Quittner, P.

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EQUADIFF 7Pavol QuittnerStability of stationary solutions of parabolic variational inequalitiesIn: Jaroslav Kurzweil (ed.): Equadiff 7, Proceedings of the 7th Czechoslovak Conferenceon Differential Equations and Their Applications held in Prague, 1989. BSB B.G.Teubner Verlagsgesellschaft, Leipzig, 1990. Teubner-Texte zur Mathematik, Bd. 118.pp. 191--194.Persistent URL: http://dml.cz/dmlcz/702372Terms of use:© BSB B.G. Teubner Verlagsgesellschaft, 1990Institute of Mathematics of the Academy of Sciences of the Czech Republic providesaccess to digitized documents strictly for personal use. Each copy of any part of thisdocument must contain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The CzechDigital Mathematics Library http://project.dml.czSTABILITY OF STATIONARY SOLUTIONS OF PARABOLIC VARIATIONAL INEQUALITIES QUITTNER P., BRATISLAVA, Czechoslovakia The stability of solutions of parabolic variational inequalities was studied by many authors (see e.g. [1,4,5,6,7,15,16]). In the present paper we study the Lyapunov stability of stationary solutions of semilinear parabolic variational inequalities in Hilbert spaces. In comparison to (some of) the papers cited above our assumptions are rather special, nevertheless we are able to prove exponential stability of a given stationary solution and also the instability results seem to be new. In our abstract setting we use the existence and regularity results of Brezis [2,3]; similar results should be possible to obtain also under another assumptions guaranteeing the necessary existence and regularity properties of the solution of the parabolic inequality. Throughout this paper we assume that V or H is a real Hilbert space with the scalar product (•, •) or (•, •) and the norm || • || or | • |, respectively. We assume also that V C H, where the imbedding is dense and compact, and we denote by (•,•) also the duality between V and its dual space V1 induced by the imbedding V C H = H' C V1. Finally, let K be a closed convex set in V and F : V —• H a (locally) Lipschitz continuous map. We define the operator A : V —• V' by (Au} v) := («, v) for any u,v £V and denote by A the H-realization of A, i.e. the domain of A is V(A) := {u G V \Au £ H} and Au := Au for u £ V(A). Then A is a positive self-adjoint operator in H with a compact resolvent and the operator F := A"lF : V —• V is compact. By Xa we denote the domains of the fractional powers A° (particularly, Xij2 = V). We shall suppose that u0 € K is a stationary solution of the inequality at (1) t-(0) = u0 ( « 0 - F K ) , i i - t i 0 ) > o VveK The inequality (1) can be written also in the form u(t)£K ( ^ + > U * - F ( t i ) . v - u ) > 0 Vt/GK Analogously as for equations one can try to "linearize" this inequality and to study the stability 191 of the stationary solution 0 for the "linearized" inequality du u(t)£K0: ( - - + . A u - L u + F o , t / - u ) > 0 Vt/£K0 , (1)L at where K0 is the closure of the set \Ja>0oc(K — u0), L : V —• H is a continuous linear operator •uch that ||F(u) - F(u0) - L(u - u0)\\v> = o(||u - u0||) for u -> u0, F0 := ,4u0 - F(u0). This approach does not yield always satisfactory results , however the following Theorems 1, IB and 2B are true. Theorem 1 ([12]). Let A- := inf{(-4u - Lu, u) | u G K0i \u\ = 1,(F0iu) = 0} > 0. Then u0 is an exponentially stable solution of (1) in V. If F0 = 0 and L is symmetric, then the number Ax defined in Theorem 1 is the least eigenvalue of the inequality «GKo (Au-Lu}v-u)> A(u, t / -u) Vt/€ Ko, (2) i.e. there exists a nontrivial solution Ui of (2). The function u(t) := e~*xiui is a solution of the inequality (1)^, hence Ai < 0 implies an instability result for (l)z,. However, if L is not symmetric then the condition Ai > 0 is, in general, not necessary for the (exponential) stability of u0- A possible generalization of Theorem 1 for nonsymmetric L is the following Theorem IB ([13]). Let B : H —• H be a strictly positive, self-adjoint, continuous linear operator such that B(V) C V, B2(K - u0) C K - u0, (id - B2)(d(K - u0)) C K - u0, (utB2u) > a||u[|2 — C|u|2, (F0iB2u) > c(Fotu) for some positive constants a,c, C and any u £ K. Then the condition Af := inf{(.4u - Lu,H2u)|u G K0,|-9u| = l ,(F0 ,u) = 0} > 0 is sufficient for the exponential stability of the stationary solution u0 of (1) in V. Similarly, the instability result [12, Theorem 2] can be generalized to the following Theorem 2B. Let, in addition to our general assumptions, K be a cone with its vertex at uot F0 = 0 , | F ( u ) - F ( u 0 ) - L ( u - u 0 ) | < C | | u - u 0 | |2 / o r u in a neighbourhood of u0 and let B fulfill the assumptions of Theorem IB. Let u- € K0, \ui\ = 1, be an eigenvector of the inequality (I) with an eigenvalue X1 < 0 and let (Ay — LyyB2y) > \x\By\2 for any y 6 K0' — «i. Then the stationary solution u0 of (1) is unstable in both of the topologies of V and H. The assumptions of Theorem 2B are rather restrictive, however in general one can not expect that the instability result for (1)L would imply an instability result for (1) (see [12, Example 1]). Thus we are led to the study of the nonlinear stationary inequality u G K { u - F ( u ) , t / - u ) > 0 Vt /€K (3) 191 which is obviously equivalent to the equation u - PKF(u) = 0, where PK is the projection in V onto K. We need some additional regularity hypothesis on F and K\ we suppose that there exists a neighbourhood U of u0 in V, a > 0 and a bounded (nonlinear) operator G :U -+ X(1+f t)/2 such that u + AAa(u + G(u)) eKQU implies u £ K for small A > 0 (cf. [14]) and F : U C J-> -~,K is Lipschitz continuous for some j3 < 1/2. By d(u0) we denote the Leray-Schauder degree deg(id - P/cF, 0,u). Supposing u ^ PKF(u) for any u G W \ {u0} we have Theorem 3 ([14]). Let d(u0) -̂  1. Then u0 is not asymptotically stable (neither in the topology of V nor in the topology of H). If, moreover, F has a potential, then u0 is not stable. The assumption d(u0) ^ 1 is, of course, not necessary for the instability of u0. However, if F has a potential T and the functional §(u) := | | |« | | 2 — F(u) attains its local minimum at u0 with respect to K, then d(u0) = 1 (if this degree exists) and u0 is an asymptotically stable solution of (1). The degree d(u0) was studied by the author in [8,9,10,11] and an application of Theorem 3 is given in [14]. Applications of Theorems 1 and IB can be found in [12,13]; here we apply Theorem 2B to the system of reaction-diffusion equations (4) where ft is a smoothly bounded domain in 1\N} d > 0 and ft : 1R2 —> fft are C1 functions such \§lL(mt\\ S f(l _i_L,m ~ * 2 .f M v. O T.pf „ __ (9ll , .2\ ^-dAu^MuW) Џ-Au2-f2(u\u2) in (0,T)x ft дu1 _ дu2 __ дn " õn on (0, T) x дӣ ti1(0,.) = tiił гг2(0,) = tz2 that §£•(«) < C(l + |ttp)f 7 < j?2^ if-V > 2. Let u0 = («J,u2) G V := WJ'2(ft) x W^2(ft) be a stationary, spatially homogenous solution of (4) and denote 6,, := 3J~(*-O). Assuming &n > 0, 621 > 0, 6n 4- 622 < 0, bub22 > bi2&2i one can easily check (cf. [6,10]) that the condition d > d° := maxAi>0 4~(&ii + A - 6 )» where A, is the t-th eigenvalue of - A with Neumann boundary conditions, is sufficient for the exponential stability of u0 and the condition d <d° is sufficient for the instability of u0. Now suppose that we have an unilateral constraint for the solution u of (4), which leads to an inequality on a convex cone K = (K_+u\) x (K2+«2) with its vertex at u0 and suppose K2 C K+ and -11 € K_, where Ar+ is the positive cone in Wlj2(ft) and 11 is the function l!'v.r) := 1. Then one can easily check that u_ := (-11,0) is the eigenvector of (2) with A! = -bu < 0 and that 13 Kurzweil, Equadiff 7 193 the assumption (Ay - Ly} B2y) > Ai|l.ty|2 is fulfilled for any y £ V if we choose B := ( Q J )., where b = —&21/&12 and I is the identity in L2(to). Hence the solution u0 of the corresponding inequality (1) is unstable for any d > 0. REFERENCES 1. Boccardo L. & Donati F., Existence and stability results for solutions of some strongly nonlinear constrained problems, Nonl. Analysis (TM&cA) 5, 975-988 (1981). 2. Brews H., Problemes unilat^raux, J. Math, pures et appl. 51, 1-168 (1972). 3. Brezis H., ProprtetSs re*gularisantes de certains semi-groupes non lin^aires, Isr. J. Math. 9, 513-534 (1971). 4. Conrad F. & Cortey-Dumont Ph., Nonlinear eigenvalue problems in elliptic variational inequalities: Some results for the maximal branch, Numer. Funct. Anal, and Optimiz. 9, 1059-1114 (1987). 5. Conrad F. & Issard-Roch F., Inequations variationnelles avec points de retournement. Echange de stability C. R. Acad. Sci., Paris, Ser. 1300, 481-484 (1985). 6. Dr&bek P. & Kucera M., Reaction-diffusion systems: Destabilizing effect of unilateral con-ditions, Nonl. Analysis (TM&A) 12, 1173-1192 (1988). 7. Furuya H. & Miyashiba K. & Kenmochi N., Asymptotic behavior of solutions to a class of nonlinear evolution equations, J. Differ. Equations 62, 73-94 (1986). 8. Quittner P., Spectral analysis of variational inequalities, Comment. Math. Univ. Carolin. 27, 605-629 (1986). 9. Quittner P., Solvability and multiplicity results for variational inequalities, To appear in Comment. Math. Univ. Carolin. 30 (1989). 10. Quittner P., Bifurcation points and eigenvalues of inequalities of reaction-diffusion type, J. Reine Angew. Math. 380, 1-13 (1987). 11. Quittner P., A note on the degree theory for variational inequalities involving gradient operators, Preprint. 12. Quittner P., On the principle of linearized stability for variational inequalities, Math. Ann. 283, 257-270 (1989). 13. Quittner P., A remark on the stability of stationary solutions of parabolic variational in-equalities, Submitted to Czechoslovak Math. J. 14. Quittner P., An instability criterion for variational inequalities, Submitted to Nonl. Analysis (TM&A). 15. Rodrigues J. P., On a class of parabolic unilateral problems, Nonl. Analysis (TM&A) 10, 1357-1366 (1986). 16. Santolini A. T., Comportamento asintotico per la soluzione di una disequazione parabolica quasi lineare, Riv. Mat. Univ. Parma, IV. Ser. 7, 251-265 (1981). 194 

Stability of stationary solutions of parabolic variational inequalities

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Stability of stationary solutions of parabolic variational inequalities

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Institute of Mathematics AS CR, v. v. i.