Lp Decay problem for the dissipative wave equation in even dimensions

We study L^p decay estimates of the solution to the Cauchy problem for the dissipative wave equation in even dimensions: (□+?_t)u=0 in ?^N × (0,∞) for even N=2n?2 with initial data (u,?_tu)∣_t=0 =(u₀,u₁). The representation formulas of the solution u(t)=?_tS(t)u₀ + S(t)(u₀+u₁) provide the sharp estimates on L^p norms with p?1. Copyright © 2004 John Wiley & Sons, Ltd.     

decay problem for the dissipative wave equation in odd dimensions

Consider the Cauchy problem in odd dimensions for the dissipative wave equation: (□+∂_t)u=0 in with (u,∂_tu)|_t=0=(u₀,u₁). Because the L² estimates and the L^∞ estimates of the solution u(t) are well known, in this paper we pay attention to the L^p estimates with 1p<2 (in particular, p=1) of the solution u(t) for t0. In order to derive L^p estimates we first give the representation formulas of the solution u(t)=∂_tS(t)u₀+S(t)(u₀+u₁) and then we directly estimate the exact solution S(t)g and its derivative ∂_tS(t)g of the dissipative wave equation with the initial data (u₀,u₁)=(0,g). In particular, when p=1 and n1, we get the L¹ estimate: u(t)_L¹Ce^−t/4(u_0W^n,1+u_1W^n−1,1)+C(u_0L¹+u_1L¹) for t0.     

Decay estimates for dissipative wave equations in exterior domains

Consider the initial boundary value problem for the linear dissipative wave equation (□+∂_t)u=0 in an exterior domain . Using the so-called cut-off method together with local energy decay and L² decays in the whole space, we study decay estimates of the solutions. In particular, when N?3, we derive L^p decays with p?1 of the solutions. Next, as an application of the decay estimates for the linear equation, we consider the global solvability problem for the semilinear dissipative wave equations (□+∂_t)u=f(u) with f(u)=|u|^α+1,|u|^αu in an exterior domain.     

Lp-Lq DECAY ESTIMATES FOR HYPERBOLIC EQUATIONS WITH OSCILLATIONS IN COEFFICIENTS

This work is concerned with the proof of Lp -Lq decay estimates for solutions of the Cauchy problem for utt -λ2(t)b2(t)/Δu =0. The coefficient consists of an increasing smooth function λ and an oscillating smooth and bounded function b which are uniformly separated from zero. The authors‘ main interest is devoted to the critical case where one has an interesting interplay between the growing and the oscillating part.     

Longtime behavior for a nonlinear wave equation arising in elasto‐plastic flow

The paper studies the longtime behavior of solutions to the initial boundary value problem (IBVP) for a nonlinear wave equation arising in elasto‐plastic flow u_tt?div{|?u|^m?1?u}?λΔu_t+Δ²u+g(u)=f(x). It proves that under rather mild conditions, the dynamical system associated with above‐mentioned IBVP possesses a global attractor, which is connected and has finite Hausdorff and fractal dimension in the phase spaces X₁=H(Ω) × L²(Ω) and X=(H³(Ω)∩H(Ω)) × H(Ω), respectively. Copyright © 2008 John Wiley & Sons, Ltd.     

Global existence,asymptotic behaviour,and global non‐existence of solutions for damped non‐linear wave equations of Kirchhoff type in the whole space

We study the global existence, asymptotic behaviour, and global non‐existence (blow‐up) of solutions for the damped non‐linear wave equation of Kirchhoff type in the whole space: u_tt+u_t=(a+b∥∇u∥^2γ)Δu+∣u∣^αu in ℝ^N×ℝ₊ for a, b⩾0, a+b>0, γ⩾1, and α>0, with initial data u(x, 0)=u₀(x) and u_t(x, 0)=u₁(x). Copyright © 2000 John Wiley & Sons, Ltd.     

Divergence of Infinite-Variance Nonradial Solutions to the 3D NLS Equation

We consider solutions u(t) to the 3d NLS equation i? _t u + Δu + |u|² u = 0 such that ‖xu(t)‖ _{L ²}  = ∞ and u(t) is nonradial. Denoting by M[u] and E[u], the mass and energy, respectively, of a solution u, and by Q(x) the ground state solution to ?Q + ΔQ + |Q|² Q = 0, we prove the following: if M[u]E[u] < M[Q]E[Q] and ‖u ₀‖ _{L ²} ‖?u ₀‖ _{L ²}  > ‖Q‖ _{L ²} ‖?Q‖ _{L ²} , then either u(t) blows-up in finite positive time or u(t) exists globally for all positive time and there exists a sequence of times t _n  → + ∞ such that ‖?u(t _n )‖ _{L ²}  → ∞. Similar statements hold for negative time.     

An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation

We report a new unconditionally stable implicit alternating direction implicit (ADI) scheme of O(k² + h²) for the difference solution of linear hyperbolic equation u_tt + 2αu_t + β²u = u_xx + u_yy + f(x, y, t), αβ ≥ 0, 0 < x, y < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where α > 0 and β ≥ 0 are real numbers. The resulting system of algebraic equations is solved by split method. Numerical results are provided to demonstrate the efficiency and accuracy of the method. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 684–688, 2001     

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.     

Solution representations for a wave equation with weak dissipation

We consider the Cauchy problem for the weakly dissipative wave equation □v+μ/1+t v_t=0, x∈?ⁿ, t≥0 parameterized by μ>0, and prove a representation theorem for its solutions using the theory of special functions. This representation is used to obtain L_p–L_q estimates for the solution and for the energy operator corresponding to this Cauchy problem. Especially for the L₂ energy estimate we determine the part of the phase space which is responsible for the decay rate. It will be shown that the situation depends strongly on the value of μand that μ=2 is critical. Copyright © 2004 John Wiley & Sons, Ltd.     

Stability of planar diffusion wave for nonlinear evolution equation

It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f’(u) > 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method.     

On the Lp–Lq maximal regularity for Stokes equations with Robin boundary condition in a bounded domain

We obtain the L_p–L_q maximal regularity of the Stokes equations with Robin boundary condition in a bounded domain in ?ⁿ (n?2). The Robin condition consists of two conditions: v ? u=0 and αu+β(T(u, p)v – 〈T(u, p)v, v〉v)=h on the boundary of the domain with α, β?0 and α+β=1, where u and p denote a velocity vector and a pressure, T(u, p) the stress tensor for the Stokes flow and v the unit outer normal to the boundary of the domain. It presents the slip condition when β=1 and non‐slip one when α=1, respectively. The slip condition is appropriate for problems that involve free boundaries. Copyright © 2006 John Wiley & Sons, Ltd.     

On decay properties of solutions for degenerate strongly damped wave equations of Kirchhoff type

We consider the initial-boundary value problem for the degenerate strongly damped wave equations of Kirchhoff type: . For all t?0, we will give the optimal decay estimate C−1(1+t)^−1/γ?‖A1/2u(t)^‖2?C(1+t)^−1/γ, when either the coefficient ρ is appropriately small or the initial data are appropriately small. And, we will show a decay property of the norm ‖Au(t)^‖2 for t?0.     

Non-homogeneous non-linear damped wave equations in unbounded domains

We present a global existence theorem for solutions of u^tt ? ?_ia_ik (x)?_ku + u_t = ?(t, x, u, u_t, ?u, ?u_t, ?²u), u(t = 0) = u⁰, u(=0)=u¹, u(t, x), t ? 0, x?Ω.Ω equals ?³ or Ω is an exterior domain in ?³ with smoothly bounded star-shaped complement. In the latter case the boundary condition u|_?Ω = 0 will be studied. The main theorem is obtained for small data (u⁰, u¹) under certain conditions on the coefficients a_ik. The L^p - L^q decay rates of solutions of the linearized problem, based on a previously introduced generalized eigenfunction expansion ansatz, are used to derive the necessary a priori estimates.     

退化抛物型方程弱解的存在性

讨论初值为u_0,v_0∈L_+~4(Ω),w∈W~(1,p)(Ω)(p≥2)时退化抛物型方程弱解的存在性.首先利用截断的方法将原问题正则化,化为u_0,v_0∈L_+~∞(Ω)的退化问题,接着对正则化问题的解做估计(这里的估计与具体的截断无关),最后利用弱收敛性,通过取极限的方法证明了原问题解的存在性.     

ON THE DIFFUSION PHENOMENON OF QUASILINEAR HYPERBOLICWAVES

Introduction1.1.ConsiderthefollowingquasilinearhyperbolicCauchyproblemwithlineardamping{:;;!OTt=-:i<:,;>>L06,(11)wherexER",t20,anda(.)isasmoothfunctionsatisfyinga(y)~1 O(lyl")aslyl-0,orEN.(1.2)Thepurposeofthispaperistoshowthat,atleastwhenn53,theasymptoticprofileofthesolutionu(x,t)of(l.1)isgivenbythesolutionv(x,t)ofthecorrespondingparabolicproblem{:;.t>ivj:相似文献   

A new off-step high order approximation for the solution of three-space dimensional nonlinear wave equations

In this paper, we propose a new high accuracy numerical method of O(k² + k²h² + h⁴) based on off-step discretization for the solution of 3-space dimensional non-linear wave equation of the form u_tt = A(x,y,z,t)u_xx + B(x,y,z,t)u_yy + C(x,y,z,t)u_zz + g(x,y,z,t,u,u_x,u_y,u_z,u_t), 0 < x,y,z < 1,t > 0 subject to given appropriate initial and Dirichlet boundary conditions, where k > 0 and h > 0 are mesh sizes in time and space directions respectively. We use only seven evaluations of the function g as compared to nine evaluations of the same function discussed in  and . We describe the derivation procedure in details of the algorithm. The proposed numerical algorithm is directly applicable to wave equation in polar coordinates and we do not require any fictitious points to discretize the differential equation. The proposed method when applied to a telegraphic equation is also shown to be unconditionally stable. Comparative numerical results are provided to justify the usefulness of the proposed method.     

One application of Floquet's theory to Lp–Lq estimates for hyperbolic equations with very fast oscillations

In this paper we consider the strictly hyperbolic equation u_tt−λ²(t)b²(t)Δu=0. The coefficient consists of an increasing function λ=λ(t) and a non‐constant periodic function b=b(t). We study the question for the influence of these parts on L_p–L_q decay estimates for the solution of the Cauchy problem. A fairly wide class of equations will be described for which the influence of the oscillating part dominates. This implies, on the one hand, that there exist no L_p–L_q decay estimates and, on the other hand, that the energy estimate from Gronwall's inequality is near to an optimal one. Copyright © 1999 John Wiley & Sons, Ltd.