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1.
We study the large time behavior of non‐negative solutions to the nonlinear fractional reaction–diffusion equation ?tu = ? tσ( ? Δ)α ∕ 2u ? h(t)up (α ∈ (0,2]) posed on and supplemented with an integrable initial condition, where σ ≥ 0, p > 1, and h : [0, ∞ ) → [0, ∞ ). Defining the mass , under certain conditions on the function h, we show that the asymptotic behavior of the mass can be classified along two cases as follows:
  • if , then there exists M ∈ (0, ∞ ) such that ;
  • if , then .
Copyright © 2014 John Wiley & Sons, Ltd.  相似文献   

2.
In this paper, we study the existence and concentration behavior of positive solutions for the following Kirchhoff type equation: where ɛ is a positive parameter, a and b are positive constants, and 3<p<5. Let denotes the ground energy function associated with , , where is regard as a parameter. Suppose that the potential V(x) decays to zero at infinity like |x|α with 0<α≤2, we prove the existence of positive solutions uɛ belonging to for vanishing or unbounded K(x) when ɛ > 0 small. Furthermore, we show that the solution uɛ concentrates at the minimum points of as ɛ→0+.  相似文献   

3.
In this paper, we are interested in the nonlinear Schrödinger equation with non‐local regional diffusion (1) where 0 < α < 1 and is a variational version of the regional Laplacian, whose range of scope is a ball with radius ρ(x) > 0. The novelty of this paper is that, assuming f is of subquadratic growth as |u|→+, we show that 1 possesses infinitely many solutions via the genus properties in critical point theory. Furthermore, if f(x,u) = γa(x)|u|γ ? 1, where is a nonincreasing radially symmetric function, then the solution of 1 is radially symmetric. Copyright © 2015 John Wiley & Sons, Ltd.  相似文献   

4.
Horváth and Kiss (Proc. Amer. Math. Soc., 2005) proved the upper bound estimate for Dirichlet eigenvalue ratios of the Schrödinger problem ?y + q(x)y = λy with nonnegative and single‐well potential q. In this paper, we prove that if q(x) is a nonpositive, continuous, and single‐barrier potential, then for λn > λm≥ ? 2q?, where . In particular, if q(x) satisfies the additional condition , then λ1 > 0 and for n > m ≥ 1. For this result, we develop a new approach to study the monotonicity of the modified Prüfer angle function.  相似文献   

5.
This paper deals with the quasilinear Keller–Segel system with rotation where is a bounded domain with smooth boundary, D(u) is supposed to be sufficiently smooth and satisfies D(u)≥D0um ? 1(m≥1) and D(u)≤D1(u + 1)K ? mum ? 1(K≥1) for all u≥0 with some positive constants D0 and D1, and f(u) is assumed to be smooth enough and non‐negative for all u≥0 and f(0) = 0, while S(u,v,x) = (sij)n × n is a matrix with and with l≥2, where is nondecreasing on [0,). It is proved that when , the system possesses at least one global and bounded weak solution for any sufficiently smooth non‐negative initial data. Copyright © 2015 John Wiley & Sons, Ltd.  相似文献   

6.
We study a parabolic‐elliptic chemotactic PDEs system, which describes the evolution of a biological population “u” and a chemical substance “v” in a bounded domain . We consider a growth term of logistic type in the equation of “u” in the form μu(1 ? u + f(t,x)). The function “f,” describing the resources of the systems, presents a periodic asymptotic behavior in the sense where f ? is independent of x and periodic in time. We study the global existence of solutions and its asymptotic behavior. Under suitable assumptions on the initial data and f ?, if the constant chemotactic sensitivity χ satisfies we obtain that the solution of the system converges to a homogeneous in space and periodic in time function.  相似文献   

7.
We consider the linearized thermoelastic plate equation with the Dirichlet boundary condition in a general domain Ω, given by with the initial condition u|(t=0)=u0, ut|(t=0)=u1, and θ|(t=0)=θ0 in Ω and the boundary condition u=νu=θ=0 on Γ, where u=u(x,t) denotes a vertical displacement at time t at the point x=(x1,⋯,xn)∈Ω, while θ=θ(x,t) describes the temperature. This work extends the result obtained by Naito and Shibata that studied the problem in the half‐space case. We prove the existence of ‐bounded solution operators of the corresponding resolvent problem. Then, the generation of C0 analytic semigroup and the maximal LpLq‐regularity of time‐dependent problem are derived.  相似文献   

8.
We study the semilinear equation where 0 < s < 1, , V(x) is a sufficiently smooth non‐symmetric potential with , and ? > 0 is a small number. Letting U be the radial ground state of (?Δ)sU + U ? Up=0 in , we build solutions of the form for points ?j,j = 1,?,m, using a Lyapunov–Schmidt variational reduction. Copyright © 2014 John Wiley & Sons, Ltd.  相似文献   

9.
We consider the problem Δ2u = V(x)up + ? in with uu→0 as |x|→ + , where , N ≥ 5, V is a positive continuous potential. Our aim is to construct high‐energy solutions for this equation by applying the finite‐dimensional reduction method and the penalization method.  相似文献   

10.
Let n≥3, Ω be a strongly Lipschitz domain of and LΩ:=?Δ+V a Schrödinger operator on L2(Ω) with the Dirichlet boundary condition, where Δ is the Laplace operator and the nonnegative potential V belongs to the reverse Hölder class for some q0>n/2. Assume that the growth function satisfies that ?(x,·) is an Orlicz function, (the class of uniformly Muckenhoupt weights) and its uniformly critical lower type index , where and μ0∈(0,1] denotes the critical regularity index of the heat kernels of the Laplace operator Δ on Ω. In this article, the authors first show that the heat kernels of LΩ satisfy the Gaussian upper bound estimates and the Hölder continuity. The authors then introduce the ‘geometrical’ Musielak–Orlicz–Hardy space via , the Hardy space associated with on , and establish its several equivalent characterizations, respectively, in terms of the non‐tangential or the vertical maximal functions or the Lusin area functions associated with LΩ. All the results essentially improve the known results even on Hardy spaces with p∈(n/(n + δ),1] (in this case, ?(x,t):=tp for all x∈Ω and t∈[0,)). Copyright © 2016 John Wiley & Sons, Ltd.  相似文献   

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