that f(x)?u(t²,x) induces a bounded embedding from the Sobolev space , p∈[1,n) into the Lebesgue space , q∈(0,∞).     

Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay

In this paper, we study the existence, uniqueness and asymptotic stability of travelling wavefronts of the following equation:

u_t(x,t)=D[u(x+1,t)+u(x-1,t)-2u(x,t)]-du(x,t)+b(u(x,t-r)),     

On a stochastic nonlinear equation arising from 1D integro-differential scalar conservation laws

In this paper we study the initial problem for a stochastic nonlinear equation arising from 1D integro-differential scalar conservation laws. The equation is driven by Lévy space-time white noise in the following form:

(t_∂−A)u+x_∂q(u)=f(u)+g(u)Ft,x     

Boundedness in Nonlinear Asymmetric Oscillations

In this paper, the boundedness of all solutions of the nonlinear equation (?_p(x′))′+(p-1)[α?_p(x⁺)−β?_p(x⁻)]+f(x)+g(x)=e(t) is discussed, where e(t)∈C⁷ is 2π_p-periodic, f,g are bounded C⁶ functions, ?_p(u)=∣u∣^p−2u, p?2,α,β are positive constants, x⁺=max{x,0},x⁻=max{−x,0}.     

On the complete group classification of the reaction-diffusion equation with a delay

The reaction-diffusion delay differential equation

ut(x,t)−uxx(x,t)=g(x,u(x,t),u(x,t−τ))     

Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space

Let E be a real Banach space. Let K be a nonempty closed and convex subset of E, a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}_n?0⊂[1,+∞), limn→∞kn=1 such that F(T)≠∅. Let {αn}_n?0⊂[0,1] be such that _∑n?0αn=∞, and _∑n?0αn(kn−1)<∞. Suppose {xn}_n?0 is iteratively defined by xn+1=(1−αn)xn+αnTnxn, n?0, and suppose there exists a strictly increasing continuous function , ?(0)=0 such that 〈Tnx−x^∗,j(x−x^∗)〉?kn‖x−x^∗‖2−?(‖x−x^∗‖), ∀x∈K. It is proved that {xn}_n?0 converges strongly to x^∗∈F(T). It is also proved that the sequence of iteration {xn} defined by xn+1=anxn+bnTnxn+cnun, n?0 (where {un}_n?0 is a bounded sequence in K and {an}_n?0, {bn}_n?0, {cn}_n?0 are sequences in [0,1] satisfying appropriate conditions) converges strongly to a fixed point of T.     

On solvability of two-dimensional Lp-Minkowski problem

Given p≠0 and a positive continuous function g, with g(x+T)=g(x), for some 0<T<1 and all real x, it is shown that for suitable choice of a constant C>0 the functional has a minimizer in the class of positive functions u∈C¹(R) for which u(x+T)=u(x) for all x∈R. This minimizer is used to prove the existence of a positive periodic solution y∈C²(R) of two-dimensional L_p-Minkowski problem y^1−p(x)(y″(x)+y(x))=g(x), where p∉{0,2}.     

Hyperbolic conservation laws with nonlinear diffusion and nonlinear dispersion

We study scalar conservation laws with nonlinear diffusion and nonlinear dispersion terms (any ??1), the flux function f(u) being mth order growth at infinity. It is shown that if ε, δ=δ(ε) tend to 0, then the sequence {uε} of the smooth solutions converges to the unique entropy solution u∈L^∞(0,T_∗;Lq(R)) to the conservation law ut+fx(u)=0 in . The proof relies on the methods of compensated compactness, Young measures and entropy measure-valued solutions. Some new a priori estimates are carried out. In particular, our result includes the convergence result by Schonbek when b(λ)=λ, ?=1 and LeFloch and Natalini when ?=1.     

Standing waves for supercritical nonlinear Schrödinger equations

Let V(x) be a non-negative, bounded potential in RN, N?3 and p supercritical, . We look for positive solutions of the standing-wave nonlinear Schrödinger equation Δu−V(x)u+up=0 in RN, with u(x)→0 as |x|→+∞. We prove that if V(x)=o(⁻²|x|) as |x|→+∞, then for N?4 and this problem admits a continuum of solutions. If in addition we have, for instance, V(x)=O(|x|^−μ) with μ>N, then this result still holds provided that N?3 and . Other conditions for solvability, involving behavior of V at ∞, are also provided.     

Positive solutions for some Schrödinger equations having partially periodic potentials

This paper is concerned with the problem of finding positive solutions of the equation −Δu+(a_∞+a(x))u=|u|^q−2u, where q is subcritical, Ω is either RN or an unbounded domain which is periodic in the first p coordinates and whose complement is contained in a cylinder , a_∞>0, a∈C(RN,R) is periodic in the first p coordinates, infx∈RN(a_∞+a(x))>0 and a(x^′,x^″)→0 as |x^″|→∞ uniformly in x^′. The cases a?0 and a?0 are considered and it is shown that, under appropriate assumptions on a, the problem has one solution in the first case and p+1 solutions in the second case when p?N−2.     

Nonlinear dissipative wave equations with space-time dependent potential

We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space-time dependent friction a(t,x)u_t and nonlinear absorption |u|^p−1u (Ikawa (2000) [17]). We consider 1<p<(n+2)/(n−2) and separable a(t,x)=λ(x)η(t) with λ(x)∼(1+|x|)^−α and η(t)∼(1+t)^−β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L² and L^p+1 norms of solutions. We also observe the following behavior: if α∈[0,1), β∈(−1,1) and 0<α+β<1, there are three different regions for the decay of solutions depending on p; if α∈(−∞,0) and β∈(−1,1), there are only two different regions for the decay of the solutions depending on p.     

Large time behavior of solutions to degenerate parabolic equations with weights

We study the degenerate parabolic equation t_∂u=a(δ(x))upΔu−g(u) in Ω×(0,∞), where Ω⊂RN (N?1) is a smooth bounded domain, p?1, δ(x)=dist(x,∂Ω) and a is a continuous nondecreasing function such that a(0)=0. Under some suitable assumptions on a and g we prove the existence and the uniqueness of a classical solution and we study its asymptotic behavior as t→∞.     

On oscillatory solutions of certain forced Emden-Fowler like equations

We give a constructive proof of existence to oscillatory solutions for the differential equations x^″(t)+a(t)λ|x(t)|sign[x(t)]=e(t), where t?t₀?1 and λ>1, that decay to 0 when t→+∞ as O(t−μ) for μ>0 as close as desired to the “critical quantity” . For this class of equations, we have limt→+∞E(t)=0, where E(t)<0 and E^″(t)=e(t) throughout [t₀,+∞). We also establish that for any μ>μ^? and any negative-valued E(t)=o(t−μ) as t→+∞ the differential equation has a negative-valued solution decaying to 0 at + ∞ as o(t−μ). In this way, we are not in the reach of any of the developments from the recent paper [C.H. Ou, J.S.W. Wong, Forced oscillation of nth-order functional differential equations, J. Math. Anal. Appl. 262 (2001) 722-732].     

Self-similar singular solutions of a p-Laplacian evolution equation with absorption

We consider, for p∈(1,2) and q>1, self-similar singular solutions of the equation v_t=div(|∇v|^p−2∇v)−v^q in Rn×(0,∞); here by self-similar we mean that v takes the form v(x,t)=t^−αw(|x|t^−αβ) for α=1/(q−1) and β=(q+1−p)/p, whereas singular means that v is non-negative, non-trivial, and for all x≠0. That is, we consider the ODE problem

(0.1)     

An even-order three-point boundary value problem on time scales

We study the even-order dynamic equation (−1)ⁿx^(Δ∇)n(t)=λh(t)f(x(t)), t∈[a,c] satisfying the boundary conditions x^(Δ∇)i(a)=0 and x^(Δ∇)i(c)=βx^(Δ∇)i(b) for 0?i?n−1. The three points a,b,c are from a time scale , where 0<β(b−a)<c−a for b∈(a,c), β>0, f is a positive function, and h is a nonnegative function that is allowed to vanish on some subintervals of [a,c] of the time scale.