November  2012, 11(6): 2239-2260. doi: 10.3934/cpaa.2012.11.2239

Abstract reaction-diffusion systems with $m$-completely accretive diffusion operators and measurable reaction rates

1. 

Center of Smart Interfaces, Technical University Darmstadt, Petersenstr. 32, 64287 Darmstadt

2. 

Duisburg-Essen University, Faculty of Mathematics, Universitatsstrae 2, 45141 Essen, Germany

Received  March 2011 Revised  July 2011 Published  April 2012

We consider reaction-diffusion systems with merely measurable reaction terms to cover the possibility of discontinuities. Solutions of such problems are defined as solutions to appropriate differential inclusions which, in an abstract form, lead to evolution inclusions of the form

$u' \in - A u + F(t,u)$ on $[0,T], u(0)=u_{0},$

where $A$ is $m$-accretive and $F$ is of upper semicontinuous type. While such problems, in general, can exhibit non-existence of solutions, the present paper shows that especially for $m$-completely accretive $A$, and under reasonable assumptions on $F$, mild solutions do exist.

Citation: Dieter Bothe, Petra Wittbold. Abstract reaction-diffusion systems with $m$-completely accretive diffusion operators and measurable reaction rates. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2239-2260. doi: 10.3934/cpaa.2012.11.2239
References:
[1]

F. Andreu, N. Igbida, J. M. Mazon and J. Toledo, $L^1$ existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 61-89. doi: 10.1016/j.anihpc.2005.09.009.

[2]

H. Attouch and A. Damlamian, On multivalued evolution equations in Hilbert spaces, Israel J. Math., 12 (1972), 373-390. doi: 10.1007/BF02764629.

[3]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff, Leyden, 1976.

[4]

Ph. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Sc. Norm. Sup. Pisa, 22 (1995), 241-273.

[5]

Ph. Bénilan and M. G. Crandall, Regularizing effects of homogeneous evolution equations, in "Contributions to Analysis and Geometry" (D. N. Clark et al., eds.), Johns Hopkins Univ. Press, Baltimore/MD, (1981), 23-39.

[6]

Ph. Bénilan and M. G. Crandall, Completely accretive operators, in "Semigroup Theory and Evolution Equations," Lect. Notes Pure Appl. Math., 135 (Ph. Clément, E. Mitidieri, B. de Pagter, eds.), Marcel Dekker, New York, (1991), 41-75.

[7]

Ph. Bénilan, M. G. Crandall and A. Pazy, "Nonlinear Evolution Equations in Banach Spaces,", Preprint book., (). 

[8]

D. Bothe, Minimal solutions of multivalued differential equations, Diff. and Integral Eqs., 4 (1991), 445-447.

[9]

D. Bothe, Flow invariance for perturbed nonlinear evolution equations, Abstract and Applied Analysis, 1 (1996), 417-433 . doi: 10.1155/S1085337596000231.

[10]

D. Bothe, Reaction-diffusion systems with discontinuities. A viability approach, in "Proc. 2nd World Congress of Nonlinear Analysts, Nonlinear Analysis," 30 (1997), 677-686. doi: 10.1016/S0362-546X(97)00247-2.

[11]

D. Bothe, Multivalued perturbations of $m$-accretive differential inclusions, Israel J. Math., 108 (1998), 109-138. doi: 10.1007/BF02783044.

[12]

D. Bothe, Periodic solutions of non-smooth friction oscillators, Z. Angew. Math. Phys., 50 (1999), 779-808. doi: 10.1007/s000330050178.

[13]

D. Bothe, "Nonlinear Evolutions in Banach Spaces - Existence and Qualitative Theory with Applications to Reaction-Diffusion Systems," Habilitation thesis, University Paderborn, 1999.

[14]

D. Bothe, Nonlinear evolutions with Carathéodory forcing, J. Evol. Eqs., 3 (2003), 375-394. doi: 10.1007/s00028-003-0099-5.

[15]

D. Bothe, Flow invariance for nonlinear accretive evolutions under range conditions, J. Evol. Eqs., 5 (2005), 227-252. doi: 10.1007/s00028-005-0185-z.

[16]

K. Deimling, "Multivalued Differential Equations," De Gruyter 1992.

[17]

K. Deimling, G. Hetzer and W. Shen, Almost periodicity enforced by Coulomb friction, Advances in Diff. Eqs., 1 (1996), 265-281.

[18]

J. I. Diaz, Diffusive energy balance models in climatology, in "Studies in Mathematics and its Applications," Vol. 31 (D. Cioranescu and J. L. Lions, eds.). Elsevier Science 2002, pp. 297-328. doi: 10.1016/S0168-2024(02)80015-7.

[19]

J. I. Diaz and I. I. Vrabie, Existence for reaction diffusion systems. A compactness method approach, J. Math. Anal. Appl., 188 (1994), 521-540.

[20]

J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^1(\mu,X)$, in "Proc.Amer. Math. Soc.," 118 (1993), 447-453 . doi: 10.1090/S0002-9939-1993-1132408-x.

[21]

G. Duvaut and J. L. Lions, "Inequalities in Mechanics and Physics," Springer, 1976. doi: 10.1007/978-3-642-66165-5.

[22]

H. O. Fattorini, Infinite dimensional optimization and control theory, in "Enzyclopedia of Mathematics and its Applications," Cambridge Univ. Press, Cambridge, 1999.

[23]

E. Feireisl and J. Norbury, Some existence, uniqueness and non-uniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities, in "Proc. Roy. Soc.," Edinburgh, 119A (1991), 1-17.

[24]

A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Sides," Kluwer, 1988. doi: 10.1016/0378-4754(89)90171-7.

[25]

L. Górniewicz, A. Granas and W. Kryszewski, Sur la méthode de l'homotopie dans la théorie des point fixes pour les applications multivoques. Partie 2: L'indice dans les ANRs compacts, C. R. Acad. Sci. Paris, 308 (1989), 449-452.

[26]

V. G. Jakubowski and P. Wittbold, Regularity of solutions of nonlinear Volterra equations, J. Evol. Equ., 3 (2003), 303-319. doi: 10.1007/s00028-003-0096-9.

[27]

R. H. Martin and M. Pierre, Nonlinear reaction-diffusion systems, in "Nonlinear Equations in the Applied Sciences" (W. F. Ames and C. Rogers, eds.), Math. Sci. Eng. 185, Acad. Press, New York, 1992, 363-398. doi: 10.1016/S0076-5392(08)62804-0.

[28]

I. Miyadera, "Nonlinear Semigroups," Translations of Math. Monographs 109, Amer. Math. Soc., 1992.

[29]

J. Norbury and A. M. Stuart, A model for porous medium combustion, Quart. J. Mech. Appl. Math., 42 (1987), 159-178. doi: 10.1093/qjmam/42.1.159.

[30]

M. Pierre, Un théorème général de génération de semi-groupes non linéaires, Israel J. Math., 23 (1976), 189-199.

[31]

M. Pierre, Global existence in reaction-diffusion systems with control of mass: a survey, Milan J. Math., 78 (2010), 417-455. doi: 10.1007/s00032-010-0133-4.

[32]

T. Rzezuchowski, Scorza-Dragoni type theorem for upper semicontinuous multivalued functions, Bull. Acad. Polon. Sci., 28 (1980), 61-66.

[33]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Grundlehren math. Wissenschaften 258, Springer 1983.

[34]

A. M. Stuart, The mathematics of porous medium combustion, in "Nonlinear Diffusion Equations and their Equilibrium States II" (W. M. Ni, L. A. Peletier and J. Serrin, eds.), Springer 1988, 295-313.

[35]

A. A. Tolstonogov and Y. I. Umanskii, On solutions of evolution inclusions II, Sib. Math. J., 33 (1992), 693-702. doi: 10.1007/BF00971135.

[36]

M. Valencia, On invariant regions and asymptotic bounds for semilinear partial differential equations, Nonlinear Analysis, 14 (1990), 217-230. doi: 10.1016/0362-546X(90)90030-K.

[37]

A. Visintin, "Differential Models of Hysteresis," Springer, Berlin, 1994.

[38]

I. I. Vrabie, "Compactness Methods for Nonlinear Evolutions," $2^{nd}$ edition, Pitman, 1995.

show all references

References:
[1]

F. Andreu, N. Igbida, J. M. Mazon and J. Toledo, $L^1$ existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 61-89. doi: 10.1016/j.anihpc.2005.09.009.

[2]

H. Attouch and A. Damlamian, On multivalued evolution equations in Hilbert spaces, Israel J. Math., 12 (1972), 373-390. doi: 10.1007/BF02764629.

[3]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff, Leyden, 1976.

[4]

Ph. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Sc. Norm. Sup. Pisa, 22 (1995), 241-273.

[5]

Ph. Bénilan and M. G. Crandall, Regularizing effects of homogeneous evolution equations, in "Contributions to Analysis and Geometry" (D. N. Clark et al., eds.), Johns Hopkins Univ. Press, Baltimore/MD, (1981), 23-39.

[6]

Ph. Bénilan and M. G. Crandall, Completely accretive operators, in "Semigroup Theory and Evolution Equations," Lect. Notes Pure Appl. Math., 135 (Ph. Clément, E. Mitidieri, B. de Pagter, eds.), Marcel Dekker, New York, (1991), 41-75.

[7]

Ph. Bénilan, M. G. Crandall and A. Pazy, "Nonlinear Evolution Equations in Banach Spaces,", Preprint book., (). 

[8]

D. Bothe, Minimal solutions of multivalued differential equations, Diff. and Integral Eqs., 4 (1991), 445-447.

[9]

D. Bothe, Flow invariance for perturbed nonlinear evolution equations, Abstract and Applied Analysis, 1 (1996), 417-433 . doi: 10.1155/S1085337596000231.

[10]

D. Bothe, Reaction-diffusion systems with discontinuities. A viability approach, in "Proc. 2nd World Congress of Nonlinear Analysts, Nonlinear Analysis," 30 (1997), 677-686. doi: 10.1016/S0362-546X(97)00247-2.

[11]

D. Bothe, Multivalued perturbations of $m$-accretive differential inclusions, Israel J. Math., 108 (1998), 109-138. doi: 10.1007/BF02783044.

[12]

D. Bothe, Periodic solutions of non-smooth friction oscillators, Z. Angew. Math. Phys., 50 (1999), 779-808. doi: 10.1007/s000330050178.

[13]

D. Bothe, "Nonlinear Evolutions in Banach Spaces - Existence and Qualitative Theory with Applications to Reaction-Diffusion Systems," Habilitation thesis, University Paderborn, 1999.

[14]

D. Bothe, Nonlinear evolutions with Carathéodory forcing, J. Evol. Eqs., 3 (2003), 375-394. doi: 10.1007/s00028-003-0099-5.

[15]

D. Bothe, Flow invariance for nonlinear accretive evolutions under range conditions, J. Evol. Eqs., 5 (2005), 227-252. doi: 10.1007/s00028-005-0185-z.

[16]

K. Deimling, "Multivalued Differential Equations," De Gruyter 1992.

[17]

K. Deimling, G. Hetzer and W. Shen, Almost periodicity enforced by Coulomb friction, Advances in Diff. Eqs., 1 (1996), 265-281.

[18]

J. I. Diaz, Diffusive energy balance models in climatology, in "Studies in Mathematics and its Applications," Vol. 31 (D. Cioranescu and J. L. Lions, eds.). Elsevier Science 2002, pp. 297-328. doi: 10.1016/S0168-2024(02)80015-7.

[19]

J. I. Diaz and I. I. Vrabie, Existence for reaction diffusion systems. A compactness method approach, J. Math. Anal. Appl., 188 (1994), 521-540.

[20]

J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^1(\mu,X)$, in "Proc.Amer. Math. Soc.," 118 (1993), 447-453 . doi: 10.1090/S0002-9939-1993-1132408-x.

[21]

G. Duvaut and J. L. Lions, "Inequalities in Mechanics and Physics," Springer, 1976. doi: 10.1007/978-3-642-66165-5.

[22]

H. O. Fattorini, Infinite dimensional optimization and control theory, in "Enzyclopedia of Mathematics and its Applications," Cambridge Univ. Press, Cambridge, 1999.

[23]

E. Feireisl and J. Norbury, Some existence, uniqueness and non-uniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities, in "Proc. Roy. Soc.," Edinburgh, 119A (1991), 1-17.

[24]

A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Sides," Kluwer, 1988. doi: 10.1016/0378-4754(89)90171-7.

[25]

L. Górniewicz, A. Granas and W. Kryszewski, Sur la méthode de l'homotopie dans la théorie des point fixes pour les applications multivoques. Partie 2: L'indice dans les ANRs compacts, C. R. Acad. Sci. Paris, 308 (1989), 449-452.

[26]

V. G. Jakubowski and P. Wittbold, Regularity of solutions of nonlinear Volterra equations, J. Evol. Equ., 3 (2003), 303-319. doi: 10.1007/s00028-003-0096-9.

[27]

R. H. Martin and M. Pierre, Nonlinear reaction-diffusion systems, in "Nonlinear Equations in the Applied Sciences" (W. F. Ames and C. Rogers, eds.), Math. Sci. Eng. 185, Acad. Press, New York, 1992, 363-398. doi: 10.1016/S0076-5392(08)62804-0.

[28]

I. Miyadera, "Nonlinear Semigroups," Translations of Math. Monographs 109, Amer. Math. Soc., 1992.

[29]

J. Norbury and A. M. Stuart, A model for porous medium combustion, Quart. J. Mech. Appl. Math., 42 (1987), 159-178. doi: 10.1093/qjmam/42.1.159.

[30]

M. Pierre, Un théorème général de génération de semi-groupes non linéaires, Israel J. Math., 23 (1976), 189-199.

[31]

M. Pierre, Global existence in reaction-diffusion systems with control of mass: a survey, Milan J. Math., 78 (2010), 417-455. doi: 10.1007/s00032-010-0133-4.

[32]

T. Rzezuchowski, Scorza-Dragoni type theorem for upper semicontinuous multivalued functions, Bull. Acad. Polon. Sci., 28 (1980), 61-66.

[33]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Grundlehren math. Wissenschaften 258, Springer 1983.

[34]

A. M. Stuart, The mathematics of porous medium combustion, in "Nonlinear Diffusion Equations and their Equilibrium States II" (W. M. Ni, L. A. Peletier and J. Serrin, eds.), Springer 1988, 295-313.

[35]

A. A. Tolstonogov and Y. I. Umanskii, On solutions of evolution inclusions II, Sib. Math. J., 33 (1992), 693-702. doi: 10.1007/BF00971135.

[36]

M. Valencia, On invariant regions and asymptotic bounds for semilinear partial differential equations, Nonlinear Analysis, 14 (1990), 217-230. doi: 10.1016/0362-546X(90)90030-K.

[37]

A. Visintin, "Differential Models of Hysteresis," Springer, Berlin, 1994.

[38]

I. I. Vrabie, "Compactness Methods for Nonlinear Evolutions," $2^{nd}$ edition, Pitman, 1995.

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