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 Fakultät für Mathematik, Universität 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 &amp; Applied Analysis, 2012, 11 (6) : 2351-2369. doi: 10.3934/cpaa.2012.11.2351
##### References:
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Soc., 236 (1978), 173-191. doi: 10.1090/S0002-9947-1978-0466838-2. Google Scholar  D. W. Brewer, Locally Lipschitz continuous functional differential equations and nonlinear semigroups, Illinois J. Math., 26 (1982), 374-381. Google Scholar  M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math., 11 (1972), 57-94. doi: 10.1007/BF02761448. Google Scholar  J. Dyson and R. Villella-Bressan, Functional differential equations and non-linear evolution operators,, Proc. Royal Soc., 75A (): 223.   Google Scholar  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. Google Scholar  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. Google Scholar  L. Evans, Nonlinear evolution equations in an arbitrary Banach space, Israel J. Math., 26 (1977), 1-42. doi: 10.1007/BF03007654. Google Scholar  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. Google Scholar  S. M. Ghavidel, Flow invariance for solutions to nonlinear nonautonomous evolution equations,, in preparation., ().   Google Scholar  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. Google Scholar  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. Google Scholar  J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  R. H. Martin, "Nonlinear Operators and Differential Equations in Banach Spaces," Wiley, New York, 1976. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  I. Miyadera, "Nonlinear Semigroups," Transl. of Math. Monographs 109, Amer. Math. Soc., Providence, RI, 1992. Google Scholar  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. Google Scholar  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. Google Scholar  N. H. Pavel, "Differential Equations, Flow Invariance and Applications," Research Notes Math. 113, Pitman, Boston, London, Melbourne, 1984. Google Scholar  N. Pavel, "Nonlinear Evolution Operators and Semigroups," Lecture Notes Math. 1260, Springer, Berlin, 1987. Google Scholar  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. Google Scholar  M. Pierre, Invariant closed subsets for nonlinear semigroups, Nonlinear Analysis TMA, 2 (1978), 107-117. doi: 10.1016/0362-546X(78)90046-9. Google Scholar  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. Google Scholar  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. Google Scholar  W. M. Ruess, The evolution operator approach to functional differential equations with delay, Proc. Amer. Math. Soc., 119 (1993), 783-791. Google Scholar  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. Google Scholar  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. Google Scholar  W. M. Ruess, Existence and stability of solutions to partial functional differential equations with delay, Adv. Differential Equations, 4 (1999), 843-876. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  A. Schiaffino, On a diffusion Volterra equation, Nonlinear Analysis TMA, 3 (1979), 595-600. doi: 10.1016/0362-546X(79)90088-9. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  P. Wittbold, "Absorptions nonlinéaires," Thèse Doctorat, Université de Besançon, 1994. Google Scholar  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 Google Scholar  P. Wittbold, Nonlinear diffusion with absorption, Potential Anal., 7 (1997), 437-465. doi: 10.1023/A:1017998221347. Google Scholar  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. Google Scholar

show all references

##### References:
  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. Google Scholar  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. Google Scholar  P. Bénilan, M. G. Crandall and A. Pazy, Evolution equations governed by accretive operators,, Monograph, ().   Google Scholar  D. Bothe, Nonlinear evolutions with Carathéodory forcing, J. Evol. Equ., 3 (2003), 375-394. doi: 10.1007/s00028-003-0099-5. Google Scholar  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. Google Scholar  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. Google Scholar  D. W. Brewer, Locally Lipschitz continuous functional differential equations and nonlinear semigroups, Illinois J. Math., 26 (1982), 374-381. Google Scholar  M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math., 11 (1972), 57-94. doi: 10.1007/BF02761448. Google Scholar  J. Dyson and R. Villella-Bressan, Functional differential equations and non-linear evolution operators,, Proc. Royal Soc., 75A (): 223.   Google Scholar  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. Google Scholar  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. Google Scholar  L. Evans, Nonlinear evolution equations in an arbitrary Banach space, Israel J. Math., 26 (1977), 1-42. doi: 10.1007/BF03007654. Google Scholar  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. Google Scholar  S. M. Ghavidel, Flow invariance for solutions to nonlinear nonautonomous evolution equations,, in preparation., ().   Google Scholar  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. Google Scholar  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. Google Scholar  J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  R. H. Martin, "Nonlinear Operators and Differential Equations in Banach Spaces," Wiley, New York, 1976. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  I. Miyadera, "Nonlinear Semigroups," Transl. of Math. Monographs 109, Amer. Math. Soc., Providence, RI, 1992. Google Scholar  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. Google Scholar  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. Google Scholar  N. H. Pavel, "Differential Equations, Flow Invariance and Applications," Research Notes Math. 113, Pitman, Boston, London, Melbourne, 1984. Google Scholar  N. Pavel, "Nonlinear Evolution Operators and Semigroups," Lecture Notes Math. 1260, Springer, Berlin, 1987. Google Scholar  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. Google Scholar  M. Pierre, Invariant closed subsets for nonlinear semigroups, Nonlinear Analysis TMA, 2 (1978), 107-117. doi: 10.1016/0362-546X(78)90046-9. Google Scholar  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. Google Scholar  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. Google Scholar  W. M. Ruess, The evolution operator approach to functional differential equations with delay, Proc. Amer. Math. Soc., 119 (1993), 783-791. Google Scholar  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. Google Scholar  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. Google Scholar  W. M. Ruess, Existence and stability of solutions to partial functional differential equations with delay, Adv. Differential Equations, 4 (1999), 843-876. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  A. Schiaffino, On a diffusion Volterra equation, Nonlinear Analysis TMA, 3 (1979), 595-600. doi: 10.1016/0362-546X(79)90088-9. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  P. Wittbold, "Absorptions nonlinéaires," Thèse Doctorat, Université de Besançon, 1994. Google Scholar  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 Google Scholar  P. Wittbold, Nonlinear diffusion with absorption, Potential Anal., 7 (1997), 437-465. doi: 10.1023/A:1017998221347. Google Scholar  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. Google Scholar
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