in this paper, we consider the Cauchy problem for the wave equation with time dependent damping b(t)u(t) and absorbed semilinear term vertical bar u vertical bar(rho-1)u. Here, b(t) = b(0)(1+ t)(-beta) with -1 < beta < 1 and b(0) > 0. Using the weighted energy method, we obtain the L(1) and L(2) decay rates of the solution. which coincide to those for self-similar solutions to the corresponding parabolic equation when 1 < rho < rho(F)(N): = 1 + 2/N @ 2009 Elsevier Inc. All rights reserved.

We consider the Cauchy problem for the damped wave equation
u(tt) - Delta u + u(t) = vertical bar u vertical bar(rho-1)u, (t, x) epsilon R+ x R-N
and the heat equation
phi(t) - Delta phi = vertical bar phi vertical bar(rho-1)phi, (t, x) epsilon R+ x R-N
If the data is small and slowly decays likely c(1) (1 + vertical bar x vertical bar)(-kN), 0 < k <= 1, then the critical exponent is rho(c)(k) = 1 + 2/kN for the semilinear heat equation. In this paper it is shown that in the supercrifical case there exists a unique time global solution to the Cauchy problem for the semilinear heat equation in any dimensional space R-N, whose asymptotic profile is given by
Phi(0)(t, x) = integral(RN)e(-vertical bar x-y vertical bar 2/4t)/(4 pi t)(N/2) c(1)/(1+vertical bar y vertical bar(2))(kN/2)dy
provided that the data phi(0) satisfies lim(vertical bar x vertical bar ->infinity)< x >(kN)phi(0)(x) = c(1) (not equal 0). Even in the semilinear damped wave equation in the supercritical case a time global solution u with the data (u, u(t))(0, x) = (u(0), u(1))(x) is shown in low dimensional spaces R-N, N = 1, 2, 3, to have the same asymptotic profile Phi(0)(t, x) provided that lim(vertical bar x vertical bar ->infinity)< x >(kN)(u(0) + u(1))(x) = c(1) (not equal 0). Those proofs are given by elementary estimates on the explicit formulas of solutions. (C) 2007 Elsevier Inc. All rights reserved.

We consider the Cauchy problem for the damped wave equation with absorption
u(tt) - Delta u + u(t) + vertical bar u vertical bar(p-1) u = 0, (t,x) is an element of R+ x R-N.
The behavior of u as t -> infinity is expected to be same as that for the corresponding heat equation phi(t) - Delta phi + vertical bar phi vertical bar(p-1) phi = 0, (t, x) is an element of R+ x R-N. In the subcritical case 1 < p < p(c) (N): = 1 + 2/N there exists a similarity solution W-b (t, x) with the form t(-1/(p-1)) f (x/root t) depending on b = lim(vertical bar x vertical bar ->infinity vertical bar x vertical bar)(2/(p- 1)) f (vertical bar x vertical bar) >= 0. Our first aim is to show the decay rates
(parallel to u(t)parallel to(L2), parallel to u(t)parallel to(Lp+1), parallel to del u(t)parallel to(L2)) = 0 (t (-) 1/p-1+N/4, t (-) 1/p-1+N/2(p+1), t (-) 1/p-1-1/2+N/4) (**)
provided that the initial data without initial data size restriction spatially decays with reasonable polynomial order. The decay rates (**) are sharp in the sense that they are same as those of the similarity solution. The second aim is to show that the Gauss kernel is the asymptotic profile in the supercritical case, which has been shown in case of one-dimensional space by Hayashi. Kaikina and Naumkin [N. Hayashi, E.I. Kaikina, P.I. Naumkin, Asymptotics for nonlinear damped wave equations with large initial data, preprint, 2004]. We C-energy show this assertion in two- and three-dimensional space. To prove our results, both the weighted L-2-energy method and the explicit formula of solutions will be employed. The weight is an improved one originally developed in [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489]. (c) 2006 Elsevier Inc. All rights reserved.

We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space
psi(t) = -(1 - alpha) psi - theta(x) + alpha psi(xx), (t, x) is an element of (0, infinity) x R theta(t) = -(1 - alpha) theta + nu(2)psi(x) + alpha theta(xx) + 2 psi theta(x),
with 0 < nu(2) < 4 alpha(1 - alpha), 0 < alpha < 1. S. Q. Tang and H. Zhao [4] have considered the problem and obtained the optimal decay property for suitably small data. In this paper we derive the asymptotic profile using the Gauss kernel G(t, x), which shows the precise behavior of solution as time tends to infinity. In fact, we will show that the asymptotic formula
[GRAPHICS]
holds, where D-0, beta(0) are determined by the data. It is the key point to reformulate the system to the nonlinear parabolic one by suitable changing variables.

We consider the Cauchy problem for the damped wave equation with absorption
u(tt) - Delta u + u(t) + \u\(rho-1) u = 0, (t, x) is an element of R+ x R-N, (*)
with N = 3,4. The behavior of u as t --> infinity is expected to be the Gauss kernel in the supercritical case rho > rho(c) (N) := 1 + 2/N. In fact, this has been shown by Karch [12] (Studia Math., 143 (2000), 175-197) for rho > 1+ 4/N (N = 1, 2,3), Hayashi, Kaikina and Naumkin [8] (preprint (2004)) for rho > rho(c) (N)(N = 1) and by Ikehata, Nishihara and Zhao [11] (J. Math. Anal. Appl., 313 (2006), 598-610) for rho(c)(N) < rho <= 1 + 4/N (N = 1, 2) and rho(c) (N) < rho < 1 + 3/N (N = 3). Developing their result, we will show the behavior of solutions for rho(c) (N) < rho <= 1 + 4/N (N = 3), rho(c) (N) < rho < 1 + 4/N (N = 4). For the proof, both the weighted L-2-energy method with an improved weight developed in Todorova and Yordanov [22] (J. Differential Equations, 174 (2001), 464-489) and the explicit formula of solutions are still usefully used. This method seems to be not applicable for N = 5, because the semilinear term is not in C-2 and the second derivatives are necessary when the explicit formula of solutions is estimated.

On the half-line R+ = (0, infinity) the initial-boundary value problems with null-Dirichlet boundary for both the semilinear heat equation and damped wave equation are considered. The critical exponent p(c) (N, k) of semilinear term for the existence and nonexistence about the semilinear heat equation on the halved space D-N,D-k = R-+(k) = RN-k is given by p(c)(N, k) = 1 + 2/(N + k) (J. Appl. Math. Phys. 39 (1988) 135-149; Arch. Rational Mech. Anal. 109 (1990) 63-71). Since the damped wave equation is expected to be close to the heat equation (J. Differential Equations 191 (2003) 445-469; Math. Z. 244 (2003) 631-649), the critical exponent for the semilinear damped wave equation is expected to be same as that of the semilinear heat equation. However, there is no blow-up result on the halved space for the damped wave equation. In this paper, the exponent pc(l, 1) = 2 is shown to be critical for the existence and nonexistence of time-global solution to both the semilinear heat equation and damped wave equation on the half-line R+, together with the derivation of the blow-up time. For the proof the explicit formulas of solutions are used in a similar fashion to those in Li and Zhou (Discrete Continuous Dynamic Systems 1 (1995) 503-520). (c) 2005 Elsevier Ltd. All rights reserved.

This paper is concerned with global stability of strong rarefaction waves of the Jin-Xin relaxation model for the p-system. The proofs are given by an elementary energy method and the existence of a positively invariant region obtained by Serre [Serre, D. (2000). Relaxations semi-lineaire et cinetique des systemes de Lois de conservation. Ann. Inst. H. Poincare Anal. Non Lineaire 17(2):169-1921 plays an important role in our analysis.

This paper is concerned with the time-asymptotic behavior toward strong rarefaction waves of solutions to one-dimensional compressible Navier-Stokes equations. Assume that the corresponding Riemann problem to the compressible Euler equations can be solved by rarefaction waves (V-R, U-R, S-R)(t, x). If the initial data (v(0), u(0), s(0))(x) to the nonisentropic compressible Navier-Stokes equations is a small perturbation of an approximate rarefaction wave constructed as in [ S. Kawashima, A. Matsumura, and K. Nishihara, Proc. Japan Acad. Ser. A, 62 (1986), pp. 249-252], then we show that, for the general gas, the Cauchy problem admits a unique global smooth solution (v, u, s)(t, x) which tends to (V-R, U-R, S-R)(t, x) as t tends to infinity. A global stability result can also be established for the nonisentropic ideal polytropic gas, provided that the adiabatic exponent gamma is close to 1. Furthermore, we show that for the isentropic compressible Navier-Stokes equations, the corresponding global stability result holds, provided that the resulting compressible Euler equations are strictly hyperbolic and both characteristical fields are genuinely nonlinear. Here, global stability means that the initial perturbation can be large. Since we do not require the strength of the rarefaction waves to be small, these results give the nonlinear stability of strong rarefaction waves for the one-dimensional compressible Navier-Stokes equations.

We first obtain the L-p-L-q estimates of solutions to the Cauchy problem for one-dimensional damped wave equation
V-tt - V-xx + V-t = 0, (V, V-t)\(t=0) = (V-0, V-1)(x), (x, t) cis an element of R x R+,
corresponding to that for the parabolic equation
phi(t) - phi(xx) = 0 phi\(t=0) = (V-0 + V-1)(x).
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etc. for 1less than or equal toqless than or equal topless than or equal toinfinity. To show (*), the explicit formula of the damped wave equation will be used. To apply the estimates to nonlinear problems is the second aim. We will treat the system of a compressible flow through porous media. The solution is expected to behave as the diffusion wave, which is the solution to the porous media equation due to the Darcy law. When the initial data has the same constant state at +/- infinity, a sharp L-p-convergence rate for pgreater than or equal to2 has been recently obtained by Nishihara (Proc. Roy. Soc. Edinburgh, Sect. A, 133A (2003), 1-20) by choosing a suitably located diffusion wave. We will show the L-1 convergence, ;applying (*). (C) 2003 Elsevier Science (USA). All rights reserved.

It has been asserted that the damped wave equation has the diffusive structure as t --> infinity. 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.

We present an abstract theory of the diffusion phenomenon for second order linear evolution equations in a Hilbert space. To derive the diffusion phenomenon, a new device developed in Ikehata-Matsuyama [5] is applied. Several applications to damped linear wave equations in unbounded domains are also given.

This paper is concerned with the convergence rates to viscous shock profile for general scalar viscous conservation laws. Compared with former results in this direction, the main novelty in this paper lies in the fact that the initial disturbance can be chosen arbitrarily large, This answers positively an open problem proposed by A. Matsumura in [12] and K. Nishihara in [16]. Our analysis is based on the L-1-stability results obtained by H. Freistuhler and D. Serre in [1].

The "inflow problems" for a one-dimensional compressible barotropic flow on the half-line R+ = (0, +infinity) is investigated. Not only classical waves but also the new wave, which is called the "boundary layer solution", arise. Large time behaviors of the solutions to be expected have been classified in terms of the boundary values by [A. Matsumura, Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas, to appear in Proceedings of IMS Conference on Differential Equations from Mechanics, Hong Kong, 1999]. In this paper we give the rigorous proofs of the stability theorems on both the boundary layer solution and a superposition of the boundary layer solution and the rarefaction wave.

Hsiao and Serre in [Chinese Ann. Math. Ser. B, 16B (1995), pp. 1-14] showed the solution to the system
[GRAPHICS]
with initial data
(v, u, s) (0, x) = (v(0), u(0), s(0)) (x) ((v) under bar, u +/-, (s) under bar) as x --> +/-infinity
tends to the following nonlinear parabolic equation time-asymptotically:
[GRAPHICS]
In this paper we find its convergence rate, which will be optimal.

The initial-boundary value problem on the negative half-line
[GRAPHICS]
is considered, subsequently to T.-P. Liu and K. Nishihara (1997, J. Differential Equations 133, 296-320). Here, the flux f is a smooth function satisfying flu,) = 0 and the Oleinik shock condition f(phi) < 0 for u(+) < phi < u(-) if u(+) < u(-) or f(phi) > 0 for u(+) > phi > u(-) if u(+) < u(-). In this situation the corresponding Cauchy problem on the whole line R = (-<infinity>,infinity) to (*) has a stationary viscous shock wave phi (x + x,) for any fixed x,. Our aim in this paper is to show that the solution U(x, t) to (*) behaves as phi (x + d(t)) with d(t) = O(In t) as t --> infinity under the suitable smallness conditions. When f = u(2)/2, the fact was shown by T.-P. Liu and S.-H. Yu (1997, Arch. Rational Mech. Anal. 139, 57-82), based on the Hopf-Cole transformation. Our proof is based on the weighted energy method. (C) 2001 Academic Press.

In this paper we investigate the large time behavior of solutions to the Cauchy problem on R for a one-dimensional thermoelastic system with dissipation. When the initial data is suitably small, (S. Zheng, Chin. Ann. Math. 8B(1987), 142-155) established the global existence and the decay properties of the solution. Our aim is to improve the results and to obtain the sharper decay properties, which seems to be optimal. The proof is given by the energy method and the Green function method.

The initial-boundary value problem on the half-line R+. = (0, infinity) for a system of barotropic viscous flow b(t) - u(x) = 0, u(t) + p(v)(x) = mu ux/v)(x) is investigated. where the pressure p(v) = v(-gamma) (gamma greater than or equal to 1) for the specific volume v > 0. Note that the boundary value at x = 0 is given only for the velocity u, say u(-), and that the initial data (v(0), u(0)) (x) have the constant states (v(+), u(+)) at x = +infinity with v(0)(x) > 0, v(+) > 0. If u(-) < u(+), then there is a unique v(-) such that (v(+),u(+)) is an element of R-2(v(-),u(-)) (the 2-rarefaction curve) and hence there exists the 2-rarefaction wave (v(2)(R), u(2)(R)) (x/T) connecting (v(-), u(-)) with (v(+), u(+)). Our assertion is that, if u(-) < u(+), then there exists a global solution (v, u) (t, x) in C-0 ([0, infinity); H-1 (R+)), which tends to the 2-rarefaction wave (v(2)(R), u(2)(R))(x/t)\(x greater than or equal to 0) as t --> infinity in the maximum norm, with no smallness condition on \u(+) - u(-)\ and parallel to(v(0) - v(+), u(0) - u(+))parallel to(H1), nor restriction on gamma (greater than or equal to 1). A similar result to the corresponding Cauchy problem is also obtained. The proofs are given by an elementary L-2-energy method.

In this paper, we study the p-system with frictional damping and show that the solutions time-asymptotically tend to the nonlinear diffusion waves governed by the classical Darcy's law. By introducing an approximate Green function we obtain the optimal L-p, 2 less than or equal to p less than or equal to + infinity, convergence rate of the solution, which is a perturbation of the nonlinear diffusion wave, to the hyperbolic system. (C) 2000 Academic Press.

This paper is concerned with asymptotic behavior of solutions of a one-dimensional barotropic flow governed by v(t) - u(x) = 0, u(t) + p(v)(x) = mu(u(x)/v)(x) on R-+(1) with boundary. The initial data of (v, u) have constant states (v(+), u(+)) at +infinity and the boundary condition at x = 0 is given only on the velocity u, say u_. By virtue of the boundary effect the solution is expected to behave as outgoing wave. Therefore, when u_ < u(+), v(-) is determined as (v(+), u(+)) is an element of R-2(v(-), u(-)), 2-rarefaction curve for the corresponding hyperbolic system, which admits the a-rarefaction wave (v(r), u(r))(x/t) connecting two constant states (v(-), u(-)) and (v(+), u(+)). Our assertion is that the solution of the original system tends to the restriction of (v(r), u(r))(x/t) to R-+(1) as t --> infinity provided that both the initial perturbations and \(v(+) - v(-), u(+) - u(-))\ are small. The result is given by an elementary L-2 energy method.

We consider the asymptotic behaviour of solutions to the p-system with linear damping on the half-line R+ = (0, infinity),
v(t) - u(x) = 0, u(t) + p(v)(x) = -alpha u,
with the Dirichlet boundary condition u/(x=0) =0 or the Neumann boundary condition u(x)/(x=0) =0 The initial date (v(0),u(0))(x) has the constant state (v(+),u(+)) at x = infinity. L. Hsiao and T.-P. Liu [ Commun. Math. Phys. 143 (1992), 599-605] have shown that the solution to the corresponding Cauchy problem behaves like diffusion wave, and K. Nishihara [J. Differential Equations 131 (1996), 171-188; 137 (1997), 384-395] has proved its optimal convergence rate. Our main concern in this paper is the boundary effect. In the case of null-Dirichlet boundary condition on u, the solution (u, u) is proved to lend to (v(+), 0) as t tends to infinity. Its optimal convergence rate is also obtained by using the Green function of the diffusion equation with constant coefficients. In the case of null-Neumann boundary condition on u, v(0, t) is conservative and v(0, t) equivalent to v(0)(0) by virtue of the first equation, so that v(x, t) is expected to tend to the diffusion wave (v) over bar(x, t) connecting v(0)(0) and u,. In fact the solution in (v, u)(x, t) is proved to tend to ((v) over bar(x, t), 0) In the special case v(0)(0)=v(+), the optimal convergence rate is also obtained. However, this is not known in the case v(0)(0) not equal v(+). (C) 1999 Academic Press.