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Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains. (English) Zbl 1082.35036
In this well-written paper, the existence of a global attractor for nonlinear reaction-diffusion equations in ${\Bbb R}^n$ $(n\geq 3)$ of the form $$u_t=\Delta u-\lambda u-f(u)+f_{x_i}^i+g(x) \quad\text{in }{\Bbb R}^+\times{\Bbb R}^n \tag*$$ with initial data $$u(0,x)=u_0(x) \quad\text{in }{\Bbb R}^n \tag**$$ is shown. Here, the nonlinearity $f$ is allowed to have polynomial growth of arbitrary order $p-1$ $(p\geq 2)$ and in the inhomogeneous term, $f_{x_i}^i$, $i=1,\ldots,n$, are distributional derivatives of $f\in L^2({\Bbb R}^n)$, $g\in L^2({\Bbb R}^n)$. In order to obtain this for the problem $(\ast)$, $(\ast\ast)$ two difficulties appear: (1) The regularity of its solutions is not sufficiently high to apply appropriate embedding theorems. (2) It is hard to get continuity of the associated semigroup in the $L^p({\Bbb R}^n)$-topology without restriction on $p$. Thus, for abstract semigroups in $L^2({\Bbb R}^n)$ the authors derive a sufficient criterion that a global attractor in $L^2({\Bbb R}^n)$ also attracts bounded sets of $L^2({\Bbb R}^n)$ w.r.t. the $L^p({\Bbb R}^n)$-norm. Using a new method based on a priori estimates, this criterion applies to show that the semigroup in $L^2({\Bbb R}^2)$ associated with $(\ast)$, $(\ast\ast)$ possesses a $(L^2({\Bbb R}^n),L^p({\Bbb R}^n))$-global attractor $A$ in the sense that $A$ is nonempty, compact, invariant in $L^p({\Bbb R}^n)$ and attracts every bounded subset of $L^2({\Bbb R}^n)$ in the $L^p({\Bbb R}^n)$-norm.

##### MSC:
 35B41 Attractors (PDE) 35K57 Reaction-diffusion equations 35B45 A priori estimates for solutions of PDE 35K15 Second order parabolic equations, initial value problems
##### Keywords:
measures of noncompactness; polynomial growth
Full Text:
##### References:
 [1] Arrieta, J. M.; Cholewa, J. W.; Dlotko, T.; Rodriguez-Bernal, A.: Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains. Nonlinear analysis 56, 515-554 (2004) · Zbl 1058.35102 [2] Babin, A. V.; Vishik, M. I.: Attractors of evolution equations. (1992) · Zbl 0778.58002 [3] Ball, J. M.: Global attractors for damped semilinear wave equations. Discrete contin. Dynam. systems 10, No. 1&2, 31-52 (2004) · Zbl 1056.37084 [4] J.W. Cholewa, T. Dlotko, Bi-spaces globle attractors in abstract parabolic equations, Evolution equations Banach Center Publications, vol. 60, 2003, pp. 13 -- 26. [5] Deimling, K.: Nonlinear functional analysis. (1985) · Zbl 0559.47040 [6] Hale, J. K.: Asymptatic behavior of dissippative systems. (1988) [7] Ladyzenskaya, O. A.: Attractors for semigroups and evolution equations. (1991) [8] Marion, M.: Attractors for reactions -- diffusion equations: existence and estimate of their dimension. Appl. anal. 25, 101-147 (1987) · Zbl 0609.35009 [9] Marion, M.: Approximate inertial manifolds for reaction -- diffusion equations in high space dimension. J. dynan. Differential equations 1, 245-267 (1989) · Zbl 0702.35127 [10] Ma, Q. F.; Wang, S. H.; Zhong, C. K.: Necessary and sufficient conditions for the existence of global attractors for semigroups and applications. Indiana university math. J. 51, No. 6 (2002) · Zbl 1028.37047 [11] Prizzi, M.: A remark on reaction -- diffusion equations in unbounded domains. Discrete contin. Dynam. systems 9, No. 2, 281-286 (2003) · Zbl 1029.35044 [12] Rodrigue-Bernal, A.; Wang, B.: Attractors for partly dissipative reaction diffusion systems in rn. J. math. Anal. appl. 252, 790-803 (2000) · Zbl 0977.35028 [13] Robinson, J. C.: Infinite-dimensional dynamical systems an introduction to dissipative parabolic pdes and the theory of global attractors. (2001) · Zbl 0980.35001 [14] Rosa, R.: The global attractor for the 2D Navier -- Stokes flow on some unbounded domains. Nonlinear anal. 32, 71-85 (1998) · Zbl 0901.35070 [15] Sell, G. R.; You, Y.: Dynamics of evolutionary equations. (2002) · Zbl 1254.37002 [16] C.Y. Sun, S.Y. Wang, C.K. Zhong, Global attractors for a nonclassical diffusion equation, Acta. Math. Sin., in press. · Zbl 1128.35027 [17] Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics. (1997) · Zbl 0871.35001 [18] Wang, B.: Attractors for reaction -- diffusion equations in unbounded domains. Physica D 128, 41-52 (1999) · Zbl 0953.35022 [19] Zhong, C. K.; Sun, C. Y.; Niu, M. F.: On the existence of global attractor for a class of infinite dimensional nonlinear dissipative dynamical systems. Chin. ann. Math. 26B:3, No. 1 -- 8 (2005) · Zbl 1079.35026