# 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 & 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. doi: 10.1016/0022-247X(78)90192-0. [2] P. Bénilan and M. G. Crandall, Completely accretive operators,, In, (1991), 41. [3] P. Bénilan, M. G. Crandall and A. Pazy, Evolution equations governed by accretive operators,, Monograph, (). [4] D. Bothe, Nonlinear evolutions with Carathéodory forcing,, J. Evol. Equ., 3 (2003), 375. 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. 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. 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. [8] M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces,, Israel J. Math., 11 (1972), 57. doi: 10.1007/BF02761448. [9] J. Dyson and R. Villella-Bressan, Functional differential equations and non-linear evolution operators,, Proc. Royal Soc., 75A (): 223. [10] J. Dyson and R. Villella-Bressan, Semigroups of translation associated with functional and functional differential equations,, Proc. Royal Soc. Edinburgh, 82A (1979), 171. 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. doi: 10.1007/BF01194220. [12] L. Evans, Nonlinear evolution equations in an arbitrary Banach space,, Israel J. Math., 26 (1977), 1. 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. 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. 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. 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. [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. 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. 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. 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. 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. 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. doi: 10.1090/S0002-9939-1979-0539637-7. [24] R. H. Martin, "Nonlinear Operators and Differential Equations in Banach Spaces,", Wiley, (1976). [25] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans, 321 (1990), 1. doi: 10.2307/2001590. [26] R. H. Martin and H. L. Smith, Convergence in Lotka-Volterra systems with diffusion and delay,, in, (1991), 259. [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. [28] I. Miyadera, "Nonlinear Semigroups,", Transl. of Math. Monographs 109, (1992). [29] S. Murakami, Stable equilibrium point of some diffusive functional differential equations,, Nonlinear Analysis TMA, 25 (1995), 1037. 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. doi: 10.1016/0362-546X(82)90018-9. [31] N. H. Pavel, "Differential Equations, Flow Invariance and Applications,", Research Notes Math. 113, (1984). [32] N. Pavel, "Nonlinear Evolution Operators and Semigroups,", Lecture Notes Math. 1260, (1260). [33] N. Pavel and F. Iacob, Invariant sets for a class of perturbed differential equations of retarded type,, Israel J. Math., 28 (1977), 254. doi: 10.1007/BF02759812. [34] M. Pierre, Invariant closed subsets for nonlinear semigroups,, Nonlinear Analysis TMA, 2 (1978), 107. 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. 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. 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. [38] W. M. Ruess, Existence of solutions to partial functional differential equations with delay,, in, (1996), 259. [39] W. M. Ruess, Existence of solutions to partial functional evolution equations with delay,, in, (1996), 377. [40] W. M. Ruess, Existence and stability of solutions to partial functional differential equations with delay,, Adv. Differential Equations, 4 (1999), 843. [41] W. M. Ruess, Flow invariance for nonlinear partial differential delay equations,, Trans. Amer. Math. Soc., 361 (2009), 4367. 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. 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. doi: 10.1006/jmaa.1996.0085. [44] A. Schiaffino, On a diffusion Volterra equation,, Nonlinear Analysis TMA, 3 (1979), 595. 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. 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. 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. 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. 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. doi: 10.1090/S0002-9939-1976-0402237-0. [50] P. Wittbold, "Absorptions nonlinéaires,", Thèse Doctorat, (1994). [51] P. Wittbold, Nonlinear diffusion with absorption,, in, (1996), 142. [52] P. Wittbold, Nonlinear diffusion with absorption,, Potential Anal., 7 (1997), 437. 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.

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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. doi: 10.1016/0022-247X(78)90192-0. [2] P. Bénilan and M. G. Crandall, Completely accretive operators,, In, (1991), 41. [3] P. Bénilan, M. G. Crandall and A. Pazy, Evolution equations governed by accretive operators,, Monograph, (). [4] D. Bothe, Nonlinear evolutions with Carathéodory forcing,, J. Evol. Equ., 3 (2003), 375. 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. 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. 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. [8] M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces,, Israel J. Math., 11 (1972), 57. doi: 10.1007/BF02761448. [9] J. Dyson and R. Villella-Bressan, Functional differential equations and non-linear evolution operators,, Proc. Royal Soc., 75A (): 223. [10] J. Dyson and R. Villella-Bressan, Semigroups of translation associated with functional and functional differential equations,, Proc. Royal Soc. Edinburgh, 82A (1979), 171. 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. doi: 10.1007/BF01194220. [12] L. Evans, Nonlinear evolution equations in an arbitrary Banach space,, Israel J. Math., 26 (1977), 1. 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. 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. 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. 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. [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. 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. 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. 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. 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. 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. doi: 10.1090/S0002-9939-1979-0539637-7. [24] R. H. Martin, "Nonlinear Operators and Differential Equations in Banach Spaces,", Wiley, (1976). [25] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans, 321 (1990), 1. doi: 10.2307/2001590. [26] R. H. Martin and H. L. Smith, Convergence in Lotka-Volterra systems with diffusion and delay,, in, (1991), 259. [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. [28] I. Miyadera, "Nonlinear Semigroups,", Transl. of Math. Monographs 109, (1992). [29] S. Murakami, Stable equilibrium point of some diffusive functional differential equations,, Nonlinear Analysis TMA, 25 (1995), 1037. 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. doi: 10.1016/0362-546X(82)90018-9. [31] N. H. Pavel, "Differential Equations, Flow Invariance and Applications,", Research Notes Math. 113, (1984). [32] N. Pavel, "Nonlinear Evolution Operators and Semigroups,", Lecture Notes Math. 1260, (1260). [33] N. Pavel and F. Iacob, Invariant sets for a class of perturbed differential equations of retarded type,, Israel J. Math., 28 (1977), 254. doi: 10.1007/BF02759812. [34] M. Pierre, Invariant closed subsets for nonlinear semigroups,, Nonlinear Analysis TMA, 2 (1978), 107. 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. 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. 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. [38] W. M. Ruess, Existence of solutions to partial functional differential equations with delay,, in, (1996), 259. [39] W. M. Ruess, Existence of solutions to partial functional evolution equations with delay,, in, (1996), 377. [40] W. M. Ruess, Existence and stability of solutions to partial functional differential equations with delay,, Adv. Differential Equations, 4 (1999), 843. [41] W. M. Ruess, Flow invariance for nonlinear partial differential delay equations,, Trans. Amer. Math. Soc., 361 (2009), 4367. 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. 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. doi: 10.1006/jmaa.1996.0085. [44] A. Schiaffino, On a diffusion Volterra equation,, Nonlinear Analysis TMA, 3 (1979), 595. 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. 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. 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. 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. 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. doi: 10.1090/S0002-9939-1976-0402237-0. [50] P. Wittbold, "Absorptions nonlinéaires,", Thèse Doctorat, (1994). [51] P. Wittbold, Nonlinear diffusion with absorption,, in, (1996), 142. [52] P. Wittbold, Nonlinear diffusion with absorption,, Potential Anal., 7 (1997), 437. 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.
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