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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 \({\mathbb R}^n\) \((n\geq 3)\) of the form \[ u_t=\Delta u-\lambda u-f(u)+f_{x_i}^i+g(x) \quad\text{in }{\mathbb R}^+\times{\mathbb R}^n \tag{*} \] with initial data \[ u(0,x)=u_0(x) \quad\text{in }{\mathbb R}^n \tag{**} \] 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 \(f\in L^2({\mathbb R}^n)\), \(g\in L^2({\mathbb R}^n)\).
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 \(L^p({\mathbb R}^n)\)-topology without restriction on \(p\).
Thus, for abstract semigroups in \(L^2({\mathbb R}^n)\) the authors derive a sufficient criterion that a global attractor in \(L^2({\mathbb R}^n)\) also attracts bounded sets of \(L^2({\mathbb R}^n)\) w.r.t. the \(L^p({\mathbb R}^n)\)-norm.
Using a new method based on a priori estimates, this criterion applies to show that the semigroup in \(L^2({\mathbb R}^2)\) associated with \((\ast)\), \((\ast\ast)\) possesses a \((L^2({\mathbb R}^n),L^p({\mathbb R}^n))\)-global attractor \(A\) in the sense that \(A\) is nonempty, compact, invariant in \(L^p({\mathbb R}^n)\) and attracts every bounded subset of \(L^2({\mathbb R}^n)\) in the \(L^p({\mathbb R}^n)\)-norm.

MSC:

35B41 Attractors
35K57 Reaction-diffusion equations
35B45 A priori estimates in context of PDEs
35K15 Initial value problems for second-order parabolic equations
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[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), North-Holland: North-Holland Amsterdam · Zbl 0778.58002
[3] Ball, J. M., Global attractors for damped semilinear wave equations, Discrete Contin. Dynam. Systems, 10, 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.; J.W. Cholewa, T. Dlotko, Bi-spaces globle attractors in abstract parabolic equations, Evolution equations Banach Center Publications, vol. 60, 2003, pp. 13-26. · Zbl 1024.35058
[5] Deimling, K., Nonlinear Functional Analysis (1985), Springer: Springer Berlin · Zbl 0559.47040
[6] Hale, J. K., Asymptatic Behavior of Dissippative Systems (1988), AMS: AMS Providence, RJ · Zbl 0642.58013
[7] Ladyzenskaya, O. A., Attractors for Semigroups and Evolution Equations (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0755.47049
[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, 6 (2002) · Zbl 1028.37047
[11] Prizzi, M., A remark on reaction-diffusion equations in unbounded domains, Discrete Contin. Dynam. Systems, 9, 2, 281-286 (2003) · Zbl 1029.35044
[12] Rodrigue-Bernal, A.; Wang, B., Attractors for partly dissipative reaction diffusion systems in \(R^n\), 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), Cambridge University Press: Cambridge University Press Cambridge · 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), Springer: Springer New York · 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.; 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), Springer: Springer New York · 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, 1-8 (2005)
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