# American Institute of Mathematical Sciences

November  2012, 11(6): 2351-2369. doi: 10.3934/cpaa.2012.11.2351

## Flow invariance for nonautonomous nonlinear partial differential delay equations

 1 Department of Mathematics, Razi University, Kermanshah, Iran 2 Fakultät für Mathematik, Universität Duisburg-Essen, D-45117 Essen, Germany

Received  March 2011 Revised  May 2011 Published  April 2012

Several fundamental results on existence and flow-invariance of solutions to the nonlinear nonautonomous partial differential delay equation $\dot{u}(t) + B(t)u(t) \ni F(t; u_t), 0 \leq s \leq t, u_s = \varphi,$ with $B(t)\subset X\times X$ $\omega-$accretive, are developed for a general Banach space $X.$ In contrast to existing results, with the history-response $F(t;\cdot)$ globally defined and, at least, Lipschitz on bounded sets, the results are tailored for situations with $F(t;\cdot)$ defined on -- possibly -- thin subsets of the initial-history space $E$ only, and are applied to place several classes of population models in their natural $L^1-$setting. The main result solves the open problem of a subtangential condition for flow-invariance of solutions in the fully nonlinear case, paralleling those known for the cases of (a) no delay, (b) ordinary delay equations with $B(\cdot)\equiv 0,$ and (c) the semilinear case.
Citation: Seyedeh Marzieh Ghavidel, Wolfgang M. Ruess. Flow invariance for nonautonomous nonlinear partial differential delay equations. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2351-2369. doi: 10.3934/cpaa.2012.11.2351
##### References:
 [1] H. Amann, Invariant sets and existence theorems for semilinear parabolic and elliptic systems, J. Math. Anal. Appl., 65 (1978), 432-467. doi: 10.1016/0022-247X(78)90192-0. [2] P. Bénilan and M. G. Crandall, Completely accretive operators, In "Semigroup Theory and Evolution Equations" (P. Clément, E. Mitidieri and B. de Pagter eds.), Lecture Notes Pure Appl. Math., 135, Marcel-Dekker, 1991, 41-75. [3] P. Bénilan, M. G. Crandall and A. Pazy, Evolution equations governed by accretive operators, Monograph, in preparation. [4] D. Bothe, Nonlinear evolutions with Carathéodory forcing, J. Evol. Equ., 3 (2003), 375-394. doi: 10.1007/s00028-003-0099-5. [5] D. Bothe, Flow invariance for nonlinear accretive evolutions under range conditions, J. Evol. Equ., 5 (2005), 227-252. doi: 10.1007/s00028-005-0185-z. [6] D. W. Brewer, A nonlinear semigroup for a functional differential equation, Trans. Amer. Math. Soc., 236 (1978), 173-191. doi: 10.1090/S0002-9947-1978-0466838-2. [7] D. W. Brewer, Locally Lipschitz continuous functional differential equations and nonlinear semigroups, Illinois J. Math., 26 (1982), 374-381. [8] M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math., 11 (1972), 57-94. doi: 10.1007/BF02761448. [9] J. Dyson and R. Villella-Bressan, Functional differential equations and non-linear evolution operators, Proc. Royal Soc., Edinburgh, 75A (1975/76), 223-234. [10] J. Dyson and R. Villella-Bressan, Semigroups of translation associated with functional and functional differential equations, Proc. Royal Soc. Edinburgh, 82A (1979), 171-188. doi: 10.1017/S030821050001115X. [11] J. Dyson and R. Villella-Bressan, Nonautonomous locally Lipschitz continuous functional differential equations in spaces of continuous functions, Nonlinear Diff. Eqns. Appl., 3 (1996), 127-147. doi: 10.1007/BF01194220. [12] L. Evans, Nonlinear evolution equations in an arbitrary Banach space, Israel J. Math., 26 (1977), 1-42. doi: 10.1007/BF03007654. [13] S. M. Ghavidel, Flow invariance for solutions to nonlinear nonautonomous partial differential delay equations, J. Math. Anal. Appl., 345 (2008), 854-870. doi: 10.1016/j.jmaa.2008.04.041. [14] S. M. Ghavidel, Flow invariance for solutions to nonlinear nonautonomous evolution equations, in preparation. [15] J. K. Hale, Functional differential equations with infinite delays, J. Math. Anal. Appl., 48 (1974), 276-283. doi: 10.1016/0022-247X(74)90233-9. [16] J. K. Hale, Large diffusivity and asymptotic behavior in parabolic systems, J. Math. Anal. Appl., 118 (1986), 455-466. doi: 10.1016/0022-247X(86)90273-8. [17] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. [18] A. G. Kartsatos and M. E. Parrott, Global solutions of functional evolution equations involving locally defined Lipschitzian perturbations, J. London Math. Soc., 27 (1983), 306-316. doi: 10.1112/jlms/s2-27.2.306. [19] A. G. Kartsatos and M. E. Parrott, Convergence of the Kato approximants for evolution equations involving functional perturbations, J. Diff. Eqns., 47 (1983), 358-377. doi: 10.1016/0022-0396(83)90041-4. [20] A. G. Kartsatos and M. E. Parrott, The weak solution of a functional differential equation in a general Banach space, J. Diff. Eqns., 75 (1988), 290-302. doi: 10.1016/0022-0396(88)90140-4. [21] V. Lakshmikhantam, S. Leela and V. Moauro, Existence and uniqueness of solutions of delay differential equations on a closed subset of a Banach space, Nonlinear Analysis TMA, 2 (1978), 311-327. doi: 10.1016/0362-546X(78)90020-2. [22] S. Leela and V. Moauro, Existence of solutions in a closed set for delay differential equations in Banach space, Nonlinear Analysis TMA, 2 (1978), 47-58. doi: 10.1016/0362-546X(78)90040-8. [23] J. H. Lightbourne III, Function space flow-invariance for functional differential equations of retarded type, Proc. Amer. Math. Soc., 77 (1979), 91-98. doi: 10.1090/S0002-9939-1979-0539637-7. [24] R. H. Martin, "Nonlinear Operators and Differential Equations in Banach Spaces," Wiley, New York, 1976. [25] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans, Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590. [26] R. H. Martin and H. L. Smith, Convergence in Lotka-Volterra systems with diffusion and delay, in "Differential Equations with Applications in Biology, Physics and Engineering" (J.A. Goldstein, F. Kappel, and W. Schappacher, eds.), Lecture Notes Pure Appl. Math., 133, Marcel-Dekker, 1991, 259-267. [27] R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence, J. reine angew. Math., 413 (1991), 1-35. [28] I. Miyadera, "Nonlinear Semigroups," Transl. of Math. Monographs 109, Amer. Math. Soc., Providence, RI, 1992. [29] S. Murakami, Stable equilibrium point of some diffusive functional differential equations, Nonlinear Analysis TMA, 25 (1995), 1037-1043. doi: 10.1016/0362-546X(95)00097-F. [30] M. E. Parrott, Representation and approximation of generalized solutions of a nonlinear functional differential equation, Nonlinear Analysis TMA, 6 (1982), 307-318. doi: 10.1016/0362-546X(82)90018-9. [31] N. H. Pavel, "Differential Equations, Flow Invariance and Applications," Research Notes Math. 113, Pitman, Boston, London, Melbourne, 1984. [32] N. Pavel, "Nonlinear Evolution Operators and Semigroups," Lecture Notes Math. 1260, Springer, Berlin, 1987. [33] N. Pavel and F. Iacob, Invariant sets for a class of perturbed differential equations of retarded type, Israel J. Math., 28 (1977), 254-264. doi: 10.1007/BF02759812. [34] M. Pierre, Invariant closed subsets for nonlinear semigroups, Nonlinear Analysis TMA, 2 (1978), 107-117. doi: 10.1016/0362-546X(78)90046-9. [35] A. T. Plant, Nonlinear semigroups of translations in Banach space generated by functional differential equations, J. Math. Anal. Appl., 60 (1977), 67-74. doi: 10.1016/0022-247X(77)90048-8. [36] J. Prüss, On semilinear parabolic equations on closed sets, J. Math. Anal. Appl., 77 (1980), 513-538. doi: 10.1016/0022-247X(80)90245-0. [37] W. M. Ruess, The evolution operator approach to functional differential equations with delay, Proc. Amer. Math. Soc., 119 (1993), 783-791. [38] W. M. Ruess, Existence of solutions to partial functional differential equations with delay, in "Theory and Applications of Nonlinear Operators of Accretive and Monotone Type" (A.G. Kartsatos ed.), Lecture Notes Pure Appl. Math. 178, Marcel Dekker, (1996), 259-288. [39] W. M. Ruess, Existence of solutions to partial functional evolution equations with delay, in "Functional Analysis" (S. Dierolf, S. Dineen and P. Domanski eds.), Walter de Gruyter, (1996), 377-387. [40] W. M. Ruess, Existence and stability of solutions to partial functional differential equations with delay, Adv. Differential Equations, 4 (1999), 843-876. [41] W. M. Ruess, Flow invariance for nonlinear partial differential delay equations, Trans. Amer. Math. Soc., 361 (2009), 4367-4403. doi: 10.1090/S0002-9947-09-04833-8. [42] W. M. Ruess and W. H. Summers, Operator semigroups for functional differential equations with delay, Trans. Amer. Math. Soc., 341 (1994), 695-719. doi: 10.2307/2154579. [43] W. M. Ruess and W. H. Summers, Linearized stability for abstract differential equations with delay, J. Math. Anal. Appl., 198 (1996), 310-336. doi: 10.1006/jmaa.1996.0085. [44] A. Schiaffino, On a diffusion Volterra equation, Nonlinear Analysis TMA, 3 (1979), 595-600. doi: 10.1016/0362-546X(79)90088-9. [45] G. Seifert, Positively invariant closed sets for systems of delay differential equations, J. Differential Equations, 22 (1976), 292-304. doi: 10.1016/0022-0396(76)90029-2. [46] C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418. doi: 10.1090/S0002-9947-1974-0382808-3. [47] C. C. Travis and G. F. Webb, Partial differential equations with deviating arguments in the time variable, J. Math. Anal. Appl., 56 (1976), 397-409. doi: 10.1016/0022-247X(76)90052-4. [48] G. F. Webb, Autonomous nonlinear functional differential equations and nonlinear semigroups, J. Math. Anal. Appl., 46 (1974), 1-12. doi: 10.1016/0022-247X(74)90277-7. [49] G. F. Webb, Asymptotic stability for abstract nonlinear functional differential equations, Proc. Amer. Math. Soc., 54 (1976), 225-230. doi: 10.1090/S0002-9939-1976-0402237-0. [50] P. Wittbold, "Absorptions nonlinéaires," Thèse Doctorat, Université de Besançon, 1994. [51] P. Wittbold, Nonlinear diffusion with absorption, in "Progress in Partial Differential Equations: the Metz Surveys 4" (M. Chipot and I. Shafrir eds.), Pitman Res. Notes Math. Series 345, Longman, Harlow, (1996), 142-157 [52] P. Wittbold, Nonlinear diffusion with absorption, Potential Anal., 7 (1997), 437-465. doi: 10.1023/A:1017998221347. [53] K. Yoshida, The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology, Hiroshima Math. J., 12 (1982), 321-348.

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##### References:
 [1] H. Amann, Invariant sets and existence theorems for semilinear parabolic and elliptic systems, J. Math. Anal. Appl., 65 (1978), 432-467. doi: 10.1016/0022-247X(78)90192-0. [2] P. Bénilan and M. G. Crandall, Completely accretive operators, In "Semigroup Theory and Evolution Equations" (P. Clément, E. Mitidieri and B. de Pagter eds.), Lecture Notes Pure Appl. Math., 135, Marcel-Dekker, 1991, 41-75. [3] P. Bénilan, M. G. Crandall and A. Pazy, Evolution equations governed by accretive operators, Monograph, in preparation. [4] D. Bothe, Nonlinear evolutions with Carathéodory forcing, J. Evol. Equ., 3 (2003), 375-394. doi: 10.1007/s00028-003-0099-5. [5] D. Bothe, Flow invariance for nonlinear accretive evolutions under range conditions, J. Evol. Equ., 5 (2005), 227-252. doi: 10.1007/s00028-005-0185-z. [6] D. W. Brewer, A nonlinear semigroup for a functional differential equation, Trans. Amer. Math. Soc., 236 (1978), 173-191. doi: 10.1090/S0002-9947-1978-0466838-2. [7] D. W. Brewer, Locally Lipschitz continuous functional differential equations and nonlinear semigroups, Illinois J. Math., 26 (1982), 374-381. [8] M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math., 11 (1972), 57-94. doi: 10.1007/BF02761448. [9] J. Dyson and R. Villella-Bressan, Functional differential equations and non-linear evolution operators, Proc. Royal Soc., Edinburgh, 75A (1975/76), 223-234. [10] J. Dyson and R. Villella-Bressan, Semigroups of translation associated with functional and functional differential equations, Proc. Royal Soc. Edinburgh, 82A (1979), 171-188. doi: 10.1017/S030821050001115X. [11] J. Dyson and R. Villella-Bressan, Nonautonomous locally Lipschitz continuous functional differential equations in spaces of continuous functions, Nonlinear Diff. Eqns. Appl., 3 (1996), 127-147. doi: 10.1007/BF01194220. [12] L. Evans, Nonlinear evolution equations in an arbitrary Banach space, Israel J. Math., 26 (1977), 1-42. doi: 10.1007/BF03007654. [13] S. M. Ghavidel, Flow invariance for solutions to nonlinear nonautonomous partial differential delay equations, J. Math. Anal. Appl., 345 (2008), 854-870. doi: 10.1016/j.jmaa.2008.04.041. [14] S. M. Ghavidel, Flow invariance for solutions to nonlinear nonautonomous evolution equations, in preparation. [15] J. K. Hale, Functional differential equations with infinite delays, J. Math. Anal. Appl., 48 (1974), 276-283. doi: 10.1016/0022-247X(74)90233-9. [16] J. K. Hale, Large diffusivity and asymptotic behavior in parabolic systems, J. Math. Anal. Appl., 118 (1986), 455-466. doi: 10.1016/0022-247X(86)90273-8. [17] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. [18] A. G. Kartsatos and M. E. Parrott, Global solutions of functional evolution equations involving locally defined Lipschitzian perturbations, J. London Math. Soc., 27 (1983), 306-316. doi: 10.1112/jlms/s2-27.2.306. [19] A. G. Kartsatos and M. E. Parrott, Convergence of the Kato approximants for evolution equations involving functional perturbations, J. Diff. Eqns., 47 (1983), 358-377. doi: 10.1016/0022-0396(83)90041-4. [20] A. G. Kartsatos and M. E. Parrott, The weak solution of a functional differential equation in a general Banach space, J. Diff. Eqns., 75 (1988), 290-302. doi: 10.1016/0022-0396(88)90140-4. [21] V. Lakshmikhantam, S. Leela and V. Moauro, Existence and uniqueness of solutions of delay differential equations on a closed subset of a Banach space, Nonlinear Analysis TMA, 2 (1978), 311-327. doi: 10.1016/0362-546X(78)90020-2. [22] S. Leela and V. Moauro, Existence of solutions in a closed set for delay differential equations in Banach space, Nonlinear Analysis TMA, 2 (1978), 47-58. doi: 10.1016/0362-546X(78)90040-8. [23] J. H. Lightbourne III, Function space flow-invariance for functional differential equations of retarded type, Proc. Amer. Math. Soc., 77 (1979), 91-98. doi: 10.1090/S0002-9939-1979-0539637-7. [24] R. H. Martin, "Nonlinear Operators and Differential Equations in Banach Spaces," Wiley, New York, 1976. [25] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans, Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590. [26] R. H. Martin and H. L. Smith, Convergence in Lotka-Volterra systems with diffusion and delay, in "Differential Equations with Applications in Biology, Physics and Engineering" (J.A. Goldstein, F. Kappel, and W. Schappacher, eds.), Lecture Notes Pure Appl. Math., 133, Marcel-Dekker, 1991, 259-267. [27] R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence, J. reine angew. Math., 413 (1991), 1-35. [28] I. Miyadera, "Nonlinear Semigroups," Transl. of Math. Monographs 109, Amer. Math. Soc., Providence, RI, 1992. [29] S. Murakami, Stable equilibrium point of some diffusive functional differential equations, Nonlinear Analysis TMA, 25 (1995), 1037-1043. doi: 10.1016/0362-546X(95)00097-F. [30] M. E. Parrott, Representation and approximation of generalized solutions of a nonlinear functional differential equation, Nonlinear Analysis TMA, 6 (1982), 307-318. doi: 10.1016/0362-546X(82)90018-9. [31] N. H. Pavel, "Differential Equations, Flow Invariance and Applications," Research Notes Math. 113, Pitman, Boston, London, Melbourne, 1984. [32] N. Pavel, "Nonlinear Evolution Operators and Semigroups," Lecture Notes Math. 1260, Springer, Berlin, 1987. [33] N. Pavel and F. Iacob, Invariant sets for a class of perturbed differential equations of retarded type, Israel J. Math., 28 (1977), 254-264. doi: 10.1007/BF02759812. [34] M. Pierre, Invariant closed subsets for nonlinear semigroups, Nonlinear Analysis TMA, 2 (1978), 107-117. doi: 10.1016/0362-546X(78)90046-9. [35] A. T. Plant, Nonlinear semigroups of translations in Banach space generated by functional differential equations, J. Math. Anal. Appl., 60 (1977), 67-74. doi: 10.1016/0022-247X(77)90048-8. [36] J. Prüss, On semilinear parabolic equations on closed sets, J. Math. Anal. Appl., 77 (1980), 513-538. doi: 10.1016/0022-247X(80)90245-0. [37] W. M. Ruess, The evolution operator approach to functional differential equations with delay, Proc. Amer. Math. Soc., 119 (1993), 783-791. [38] W. M. Ruess, Existence of solutions to partial functional differential equations with delay, in "Theory and Applications of Nonlinear Operators of Accretive and Monotone Type" (A.G. Kartsatos ed.), Lecture Notes Pure Appl. Math. 178, Marcel Dekker, (1996), 259-288. [39] W. M. Ruess, Existence of solutions to partial functional evolution equations with delay, in "Functional Analysis" (S. Dierolf, S. Dineen and P. Domanski eds.), Walter de Gruyter, (1996), 377-387. [40] W. M. Ruess, Existence and stability of solutions to partial functional differential equations with delay, Adv. Differential Equations, 4 (1999), 843-876. [41] W. M. Ruess, Flow invariance for nonlinear partial differential delay equations, Trans. Amer. Math. Soc., 361 (2009), 4367-4403. doi: 10.1090/S0002-9947-09-04833-8. [42] W. M. Ruess and W. H. Summers, Operator semigroups for functional differential equations with delay, Trans. Amer. Math. Soc., 341 (1994), 695-719. doi: 10.2307/2154579. [43] W. M. Ruess and W. H. Summers, Linearized stability for abstract differential equations with delay, J. Math. Anal. Appl., 198 (1996), 310-336. doi: 10.1006/jmaa.1996.0085. [44] A. Schiaffino, On a diffusion Volterra equation, Nonlinear Analysis TMA, 3 (1979), 595-600. doi: 10.1016/0362-546X(79)90088-9. [45] G. Seifert, Positively invariant closed sets for systems of delay differential equations, J. Differential Equations, 22 (1976), 292-304. doi: 10.1016/0022-0396(76)90029-2. [46] C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418. doi: 10.1090/S0002-9947-1974-0382808-3. [47] C. C. Travis and G. F. Webb, Partial differential equations with deviating arguments in the time variable, J. Math. Anal. Appl., 56 (1976), 397-409. doi: 10.1016/0022-247X(76)90052-4. [48] G. F. Webb, Autonomous nonlinear functional differential equations and nonlinear semigroups, J. Math. Anal. Appl., 46 (1974), 1-12. doi: 10.1016/0022-247X(74)90277-7. [49] G. F. Webb, Asymptotic stability for abstract nonlinear functional differential equations, Proc. Amer. Math. Soc., 54 (1976), 225-230. doi: 10.1090/S0002-9939-1976-0402237-0. [50] P. Wittbold, "Absorptions nonlinéaires," Thèse Doctorat, Université de Besançon, 1994. [51] P. Wittbold, Nonlinear diffusion with absorption, in "Progress in Partial Differential Equations: the Metz Surveys 4" (M. Chipot and I. Shafrir eds.), Pitman Res. Notes Math. Series 345, Longman, Harlow, (1996), 142-157 [52] P. Wittbold, Nonlinear diffusion with absorption, Potential Anal., 7 (1997), 437-465. doi: 10.1023/A:1017998221347. [53] K. Yoshida, The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology, Hiroshima Math. J., 12 (1982), 321-348.
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