AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/501280 501280 Research Article Error Estimates for Solutions of the Semilinear Parabolic Equation in Whole Space Hu Xiaomei 1, 2 Zhao Caidi 1 School of Mathematics and Statistics Hubei University of Science and Technology Xianning 437100 China hebust.edu.cn 2 School of Mathematics and Statistics Central China Normal University Wuhan 430079 China ccnu.edu.cn 2014 872014 2014 01 06 2014 28 06 2014 8 7 2014 2014 Copyright © 2014 Xiaomei Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is focused on the error estimates for solutions of the three-dimensional semilinear parabolic equation with initial data u0L2(3). Employing the energy methods and Fourier analysis technique, it is proved that the error between the solution of the semilinear parabolic equation and that of linear heat equation has the behavior as O((1+t)3/8).

1. Introduction

In this study we consider the Cauchy problem of the following three-dimensional semilinear parabolic equation: (1)tu-Δu+|u|p-2u=0,u(x,0)=u0. Here p>5. u(x,t) is the unknown function at the point (x,t)R3×(0,) and u0 is the initial data.

As an important partial differential equation, the well-posedness and asymptotic behavior of solutions of semilinear parabolic equation has attracted more and more attention and many important results have been investigated (see  and references therein). The mathematical model (1) can be seen as the heat equation with damping and friction effects. From the view on the mathematics point, the nonlinear damping |u|p-2u in (1) may increase the regularity of the weak solutions. However, it will be the main obstacle on the asymptotic behavior of the solutions to the semilinear parabolic equation (1). For the n-dimensional linear heat equation (2)tu-Δu=0,u(x,0)=u0. The fundamental solution is (3)etΔ=E(x-y,t)=(4πt)-n/2e-|x-y|2/4t, and the solution of (2) is expressed as (4)u(x,t)=etΔu0=RnE(x-y,t)u0(y)dy=(4πt)-n/2Rne-|x-y|2/4tu0(y)dy. In particular, the solution u(x,t) of the linear heat equation (2) exhibits the following asymptotic behavior (see ): (5)etΔu0Lq(Rn)Ct-(n/2)(1/p-1/q)u0Lp(Rn),t>0. Compared with the behavior of heat equation (2), it is an interesting problem to consider the influence of the linear damping |u|p-2u in the semilinear parabolic equation (1).

Motivated by the asymptotic results on some nonlinear differential equations in , in this study we will investigate the asymptotic error estimates between the solutions of both the semilinear parabolic equation (1) and the linear parabolic equation (2). Let us give an outline analysis of this question. On one hand, taking p=q=2 in (5), we only derive the bounds of the solution; that is, (6)etΔu0L2(Rn)Cu0L2(Rn),t>0. On the other hand, from the definition of weak solution for the semilinear parabolic equation (1) (see the definition in the next section), we also only get the L2 bounds of weak solution for the semilinear parabolic equation (1) as (7)u(t)L2(Rn)Cu0L2(Rn),t>0. By the direct computation, we only get the L2 bounds of the error u-etΔu0. It is obviously important to explore the explicit error estimates as time tends to infinity. In order to come over the main difficulty raised by the nonlinear damping |u|p-2u, we will make full use of the Fourier analysis technique to explore the lower frequency effect of the nonlinear damping |u|p-2u. Fortunately, we can control the nonlinear term |u|p-2u as (8)(0tuLp-1p-1ds)1/(p-1)(0u10/310/3ds)3θ/10(0uppds)(1-θ)/pC with (9)1p-1=θ10/3+1-θp. This observation allows us to derive the explicit error estimates.

The remainder of this paper is organized as follows. In Section 2, we first recall some fundamental preliminaries and state our main results. In Section 3, we investigate the explicit error estimates of solutions between semilinear parabolic equation (1) and linear heat equation (2).

2. Preliminaries and Main Results

In this paper, we denote by C a generic positive constant which may vary from line to line.

Let S(R3) be the Schwartz class of rapidly decreasing functions (refer to ) and given gS(R3), its Fourier transformation Fg or g^ is defined by (10)Fg(ξ)=g^(ξ)=R3e-ix·ξg(x)dx.

L p ( R 3 ) with 1p denotes the usual Lebesgue space of all Lp integral functions associated with the norm (11)gLp={(R3|g(x)|pdx)1/p,1p<,esssupxR3|g(x)|,p=.

H s ( R 3 ) with sR denotes the fractional Sobolev space with (12)gHs=(R3|ξ|2s|g^|2dξ)1/2.

To state our main results, let us firstly recall the definition of the weak solutions of the semilinear parabolic equation (1) (refer to ).

Definition 1.

Given u0L2(R3), a measurable function u(x,t) on R3×(0,T) is called a weak solution to the semilinear parabolic equation (1) provided that

uL(0,T;L2(R3))L2(0,T;H1(R3)), and uLp(0,T;Lp(R3));

for any ϕC0(R3×[0,T))(13)0TR3(u·tϕ-u·ϕ-|u|p-2uϕ)dxdt=-R3u0ϕ(0)dx;

u(x,t) also satisfies energy inequality (14)12ddtR3|u|2dx+R3|u|2dx+R3|u|pdx0.

Now our results read as follows.

Theorem 2.

Suppose u0L2(R3) and u(x,t) is a weak solution of the Cauchy problem of the semilinear parabolic equation (1); one has (15)R3|u(t)-u~(t)|2dx=O((1+t)-3/4)ast, where u~(t) is the weak solution of the linear heat equation; namely, (16)tu~-Δu~=0,u~(x,0)=u0, with the same initial date u0.

Remark 3.

The result above seems inspiring. Since according to the Lp-Lq estimates of the linear heat equation (16), we have only the L2 bounds of the solution of linear equation (5); that is, (17)u~(t)L2(Rn)Cu0L2(Rn),t>0. No asymptotic behavior of solution of linear equation (5) can be derived. Compared with the previous results on the time decay of the nonlinear partial differential equations models  where the initial data satisfies some additional conditions such as L1(R3), at the same time, for the nonlinear parabolic equation (1) with the same initial date u0, the nonlinear damping term |u|p-2u is obviously not helpful for the asymptotic behavior of the semilinear parabolic equation (1). Therefore, it seems impossible to derive the asymptotic behavior of the difference between the semilinear parabolic equation (1) and the linear heat equation (16). Fortunately, we find a new trick which is different to the Lp-Lq estimates to deal with the nonlinear term. This trick is mainly based on the Fourier analysis which allows us to explore successfully the lower frequency of the nonlinear damping term |u|p-2u.

3. Error Estimates

We are now in a position to investigate the explicit error estimate in this section. It should be mentioned that the global existence of the nonlinear parabolic equation can be proved by the standard contraction mapping principle (refer to ). Hence we only prove the error estimates. To carry out this issue, we develop some new tricks which mainly borrowed the idea in . Denote the difference w(t)=u(t)-u~(t), where u(t) and u~(t) are the solutions of the semilinear parabolic equation (1) and the linear heat equation (16), respectively. Thus w(t) satisfies the following system: (18)wt-Δw+|u|pu=0,w(x,0)=0, in the weak sense. It is worth noting that the following derivation should be stated rigorously for the smooth approximated solutions and then take the limits to get the results of the weak solution of the semilinear parabolic equation (18). For convenience, we directly discuss weak solutions.

Multiplying both sides of (18) with w and integrating in R3, it follows that (19)ddtR3|w(t)|2dx+2R3|w|2dx=-2R3|u|p-2uwdx, since (20)-2R3|u|p-2uwdx=-2R3|u|p-2u(u-u~)dx=-2R3|u|pdx+2R3|u|p-2uu~dx2R3|u|p-2uu~dx,2R3|u|p-2uu~dxC(R3|u|p-1dx)u~LC(R3|u|p-1dx)(1+t)-3/4, where we have used the Hölder inequality and the Lp-Lq estimates (5). Thus inserting the above inequalities into (19), one shows that (21)ddtR3|w(t)|2dx+2R3|w|2dxC(R3|u|p-1dx)(1+t)-3/4. Taking the Parseval inequality into consideration, it follows that (22)ddtR3|w^(ξ,t)|2dξ+2R3|ξ|2|w^(ξ,t)|2dξC(R3|u|p-1dx)(1+t)-3/4.

Now multiplying both sides of (22) by (1+t)3 together with direct computation, then we have (23)ddt((1+t)3R3|w^(ξ,t)|2dξ)+2(1+t)3R3|ξ|2|w^(ξ,t)|2dξC(1+t)2R3|w^(ξ,t)|2dξ+C(R3|u|p-1dx)(1+t)9/4. Let (24)S(t)={ξR3:|ξ|(31+t)1/2}; then (25)(1+t)3R3|ξ|2|w^(ξ,t)|2dξ=(1+t)3S(t)c|ξ|2|w^(ξ,t)|2dξ+(1+t)3S(t)|ξ|2|w^(ξ,t)|2dξ(1+t)3S(t)c|ξ|2|w^(ξ,t)|2dξ3(1+t)2R3|w^(ξ,t)|2dξ-3(1+t)2S(t)|w^(ξ,t)|2dξ.

Therefore (26)ddt((1+t)3R3|w^(ξ,t)|2dξ)(1+t)2S(t)|w^(ξ,t)|2dξ+C(R3|u|p-1dx)(1+t)9/4; then integrating in time, one shows that (27)(1+t)3R3|w^(ξ,t)|2dξC0t(1+s)2S(s)|w^(ξ,s)|2dξds+C0t(1+s)9/4(R3|u|p-1dx)dsC0t(1+s)2S(s)|w^(ξ,s)|2dξds+C(1+t)9/40t(R3|u(τ)|p-1dx)dτ. In order to estimate the first term on the right hand side of (27), we take Fourier transformation to (18) (28)w^t+|ξ|2w^=-|u|p-2u^,w^(0)=0; the solution of the above ordinary differential equation is written as (29)|w^(ξ,t)|=|0te-|ξ|2t|u|p-2u^ds|C0t||u|p-2u^|dsC0t(R3|u|p-1dx)ds.

On one hand, since u is a weak solution of the semilinear parabolic equation (1) and according to the definition of weak solution and interpolation inequality, then for (30)2r+3q=32,2q6; we have (31){0(R3|u|qdx)r/qdt}1/rCesssup0<t<(R3|u|2dx)+C{0R3|u|2dxdt}1/2C; In particular, let (32)r=q=103; that is, (33){0R3|u|10/3dxdt}3/10C.

On the other hand, since p>5, that is, (34)103<p-1<p; hence applying the interpolation inequality, (35)(0tuLp-1p-1ds)1/(p-1)(0uL10/310/3ds)3θ/10(0uLppds)(1-θ)/pC with (36)1p-1=θ10/3+1-θp.

Plugging the above estimates into (29), one shows that (37)|w^(ξ,t)|C.

Hence (38)(1+t)3R3|w^(ξ,t)|2dξC0t(1+s)2S(s)1dξds+C(1+t)9/4C0t(1+s)2(1+s)-3/2ds+C(1+t)9/4C(1+t)3/2ds+C(1+t)9/4C(1+t)9/4.

That is, (39)R3|w(t)|2dx=O((1+t)-3/4),t, which completes the proof of Theorem 2.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is partially supported by the NNSF of China with no. 11271148.

Evans L. C. Partial Differential Equations 1998 Providence, RI, USA American Mathematical Society Gmira A. Veron L. Large time behaviour of the solutions of a semilinear parabolic equation in RN Journal of Differential Equations 1984 53 2 258 276 10.1016/0022-0396(84)90042-1 MR748242 2-s2.0-0000081143 Zhang L. H. Sharp rate of decay of solutions to 2-dimensional Navier-Stokes equations Communications in Partial Differential Equations 1995 20 1-2 119 127 10.1080/03605309508821089 MR1312702 ZBL0823.35145 Zhao C. Liang Y. Zhao M. Upper and lower bounds of time decay rate of solutions to a class of incompressible third grade fluid equations Nonlinear Analysis. Real World Applications 2014 15 229 238 10.1016/j.nonrwa.2013.08.001 MR3110567 Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations 1983 Springer 10.1007/978-1-4612-5561-1 MR710486 Dong B. Li Y. Large time behavior to the system of incompressible non-Newtonian fluids in R3 Journal of Mathematical Analysis and Applications 2004 298 2 667 676 10.1016/j.jmaa.2004.05.032 MR2086982 2-s2.0-10144243979 Dong B. Jiang W. On the decay of higher order derivatives of solutions to Ladyzhenskaya model for incompressible viscous flows Science in China A: Mathematics 2008 51 5 925 934 10.1007/s11425-007-0196-z MR2395395 2-s2.0-42449122722 Schonbek M. E. L 2 decay for weak solutions of the Navier-Stokes equations Archive for Rational Mechanics and Analysis 1985 88 3 209 222 10.1007/BF00752111 MR775190 2-s2.0-0021859089 Wheeler M. F. A priori L2 error estimates for Galerkin approximations to parabolic partial differential equations SIAM Journal on Numerical Analysis 1973 10 723 759 10.1137/0710062 MR0351124 Gilbarg D. Trudinger N. Elliptic Partial Differential Equations of Second Order 2001 3rd New York, NY, USA Springer MR1814364 Henry D. Geometric Theory of Semilinear Parabolic Equations 1981 New York, NY, USA Springe Dai M. Qing J. Schonbek M. Asymptotic behavior of solutions to liquid crystal systems in R3 Communications in Partial Differential Equations 2012 37 12 2138 2164 10.1080/03605302.2012.729172 MR3005539 2-s2.0-84868700939 Jin C. Yin J. Wang C. Large time behaviour of solutions for the heat equation with spatio-temporal delay Nonlinearity 2008 21 823 840 Dong B. Chen Z. Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows Discrete and Continuous Dynamical Systems A 2009 23 3 765 784 10.3934/dcds.2009.23.765 MR2461826 2-s2.0-63049087664 Schonbek M. E. Decay of solutions to parabolic conservation laws Communications in Partial Differential Equations 1980 5 5 449 473 10.1080/0360530800882145 MR571048 Dong B. Jia Y. Chen Z. Pressure regularity criteria of the three-dimensional micropolar fluid flows Mathematical Methods in the Applied Sciences 2011 34 5 595 606 10.1002/mma.1383 MR2814514 ZBL1219.35189 2-s2.0-78049475681 Dong B. Chen Z. Asymptotic stability of the critical and super-critical dissipative quasi-geostrophic equation Nonlinearity 2006 19 12 2919 2928 10.1088/0951-7715/19/12/011 MR2273766 ZBL1109.76063 2-s2.0-33846117131 Ricci R. Large time behavior of the solution of the heat equation with nonlinear strong absorption Journal of Differential Equations 1989 79 1 1 13 10.1016/0022-0396(89)90110-1 MR997606 2-s2.0-38249021900 Secchi P. L 2 stability for weak solutions of the Navier-Stokes equations in R3 Indiana University Mathematics Journal 1987 36 3 685 691 10.1512/iumj.1987.36.36039 MR905619 Dong B. Song J. Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation Discrete and Continuous Dynamical Systems 2012 32 1 57 79 10.3934/dcds.2012.32.57 MR2837054 2-s2.0-84859524039