共查询到20条相似文献,搜索用时 31 毫秒
1.
In this paper we consider the Cauchy problem of multidimensional generalized double dispersion equations utt−Δu−Δutt+Δ2u=Δf(u), where f(u)=ap|u|. By potential well method we prove the existence and nonexistence of global weak solution without establishing the local existence theory. And we derive some sharp conditions for global existence and lack of global existence solution. 相似文献
2.
The paper studies the existence of the finite-dimensional global attractors and exponential attractors for the dynamical system associated with the Kirchhoff type equation with a strong dissipation utt−M(‖∇u‖2)Δu−Δut+h(ut)+g(u)=f(x). It proves that the above mentioned dynamical system possesses a global attractor which has finite fractal dimension and an exponential attractor. For application, the fact shows that for the concerned viscoelastic flow the permanent regime (global attractor) can be observed when the excitation starts from any bounded set in phase space, and the dimension of the attractor, that is, the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow, is finite. 相似文献
3.
Yoshikazu Kobayashi 《Journal of Mathematical Analysis and Applications》2008,338(2):852-872
A generation theorem of semigroups of locally Lipschitz operators on a subset of a real Banach space is given and applied to the problem of the well-posedness of the Carrier equation utt−κ(‖u‖2)Δu+γ|ut|p−1ut=0 in Ω×(0,∞) with acoustic boundary condition, where p>2 and Ω is a bounded domain in an arbitrary dimensional space. 相似文献
4.
Jiecheng Chen Qingquan Deng Yong Ding Dashan Fan 《Nonlinear Analysis: Theory, Methods & Applications》2012,75(5):2959-2974
We study the fractional power dissipative equations, whose fundamental semigroup is given by e−t(−Δ)α with α>0. By using an argument of duality and interpolation, we extend space-time estimates of the fractional power dissipative equations in Lebesgue spaces to the Hardy spaces and the modulation spaces. These results are substantial extensions of some known results. As applications, we study both local and global well-posedness of the Cauchy problem for the nonlinear fractional power dissipative equation ut+(−Δ)αu=|u|mu for initial data in the modulation spaces. 相似文献
5.
We consider time-independent solutions of hyperbolic equations such as ∂ttu−Δu=f(x,u) where f is convex in u. We prove that linear instability with a positive eigenfunction implies nonlinear instability. In some cases the instability occurs as a blow up in finite time. We prove the same result for parabolic equations such as t∂u−Δu=f(x,u). Then we treat several examples under very sharp conditions, including equations with potential terms and equations with supercritical nonlinearities. 相似文献
6.
In this paper we study a nonlocal equation that takes into account convective and diffusive effects, ut=J∗u−u+G∗(f(u))−f(u) in Rd, with J radially symmetric and G not necessarily symmetric. First, we prove existence, uniqueness and continuous dependence with respect to the initial condition of solutions. This problem is the nonlocal analogous to the usual local convection-diffusion equation ut=Δu+b⋅∇(f(u)). In fact, we prove that solutions of the nonlocal equation converge to the solution of the usual convection-diffusion equation when we rescale the convolution kernels J and G appropriately. Finally we study the asymptotic behaviour of solutions as t→∞ when f(u)=|u|q−1u with q>1. We find the decay rate and the first-order term in the asymptotic regime. 相似文献
7.
Tomás Caraballo Alexandre N. CarvalhoJosé A. Langa Felipe Rivero 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(6):2272-2283
In this paper we consider the strongly damped wave equation with time-dependent terms
utt−Δu−γ(t)Δut+βε(t)ut=f(u), 相似文献
8.
Grzegorz ?ysik S?awomir Michalik 《Journal of Mathematical Analysis and Applications》2008,341(1):372-385
We study formal power series solutions to the initial value problem for semilinear heat equation t∂u−Δu=f(u) with polynomial nonlinearity f and prove that they belong to the formal Gevrey class G2. Next we give counterexamples showing that the solution, in general, is not analytic in time at t=0. 相似文献
9.
This paper is concerned with the existence and nonexistence of positive solutions of the second-order nonlinear dynamic equation uΔΔ(t)+λa(t)f(u(σ(t)))=0, t∈[0,1], satisfying either the conjugate boundary conditions u(0)=u(σ(1))=0 or the right focal boundary conditions u(0)=uΔ(σ(1))=0, where a and f are positive. We show that there exists a λ∗>0 such that the above boundary value problem has at least two, one and no positive solutions for 0<λ<λ∗, λ=λ∗ and λ>λ∗, respectively. Furthermore, by using the semiorder method on cones of the Banach space, we establish an existence and uniqueness criterion for positive solution of the problem. In particular, such a positive solution uλ(t) of the problem depends continuously on the parameter λ, i.e., uλ(t) is nondecreasing in λ, limλ→0+‖uλ‖=0 and limλ→+∞‖uλ‖=+∞. 相似文献
10.
Dong Li 《Advances in Mathematics》2009,220(4):1171-1056
Consider the focusing mass-critical nonlinear Hartree equation iut+Δu=−(−2|⋅|∗2|u|)u for spherically symmetric initial data with ground state mass M(Q) in dimension d?5. We show that any global solution u which does not scatter must be the solitary wave eitQ up to phase rotation and scaling. 相似文献
11.
Marcelo Montenegro 《Journal of Mathematical Analysis and Applications》2011,384(2):591-596
We prove finite time extinction of the solution of the equation ut−Δu+χ{u>0}(u−β−λf(u))=0 in Ω×(0,∞) with boundary data u(x,t)=0 on ∂Ω×(0,∞) and initial condition u(x,0)=u0(x) in Ω, where Ω⊂RN is a bounded smooth domain, 0<β<1 and λ>0 is a parameter. For every small enough λ>0 there exists a time t0>0 such that the solution is identically equal to zero. 相似文献
12.
K.J. Brown 《Journal of Differential Equations》2003,193(2):481-499
The Nehari manifold for the equation −Δu(x)=λa(x)u(x)+b(x)|u(x)|ν−1u(x) for x∈Ω together with Dirichlet boundary conditions is investigated. Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form t→J(tu) where J is the Euler functional associated with the equation) we discuss how the Nehari manifold changes as λ changes and show how existence and non-existence results for positive solutions of the equation are linked to properties of the manifold. 相似文献
13.
Ruyun Ma 《Journal of Mathematical Analysis and Applications》2011,384(2):527-535
We consider the existence of positive ω-periodic solutions for the equation
u′(t)=a(t)g(u(t))u(t)−λb(t)f(u(t−τ(t))), 相似文献
14.
Songzhe Lian Chunling Cao Hongjun Yuan 《Journal of Mathematical Analysis and Applications》2008,342(1):27-38
The authors of this paper study the Dirichlet problem of the following equation
ut−div(|u|ν(x,t)∇u)=f−|u|p(x,t)−1u. 相似文献
15.
In this paper we study the maximal regularity property for non-autonomous evolution equations t∂u(t)+A(t)u(t)=f(t), u(0)=0. If the equation is considered on a Hilbert space H and the operators A(t) are defined by sesquilinear forms a(t,⋅,⋅) we prove the maximal regularity under a Hölder continuity assumption of t→a(t,⋅,⋅). In the non-Hilbert space situation we focus on Schrödinger type operators A(t):=−Δ+m(t,⋅) and prove Lp−Lq estimates for a wide class of time and space dependent potentials m. 相似文献
16.
Fuyi Li Yanbiao Zhang Yuhua Li 《Journal of Mathematical Analysis and Applications》2008,344(1):417-428
In this paper, we consider the fourth-order Neumann boundary value problem u(4)(t)−2u″(t)+u(t)=f(t,u(t)) for all t∈[0,1] and subject to u′(0)=u′(1)=u?(0)=u?(1)=0. Using the fixed point index and the critical group, we establish the existence theorem of solutions that guarantees the problem has at least one positive solution and two sign-changing solutions under certain conditions. 相似文献
17.
The paper studies the longtime behavior of solutions to the initial boundary value problem (IBVP) for a nonlinear wave equation arising in an elastic waveguide model utt−Δu−Δutt+Δ2u−Δut−Δg(u)=f(x). It proves that when the space dimension N≤5, under rather mild conditions the dynamical system associated with the above-mentioned IBVP possesses a global attractor which is connected and has finite fractal and Hausdorff dimension. 相似文献
18.
In this paper we study the large time behavior of non-negative solutions to the Cauchy problem of ut=Δum−uq in RN×(0,∞), where m>1 and q=qc≡m+2/N is a critical exponent. For non-negative initial value u(x,0)=u0(x)∈L1(RN), we show that the solution converges, if u0(x)(1+|x|)k is bounded for some k>N, to a unique fundamental solution of ut=Δum, independent of the initial value, with additional logarithmic anomalous decay exponent in time as t→∞. 相似文献
19.
We analyse the dynamics of the non-autonomous nonlinear reaction-diffusion equation
ut−Δu=f(t,x,u), 相似文献
20.
Bo Liang 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(11):3815-3828
The paper first study the steady-state thin film type equation
∇⋅(un|∇Δu|q−2∇Δu)−δumΔu=f(x,u) 相似文献