Large time behavior and L−L estimate of solutions of 2-dimensional nonlinear damped wave equations

We show the asymptotic behavior of the solution to the Cauchy problem of the two-dimensional damped wave equation. It is shown that the solution of the linear damped wave equation asymptotically decompose into a solution of the heat and wave equations and the difference of those solutions satisfies the L^p−L^q type estimate. This is a two-dimensional generalization of the three-dimensional result due to Nishihara (Math. Z. 244 (2003) 631). To show this, we use the Fourier transform and observe that the evolution operators of the damped wave equation can be approximated by the solutions of the heat and wave equations. By using the L^p−L^q estimate, we also discuss the asymptotic behavior of the semilinear problem of the damped wave equation with the power nonlinearity |u|^αu. Our result covers the whole super critical case α>1, where the α=1 is well known as the Fujita exponent when n=2.     

L p - L q estimates of solutions to the damped wave equation in 3-dimensional space and their application

 It has been asserted that the damped wave equation has the diffusive structure as t→∞. In this paper we consider the Cauchy problem in 3-dimensional space for the linear damped wave equation and the corresponding parabolic equation, and obtain the L ^p −L ^q estimates of the difference of each solution, which represent the assertion precisely. Explicit formulas of the solutions are analyzed for the proof. The second aim is to apply the L ^p −L ^q estimates to the semilinear damped wave equation with power nonlinearity. If the power is larger than the Fujita exponent, then the time global existence of small weak solution is proved and its optimal decay order is obtained. Received: 8 June 2001; in final form: 12 August 2002 / Published online: 1 April 2003 Mathematical Subject Classification (2000): 35L15.     

Modulation space estimates for damped fractional wave equation

Instead of the L~p estimates,we study the modulation space estimates for the solution to the damped wave equation.Decay properties for both the linear and semilinear equations are obtained.The estimates in modulation space differ in many aspects from those in L~p space.     

Qualitative theory for strongly damped wave equations

We investigate the asymptotic periodicity, L^p‐boundedness, classical (resp., strong) solutions, and the topological structure of solutions set of strongly damped semilinear wave equations. The theoretical results are well complemented with a set of very illustrating applications.     

Asymptotic behavior for the Navier–Stokes equations in 2D exterior domains

We show that the L^p spatial–temporal decay rates of solutions of incompressible flow in an 2D exterior domain. When a domain has a boundary, pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary. To overcome the difficulty, we adopt the ideas in He, Xin [C. He, Z. Xin, Weighted estimates for nonstationary Navier–Stokes equations in exterior domain, Methods Appl. Anal. 7 (3) (2000) 443–458], and our previous results [H.-O. Bae, B.J. Jin, Asymptotic behavior of Stokes solutions in 2D exterior domains, J. Math. Fluid Mech., in press; H.-O. Bae, B.J. Jin, Temporal and spatial decay rates of Navier–Stokes solutions in exterior domains, submitted for publication]. For the spatial decay rate estimate, we first extend temporal decay rate result of the Navier–Stokes solutions for general L^p space when the initial velocity is in , 1<rq<∞ (1<r<q=∞).     

The L-L estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media

We first obtain the L^p-L^q estimates of solutions to the Cauchy problem for one-dimensional damped wave equation

     

Global weak solutions of the Cauchy problem for semilinear pseudo-hyperbolic equations

The Cauchy problem for a class of semilinear pseudo-hyperbolic equations is considered. For the corresponding linear problems, we obtain L _p → L _q estimates. By using these estimates, we prove global solvability theorems. We also establish the behavior of solutions as t → + ∞.     

Partial applicability of Moser's method to nonlinear elliptic systems

We consider nonlinear elliptic systems of divergent-type second-order partial differential equations with solutionsu W _p ¹ . It is proved thatDu L _q with someq (p; +) and it is explicitly shown howq depends on the ellipticity modulus of the system. Some conditions on the ellipticity modulus are obtained under which the solutions satisfy the Hölder conditions and the Liouville theorem holds.Translated fromMatematicheskie Zametki, Vol. 58, No. 4, pp. 547–557, October, 1995.     

On well-posedness of semilinear parabolic and elliptic problems in the hyperbolic space

We investigate existence and uniqueness of solutions to semilinear parabolic and elliptic equations in bounded domains of the n-dimensional hyperbolic space (n?3). Lp→Lq estimates for the semigroup generated by the Laplace-Beltrami operator are obtained and then used to prove existence and uniqueness results for parabolic problems. Moreover, under proper assumptions on the nonlinear function, we establish uniqueness of positive classical solutions and nonuniqueness of singular solutions of the elliptic problem; furthermore, for the corresponding semilinear parabolic problem, nonuniqueness of weak solutions is stated.     

A damped hyerbolic equation with critical exponent

The aim of this paper is twofold. First, we initiate a detailed study of the so-called X^s _θ spaces attached to a partial differential operator. This include localization, duality, microlocal representation, subelliptic estimates, solvability and L^p (L^q ) estimates. Secondly, we obtain some theorems on the unique continuation of solutions to semilinear second order hyperbolic equations across strongly pseudo-convex surfaces. These results are proved using some new L^p → L^q Carleman estimates, derived using the X^s _θ spaces. Our theorems cover the subcritical case; in the critical case, the problem remains open. Similar results hold for higher order partial differential operators, provided that characteristic set satisfies a curvature conditions.     

The best trigonometric and bilinear approximations for functions of many variables from the classesB p, θ r . II

Order estimates are obtained for the best trigonometric and bilinear approximations of the classesB _p, ^r of functions of many variables in the metricL _q, wherep andq connected by certain relations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 10, pp. 1411–1423, October, 1993.     

Space-time means and solutions to a class of nonlinear parabolic equations

Cauchy problem and initial boundary value problem for nonlinear parabolic equation inCB([0,T):L ^p ) orL ^q (0,T; L ^p ) type space are considered. Similar to wave equation and dispersive wave equation, the space-time means for linear parabolic equation are shown and a series of nonlinear estimates for some nonlinear functions are obtained by space-time means. By Banach fixed point principle and usual iterative technique a local mild solution of Cauchy problem or IBV problem is constructed for a class of nonlinear parabolic equations inCB([0,T);L ^p orL ^q (0,T; L ^p ) with ϕ(x)∈L ^r . In critical nonlinear case it is also proved thatT can be taken as infinity provided that ||ϕ(x)||_r is sufficiently small, where (p,q,r) is an admissible triple. Project supported by the National Natural Science Foundation of China (Grant No. 19601005).     

Gradient estimates in Orlicz spaces for quasilinear elliptic equation

In this paper we generalize classical L^q$L^q$, q≥p$q\ge p$, estimates of the gradient to the Orlicz space for weak solutions of quasilinear elliptic equations of p-Laplacian type.     

The best trigonometric approximations and the Kolmogorov diameters of the Besov classes of functions of many variables

The order estimates for the best trigonometric approximations and the Kolmogorov diameters of the classesB _p, ^r of functions of many variables in the spaceL _q are obtained for certain values of the parametersp andq.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 5, pp. 663–675, May, 1993.     

Trigonometric Widths of the Classes L β,p Ψ of Functions of Many Variables

We obtain order estimates for the trigonometric widths of the classes L _,p of periodic functions of many variables in the space L _q for 1 < p 2 q < p/(p – 1).     

Maximal subgroups of symmetric groups defined on projective spaces over finite fields

Let PL(n, q) be a complete projective group of semilinear transformations of the projective space P(n–1, q) of projective degree n–l over a finite field of q elements; we consider the group in its natural 2-transitive representation as a subgroup of the symmetric group S(P*(n–1, q)) on the setp*(n–1),q=p(n–1,q)/{O}. In the present note we show that for arbitrary n satisfying the inequality n>4[(qⁿ–1)/(q^n–1–1)] [in particular, for n>4(q +l)] and for an arbitrary substitutiong s (p*(n–1,q))pL(n,q) the group PL(n,q), g contains the alternating group A(P* (n–1,q)). Forq=2, 3 this result is extended to all n3.Translated from Matematicheskie Zametki, Vol. 16, No. 1, pp. 91–100, July, 1974.The author expresses his sincere thanks to M. M. Glukhov for his interest in his work.     

On the periodic solutions of the second-order wave equations. V

It is established that the linear problemu _u –a ² u _xx =g(x,t),u(0,t) =u(x, t + T) =u(x,t) is always solvable in the function spaceA = {g:g(x,t) =g(x,t+T) =g( –x,t) = –g(–x,t)} provided thataTq = (2p – 1) and (2p – 1,q) = 1, wherepandq are integer numbers. To prove this statement, an exact solution is constructed in the form of an integral operator, which is used to prove the existence of a solution of a periodic boundary-value problem for a nonlinear second-order wave equation. The results obtained can be used when studying the solutions to nonlinear boundary-value problems by asymptotic methods.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1115–1121, August, 1993.     

On well-posedness of the semilinear heat equation on the sphere

We are concerned with existence, uniqueness and nonuniqueness of nonnegative solutions to the semilinear heat equation in open subsets of the n-dimensional sphere. Existence and uniqueness results are obtained using L ^p ?? L ^q estimates for the semigroup generated by the Laplace?CBeltrami operator. Moreover, under proper assumptions on the nonlinear function, we establish nonuniqueness of weak solutions, when n??? 3; to do this, we shall prove uniqueness of positive classical solutions and nonuniqueness of singular solutions of the corresponding semilinear elliptic problem.     

Quasilinearization Method for Nonlinear Elliptic Boundary-Value Problems

The quasilinearization method is developed for strong solutions of semilinear and nonlinear elliptic boundary-value problems. We obtain two monotone, L^p-convergent sequences of approximate solutions. The order of convergence is two. The tools are some results on the abstract quasilinearization method and from weakly–near operators theory.